Introduction
Let E E E K K K E ( K ) ≃ E ( K ) t o r s × Z r E(K) \simeq E(K)_{\mathrm{tors}} \times\mathbb{Z}^{r} E ( K ) ≃ E ( K ) tors × Z r [34 ] the torsion subgroup E ( K ) t o r s E(K)_{\mathrm{tors}} E ( K ) tors d = [ K : Q ] d=[K:\mathbb{Q}] d = [ K : Q ] d ≥ 1 d \ge1 d ≥ 1
Φ ( d ) : = { E ( K ) t o r s : [ K : Q ] = d , E / K an elliptic curve } \Phi(d):=\{E(K)_{\mathrm{tors}}:[K:\mathbb{Q}]=d,\ E/K\text{ an elliptic curve}\} Φ ( d ) := { E ( K ) tors : [ K : Q ] = d , E / K an elliptic curve } is finite. The set Φ ( 1 ) \Phi(1) Φ ( 1 ) [29 ] , and Φ ( 2 ) \Phi(2) Φ ( 2 ) [22 ] , building on work of Kenku and Momose [28 ] . Derickx, Etropolski, van Hoeij, Morrow and Zureick-Brown [7 ] determined Φ ( 3 ) \Phi(3) Φ ( 3 ) [21 ] , Parent [40 ] and Najman [35 ] . Derickx and Najman [11 ] determined Φ ( 4 ) \Phi(4) Φ ( 4 ) [20 ] and Bruin and Najman [4 ] . Finally, Φ ( 5 ) \Phi(5) Φ ( 5 ) [36 ] . In this paper we determine Φ ( 6 ) \Phi(6) Φ ( 6 )
Write Φ ∞ ( d ) ⊆ Φ ( d ) \Phi^{\infty}(d) \subseteq\Phi(d) Φ ∞ ( d ) ⊆ Φ ( d ) E ( K ) t o r s E(K)_{\mathrm{tors}} E ( K ) tors Q \mathbb{Q} Q E E E K K K d d d Φ ( d ) ∖ Φ ∞ ( d ) \Phi(d) \setminus\Phi^{\infty}(d) Φ ( d ) ∖ Φ ∞ ( d ) sporadic torsion group in degree $d . This should not be confused with a sporadic point on a curve X X X e e e X X X e e e X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) [37 ] . We
Date: November 4, 2026.
2020 Mathematics Subject Classification. 11G05, 11G18, 14G05.
Key words and phrases. elliptic curves, torsion, sextic fields, modular curves, sporadic points.
The author was supported by the Croatian Science Foundation under the project no. IP-2022-10-5008, by the project “Implementation of cutting-edge research and its application as part of the Scientific Center of Excellence for Quantum and Complex Systems, and Representations of Lie Algebras”, PK.1.1.10.0004, co-financed by the European Union through the European Regional Development Fund – Competitiveness and Cohesion Programme 2021–2027, and by the European Union – NextGenerationEU through the National Recovery and Resilience Plan 2021–2026, via an institutional grant of the University of Zagreb Faculty of Science, IK IA 1.1.3, Impact4Math. have Φ ∞ ( d ) = Φ ( d ) \Phi^{\infty}(d)=\Phi(d) Φ ∞ ( d ) = Φ ( d ) d = 1 , 2 , 4 d=1,2,4 d = 1 , 2 , 4 Z / 21 Z \mathbb{Z}/21\mathbb{Z} Z /21 Z [35 , 7 ] , and the sporadic torsion groups in degree 5 are Z / 28 Z \mathbb{Z}/28\mathbb{Z} Z /28 Z Z / 30 Z \mathbb{Z}/30\mathbb{Z} Z /30 Z Z / 2 Z × Z / 18 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z} Z /2 Z × Z /18 Z [36 ] . The set Φ ∞ ( 6 ) \Phi^{\infty}(6) Φ ∞ ( 6 ) [14 Theorem 1.1] , the cyclic part being due to Derickx and van Hoeij [15 Theorem 3] :
Φ ∞ ( 6 ) = { Z / n Z : 1 ≤ n ≤ 30 , n ≠ 23 , 25 , 29 } ∪ { Z / 2 Z × Z / 2 n Z : 1 ≤ n ≤ 10 } ∪ { Z / 3 Z × Z / 3 n Z : 1 ≤ n ≤ 4 } ∪ { Z / 4 Z × Z / 4 Z , Z / 4 Z × Z / 8 Z , Z / 6 Z × Z / 6 Z } . (1) \Phi^{\infty}(6)=\{\mathbb{Z}/n\mathbb{Z}:1\leq n\leq30,\ n\neq23,25,29\}
\cup\{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2n\mathbb{Z}:1\leq n\leq10\}\cup\{\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3n\mathbb{Z}:1\leq n\leq4\}
\cup\{\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z},\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/8\mathbb{Z},\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}\}. \tag*{(1)} Φ ∞ ( 6 ) = { Z / n Z : 1 ≤ n ≤ 30 , n = 23 , 25 , 29 } ∪ { Z /2 Z × Z /2 n Z : 1 ≤ n ≤ 10 } ∪ { Z /3 Z × Z /3 n Z : 1 ≤ n ≤ 4 } ∪ { Z /4 Z × Z /4 Z , Z /4 Z × Z /8 Z , Z /6 Z × Z /6 Z } . (1) This set has 44 elements. Some sporadic torsion groups in degree 6 were already known. Van Hoeij [46 ] found an elliptic curve over a sextic field with a point of order 25 and one with a point of order 37, and Derickx, Kamienny, Stein and Stoll proved that every elliptic curve over a sextic field with a point of order 37 has j j j − 9317 -9317 − 9317 [10 Proposition 1.4] . González-Jiménez and Najman [18 ] found that the base change of the rational elliptic curve 162. d 2 162.\mathrm{d}2 162. d 2 Z / 4 Z × Z / 12 Z \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z} Z /4 Z × Z /12 Z [8 Remark 1] ).
Our main result is the following.
Theorem 1.1. Let K K K E E E K K K E ( K ) t o r s E(K)_{\mathrm{tors}} E ( K ) tors
Z / n Z , n = 1 , … , 22 , 24 , … , 28 , 30 , 37 , \mathbb{Z}/n\mathbb{Z},\qquad n=1,\ldots,22,24,\ldots,28,30,37, Z / n Z , n = 1 , … , 22 , 24 , … , 28 , 30 , 37 ,
Z / 2 Z × Z / 2 n Z , n = 1 , … , 10 , 12 , \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2n\mathbb{Z},\qquad n=1,\ldots,10,12, Z /2 Z × Z /2 n Z , n = 1 , … , 10 , 12 ,
Z / 3 Z × Z / 3 n Z , n = 1 , … , 4 , \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3n\mathbb{Z},\qquad n=1,\ldots,4, Z /3 Z × Z /3 n Z , n = 1 , … , 4 ,
Z / 4 Z × Z / 4 n Z , n = 1 , 2 , 3 , \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/4n\mathbb{Z},\qquad n=1,2,3, Z /4 Z × Z /4 n Z , n = 1 , 2 , 3 ,
Z / 6 Z × Z / 6 Z \mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z} Z /6 Z × Z /6 Z
Equivalently, Φ ( 6 ) = Φ ∞ ( 6 ) ∪ { Z / 25 Z , Z / 37 Z , Z / 4 Z × Z / 12 Z , Z / 2 Z × Z / 24 Z } \Phi(6)=\Phi^{\infty}(6)\cup\{\mathbb{Z}/25\mathbb{Z},\mathbb{Z}/37\mathbb{Z},\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z},\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}\} Φ ( 6 ) = Φ ∞ ( 6 ) ∪ { Z /25 Z , Z /37 Z , Z /4 Z × Z /12 Z , Z /2 Z × Z /24 Z }
The four sporadic groups are described by the following theorem, which is proved in § 10 . For two of them we determine all the elliptic curves in question.
Theorem 1.2. Let K K K E / K E/K E / K
If E ( K ) E(K) E ( K ) E ( K ) t o r s ≃ Z / 37 Z E(K)_{\mathrm{tors}}\simeq\mathbb{Z}/37\mathbb{Z} E ( K ) tors ≃ Z /37 Z ( K , E ) (K,E) ( K , E ) K K K K 37 K_{37} K 37 E E E E 37 E_{37} E 37 1225. b 2 1225.\mathrm{b}2 1225. b 2
If Z / 2 Z × Z / 24 Z ⊆ E ( K ) \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}\subseteq E(K) Z /2 Z × Z /24 Z ⊆ E ( K ) E ( K ) t o r s ≃ Z / 2 Z × Z / 24 Z E(K)_{\mathrm{tors}}\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} E ( K ) tors ≃ Z /2 Z × Z /24 Z ( K , E ) (K,E) ( K , E ) K K K K 2 , 24 K_{2,24} K 2 , 24 E E E E 2 , 24 E_{2,24} E 2 , 24
There exist elliptic curves over sextic fields with torsion groups Z / 25 Z \mathbb{Z}/25\mathbb{Z} Z /25 Z Z / 4 Z × Z / 12 Z \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z} Z /4 Z × Z /12 Z
The curve in (2) is new. We do not determine all the elliptic curves over sextic fields with torsion Z / 25 Z \mathbb{Z}/25\mathbb{Z} Z /25 Z Z / 4 Z × Z / 12 Z \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z} Z /4 Z × Z /12 Z Z / 25 Z \mathbb{Z}/25\mathbb{Z} Z /25 Z Z / 4 Z × Z / 12 Z \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z} Z /4 Z × Z /12 Z
Outline of the proof and of the paper
As in [11 ] and [36 ] , the classification reduces to showing that certain modular curves X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) § 3 we show that 66 groups have to be analyzed. Nine of them contain Z / 37 Z \mathbb{Z}/37\mathbb{Z} Z /37 Z 37 37 37 2 2 2 Z / 2 Z × Z / 2 n Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2n\mathbb{Z} Z /2 Z × Z /2 n Z
The global method of [11 Proposition 4.3] , in the form used in [36 ] , eliminates eleven groups Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n n n § 4 ). For six other large n n n [36 Proposition 7.2] , in which a single operator ⟨ a ⟩ − 1 \langle a\rangle- 1 ⟨ a ⟩ − 1
When the Jacobian has rank 0 0 0 [11 Proposition 5.2] , which shows that the elliptic curve must have good reduction at some prime above a suitable p p p § 5 ). When p = 2 p = 2 p = 2 L ( 2 C ) L(2C) L ( 2 C ) C C C X F 2 X_{\mathbb{F}_2} X F 2 p p p [11 Proposition 5.5] , which we use both over Q \mathbb{Q} Q Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) m = 3 , 4 , 6 m = 3, 4, 6 m = 3 , 4 , 6 § 6 ).
Several cases are settled by computations in the group generated by the cuspidal divisor classes over a finite field (§ 7 ). This determines all the sextic points on X 1 ( 2 , 24 ) X_1(2,24) X 1 ( 2 , 24 ) Z / 2 Z × Z / 24 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/24\mathbb{Z} Z /2 Z × Z /24 Z Z / 48 Z \mathbb{Z}/48\mathbb{Z} Z /48 Z Z / 36 Z \mathbb{Z}/36\mathbb{Z} Z /36 Z Z / 2 Z × Z / 26 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/26\mathbb{Z} Z /2 Z × Z /26 Z Z / 3 Z × Z / 18 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} Z /3 Z × Z /18 Z Z / 2 Z × Z / 30 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/30\mathbb{Z} Z /2 Z × Z /30 Z Z / 54 Z \mathbb{Z}/54\mathbb{Z} Z /54 Z Z / 54 Z \mathbb{Z}/54\mathbb{Z} Z /54 Z 5 5 5 j j j 2 2 2 2 2 2
The groups Z / 51 Z \mathbb{Z}/51\mathbb{Z} Z /51 Z Z / 63 Z \mathbb{Z}/63\mathbb{Z} Z /63 Z Z / 65 Z \mathbb{Z}/65\mathbb{Z} Z /65 Z § 8 ) required a new idea. For these n n n n n n 2 2 2 [11 Proposition 5.2] does not apply: for n = 51 n = 51 n = 51 X 1 ( 51 ) X_1(51) X 1 ( 51 ) n = 63 n = 63 n = 63 65 65 65 n n n J 1 ( n ) ( Q ) J_1(n)(\mathbb{Q}) J 1 ( n ) ( Q ) 2 2 2 p = 2 p = 2 p = 2 2 2 2 Lemma 8.1 , which shows that T 2 − ⟨ 2 ⟩ T_2 - \langle2\rangle T 2 − ⟨ 2 ⟩ 2 2 2 X 1 ( n ) X_1(n) X 1 ( n ) F 2 \mathbb{F}_2 F 2 F 2 \mathbb{F}_2 F 2 n = 65 n = 65 n = 65 n = 57 n = 57 n = 57 2 2 2 X 0 ( 57 ) X_0(57) X 0 ( 57 ) X 0 ( 65 ) X_0(65) X 0 ( 65 ) Z / 57 Z \mathbb{Z}/57\mathbb{Z} Z /57 Z 2 2 2 J 1 ( 57 ) J_1(57) J 1 ( 57 )
§ 10 contains the four sporadic groups and the proof of Theorem 1.1 , and § 11 describes the computations.
Data availability and reproducibility
All the code and all the logs of the computations described in this paper are available at
https://github.com/F-Najman/sextic_torsion
The README there lists, statement by statement, which script establishes each claim and which log it produced. In the digital form of this article, the words reporting a computation, such as “we compute”, are hyperlinks to the script that carries it out.
AI models were extensively used in the work leading to this paper. After completing [11 ] (with Derickx) and [36 ] and giving talks about this work, I was asked several times whether we could do such a classification for number fields of degree d > 5 d > 5 d > 5
OpenAI’s GPT-6.1 Sol produced almost all of the mathematical content of the first version of the proof and carried out most of the computations. Anthropic’s Claude Opus 5.5 reviewed that version, independently recomputed the relevant computations and the explicit examples, wrote the draft of the present manuscript and organized the code repository. I carefully reviewed the paper line by line and rewrote most parts of it. I understand it completely, and take full responsibility for the content of the paper.
Future work
As noted in the AI disclosure above, I have obtained AI-generated proofs that appear to prove a complete classification of the possible torsion groups of elliptic curves over number fields of degrees 7, 8, 9 and 10. Once I have fully understood and verified the arguments in degrees 7–9, I plan to combine the results for degrees 6–9 into a single paper, which I currently expect to be around 120 pages long. Degree 10 will stay in a separate paper for mathematical reasons and because of its length.
Notation and auxiliary results
Throughout, m ∣ n m \mid n m ∣ n m n > 4 mn > 4 mn > 4 X : = X 1 ( m , n ) X := X_1(m,n) X := X 1 ( m , n ) ( E , P , Q ) (E,P,Q) ( E , P , Q ) E E E P , Q P,Q P , Q m m m n n n E [ n ] E[n] E [ n ] Z / m Z × Z / n Z \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} Z / m Z × Z / n Z P P P ( n / m ) Q (n/m)Q ( n / m ) Q ζ m \zeta_m ζ m Z [ ζ m , 1 / n ] \mathbb{Z}[\zeta_m,1/n] Z [ ζ m , 1/ n ] n > 4 n > 4 n > 4 [11 §2] and [14 §2] ). Every curve occurring below has n ≥ 5 n \ge5 n ≥ 5 Y 1 ( m , n ) Y_1(m,n) Y 1 ( m , n ) J : = J 1 ( m , n ) J := J_1(m,n) J := J 1 ( m , n ) X X X J ( Y ) J(Y) J ( Y ) Y Y Y J J J J ( X ) J(X) J ( X ) X X X X 1 ( n ) : = X 1 ( 1 , n ) X_1(n) := X_1(1,n) X 1 ( n ) := X 1 ( 1 , n ) J 1 ( n ) : = J 1 ( 1 , n ) J_1(n) := J_1(1,n) J 1 ( n ) := J 1 ( 1 , n ) m ≤ 2 m \le2 m ≤ 2 X X X Q \mathbb{Q} Q d ≥ 1 d \ge1 d ≥ 1 X ( d ) X^{(d)} X ( d ) d d d X X X X ( d ) ( Q ( ζ m ) ) X^{(d)}(\mathbb{Q}(\zeta_m)) X ( d ) ( Q ( ζ m )) Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) d d d X X X X X X F F F X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) Cl F 0 X \operatorname{Cl}_F^0 X Cl F 0 X J ( F ) J(F) J ( F ) F F F 0 0 0 Cl F 0 X \operatorname{Cl}_F^0 X Cl F 0 X F F F 0 0 0 D D D X X X k k k L ( D ) : = { f ∈ k ( X ) × : div ( f ) + D ≥ 0 } ∪ { 0 } L(D) := \{f \in k(X)^\times: \operatorname{div}(f) + D \ge0\} \cup\{0\} L ( D ) := { f ∈ k ( X ) × : div ( f ) + D ≥ 0 } ∪ { 0 } Ω ( D ) \Omega(D) Ω ( D ) 0 0 0 ω \omega ω X X X div ( ω ) ≥ D \operatorname{div}(\omega) \ge D div ( ω ) ≥ D Ω ( 0 ) = H 0 ( X , Ω 1 ) \Omega(0) = H^0(X,\Omega^1) Ω ( 0 ) = H 0 ( X , Ω 1 ) X X X D = ∑ P m P P D = \sum_P m_P P D = ∑ P m P P Ω ( D ) \Omega(D) Ω ( D ) m P m_P m P P P P dim L ( D ) = deg D + 1 − g + dim Ω ( D ) \dim L(D) = \deg D + 1 - g + \dim\Omega(D) dim L ( D ) = deg D + 1 − g + dim Ω ( D ) g g g X X X
Let K K K E / K E/K E / K Z / m Z × Z / n Z ≃ E ( K ) \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \simeq E(K) Z / m Z × Z / n Z ≃ E ( K ) Q ( ζ m ) ⊆ K \mathbb{Q}(\zeta_m) \subseteq K Q ( ζ m ) ⊆ K φ ( m ) ∣ 6 \varphi(m) \mid6 φ ( m ) ∣ 6 d m : = 6 / φ ( m ) = [ K : Q ( ζ m ) ] d_m := 6/\varphi(m) = [K : \mathbb{Q}(\zeta_m)] d m := 6/ φ ( m ) = [ K : Q ( ζ m )] P , Q P,Q P , Q x ∈ Y 1 ( m , n ) ( K ) x \in Y_1(m,n)(K) x ∈ Y 1 ( m , n ) ( K ) x ( d m ) x^{(d_m)} x ( d m ) Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) x x x [ K : Q ( ζ m ) ( x ) ] [K:\mathbb{Q}(\zeta_m)(x)] [ K : Q ( ζ m ) ( x )] Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) d m d_m d m X X X trace divisor of x x x m ≤ 2 m \le2 m ≤ 2 Q ( ζ m ) = Q \mathbb{Q}(\zeta_m)=\mathbb{Q} Q ( ζ m ) = Q d m = 6 d_m=6 d m = 6 x ( 6 ) x^{(6)} x ( 6 ) Q ( ζ m ) ( x ) \mathbb{Q}(\zeta_m)(x) Q ( ζ m ) ( x ) K K K ( K , E ) (K,E) ( K , E ) ( K ′ , E ′ ) (K',E') ( K ′ , E ′ ) σ : K → K ′ \sigma:K\to K' σ : K → K ′ E σ ≃ E ′ E^\sigma\simeq E' E σ ≃ E ′ K ′ K' K ′
The cusps of X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) k k k m ∣ k ∣ n m\mid k\mid n m ∣ k ∣ n [11 §2] ). All cusps are defined over Q ( ζ n ) \mathbb{Q}(\zeta_n) Q ( ζ n ) Gal ( Q ( ζ n ) / Q ( ζ m ) ) \operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}(\zeta_m)) Gal ( Q ( ζ n ) / Q ( ζ m )) G m \mathbb{G}_m G m Z [ ζ m , 1 n ] \mathbb{Z}[\zeta_m,\frac{1}{n}] Z [ ζ m , n 1 ] c 0 c_0 c 0 n n n ( a , b ) ↦ ( ζ m a , b ) (a,b)\mapsto(\zeta_m^a,b) ( a , b ) ↦ ( ζ m a , b ) [11 Lemma 2.8] ), which is Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) X 1 ( n ) X_1(n) X 1 ( n ) [11 §2] it follows that a cusp of X 1 ( n ) X_1(n) X 1 ( n ) d d d ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ a ≡ ε ( m o d d ) a\equiv\varepsilon\pmod d a ≡ ε ( mod d ) a ≡ ε ( m o d n / d ) a\equiv\varepsilon\pmod{n/d} a ≡ ε ( mod n / d ) ε = ± 1 \varepsilon=\pm1 ε = ± 1 a a a ε \varepsilon ε n n n
For a congruence subgroup Γ ⊆ SL 2 ( Z ) \Gamma\subseteq\operatorname{SL}_2(\mathbb{Z}) Γ ⊆ SL 2 ( Z ) Γ ‾ \overline{\Gamma} Γ PSL 2 ( Z ) \operatorname{PSL}_2(\mathbb{Z}) PSL 2 ( Z ) X 1 ( m , n ) C X_1(m,n)_{\mathbb{C}} X 1 ( m , n ) C Γ 1 ( n ) ∩ Γ ( m ) \Gamma_1(n)\cap\Gamma(m) Γ 1 ( n ) ∩ Γ ( m ) n > 2 n>2 n > 2
[ PSL 2 ( Z ) : Γ 1 ( n ) ∩ Γ ( m ) ‾ ] = m 2 n 2 ∏ ℓ ∣ n ( 1 − ℓ − 2 ) , [\operatorname{PSL}_2(\mathbb{Z}):\overline{\Gamma_1(n)\cap\Gamma(m)}]=\frac{m}{2}n^2\prod_{\ell\mid n}(1-\ell^{-2}), [ PSL 2 ( Z ) : Γ 1 ( n ) ∩ Γ ( m ) ] = 2 m n 2 ℓ ∣ n ∏ ( 1 − ℓ − 2 ) , the product being over the primes dividing n n n
For a ∈ ( Z / n Z ) × a\in(\mathbb{Z}/n\mathbb{Z})^\times a ∈ ( Z / n Z ) × ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ ( E , P , Q ) ↦ ( E , a − 1 P , a Q ) (E,P,Q)\mapsto(E,a^{-1}P,aQ) ( E , P , Q ) ↦ ( E , a − 1 P , a Q ) X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) m ≤ 2 m\le2 m ≤ 2 ( E , P , Q ) ↦ ( E , a P , a Q ) (E,P,Q)\mapsto(E,aP,aQ) ( E , P , Q ) ↦ ( E , a P , a Q ) a a a ± 1 \pm1 ± 1 Δ ( n ) : = ( Z / n Z ) × / { ± 1 } \Delta(n):=(\mathbb{Z}/n\mathbb{Z})^\times/\{\pm1\} Δ ( n ) := ( Z / n Z ) × / { ± 1 } Z [ Δ ( n ) ] \mathbb{Z}[\Delta(n)] Z [ Δ ( n )] J J J a 1 , … , a r ∈ ( Z / n Z ) × a_1,\ldots,a_r\in(\mathbb{Z}/n\mathbb{Z})^\times a 1 , … , a r ∈ ( Z / n Z ) × X / ⟨ a 1 , … , a r ⟩ X/\langle a_1,\ldots,a_r\rangle X / ⟨ a 1 , … , a r ⟩ X X X Δ ( n ) \Delta(n) Δ ( n ) ⟨ a 1 ⟩ , … , ⟨ a r ⟩ \langle a_1\rangle,\ldots,\langle a_r\rangle ⟨ a 1 ⟩ , … , ⟨ a r ⟩ q ∤ n q\nmid n q ∤ n q ≡ 1 ( m o d m ) q\equiv1\pmod m q ≡ 1 ( mod m ) T q T_q T q
T q ( E , P , Q ) = ∑ G ( E / G , P m o d G , Q m o d G ) , T_q(E,P,Q)=\sum_{G}(E/G,P\bmod G,Q\bmod G), T q ( E , P , Q ) = G ∑ ( E / G , P mod G , Q mod G ) , the sum being over the q + 1 q+1 q + 1 G ⊆ E [ q ] G\subseteq E[q] G ⊆ E [ q ] q q q
A q : = T q − q ⟨ q ⟩ − 1. A_q:=T_q-q\langle q\rangle-1. A q := T q − q ⟨ q ⟩ − 1. The congruence q ≡ 1 ( m o d m ) q\equiv1\pmod m q ≡ 1 ( mod m ) T q T_q T q q q q Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) A q A_q A q Z [ Δ ( n ) ] \mathbb{Z}[\Delta(n)] Z [ Δ ( n )]
Proposition 2.1. Let q ∤ 2 n q\nmid2n q ∤ 2 n q ≡ 1 ( m o d m ) q\equiv1\pmod m q ≡ 1 ( mod m )
(Eichler–Shimura) T q , F q = Frob q + ⟨ q ⟩ F q , ∗ Ver q T_{q,\mathbb{F}_q}=\operatorname{Frob}_q+\langle q\rangle_{\mathbb{F}_q,*}\operatorname{Ver}_q T q , F q = Frob q + ⟨ q ⟩ F q , ∗ Ver q J F q J_{\mathbb{F}_q} J F q F q \mathbb{F}_q F q Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) q q q
If L ⊇ Q ( ζ m ) L\supseteq\mathbb{Q}(\zeta_m) L ⊇ Q ( ζ m ) q q q A q ( J ( L ) t o r s ) = 0 A_q(J(L)_{\mathrm{tors}})=0 A q ( J ( L ) tors ) = 0
A q ( c 0 ) = 0 A_q(c_0)=0 A q ( c 0 ) = 0
If rk J ( Q ( ζ m ) ) = 0 \operatorname{rk}J(\mathbb{Q}(\zeta_m))=0 rk J ( Q ( ζ m )) = 0 [ A q ( D ) ] = 0 [A_q(D)]=0 [ A q ( D )] = 0 D ∈ X ( d ) ( Q ( ζ m ) ) D\in X^{(d)}(\mathbb{Q}(\zeta_m)) D ∈ X ( d ) ( Q ( ζ m )) d d d
Proof. For m ≤ 2 m\le2 m ≤ 2 [11 (2.2)] , [11 Proposition 2.6] , [11 Lemma 2.8] and [11 Lemma 5.4] . For m ≥ 3 m\ge3 m ≥ 3 X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) X 1 ( m n ) X_1(mn) X 1 ( mn ) 1 1 1 n n n [14 §2] , and this isomorphism is compatible with T q T_q T q ⟨ q ⟩ \langle q\rangle ⟨ q ⟩ q ≡ 1 ( m o d m ) q\equiv1\pmod m q ≡ 1 ( mod m ) X 1 ( m n ) X_1(mn) X 1 ( mn ) [9 (10.2.3)] , and the proofs of [11 Proposition 2.6, Lemma 2.8 and Lemma 5.4] carry over verbatim. □
We will also use the following results.
(Manin–Drinfeld [31 Dri73] ) the class of a degree 0 cuspidal divisor on X X X J ( Q ) J(\mathbb{Q}) J ( Q )
(Katz [24 Appendix] ) if p > 2 p > 2 p > 2 J J J ℘ \wp ℘ L L L p p p L L L ℘ \wp ℘ J ( L ) t o r s J(L)_{\mathrm{tors}} J ( L ) tors
(Parent [40 Proposition 2.4] ) if A A A L L L λ \lambda λ L L L λ \lambda λ A ( L ) t o r s A(L)_{\mathrm{tors}} A ( L ) tors A A A λ \lambda λ λ \lambda λ J J J 2 ∤ n 2 \nmid n 2 ∤ n X 1 ( n ) X_1(n) X 1 ( n )
(Abramovich [1 Theorem 0.1] , with the bound λ 1 ≥ 975 / 4096 \lambda_1 \ge975/4096 λ 1 ≥ 975/4096 Γ ⊆ PSL 2 ( Z ) \Gamma\subseteq\operatorname{PSL}_2(\mathbb{Z}) Γ ⊆ PSL 2 ( Z )
gon C X Γ ≥ 325 2 15 [ PSL 2 ( Z ) : Γ ] ; \operatorname{gon}_{\mathbb{C}} X_\Gamma\ge\frac{325}{2^{15}}[\operatorname{PSL}_2(\mathbb{Z}):\Gamma]; gon C X Γ ≥ 2 15 325 [ PSL 2 ( Z ) : Γ ] ; (Kato [25 Corollary 14.3] ) if A A A J 1 ( n ) J_1(n) J 1 ( n ) Q \mathbb{Q} Q L ( A , 1 ) ≠ 0 L(A,1) \ne0 L ( A , 1 ) = 0 A ( Q ) A(\mathbb{Q}) A ( Q )
(Clark–Genao–Pollack–Saia [6 ] ) the exact values of the least degree d C M X 1 ( n ) d_{\mathrm{CM}}X_1(n) d CM X 1 ( n ) X 1 ( n ) X_1(n) X 1 ( n ) dcm_list_all_min_orders_X1_10k.m in the repository https://github.com/fsaia/least-cm-degree accompanying [6 ] .
We compute all the values of Abramovich’s bound and all the orders of diamond operators that we quote below.
Finally, we state the two standard reduction lemmas.
Lemma 2.2. Let K K K E / K E/K E / K Z / m Z × Z / n Z ⊆ E ( K ) \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \subseteq E(K) Z / m Z × Z / n Z ⊆ E ( K ) ℘ \wp ℘ K K K p ∤ n p \nmid n p ∤ n E E E ℘ \wp ℘
Proof. This is contained in the proof of Proposition 5.2 of [11 ] . If E E E ℘ \wp ℘ a ⋅ # F ℘ a \cdot\#\mathbb{F}_{\wp} a ⋅ # F ℘ F ℘ \mathbb{F}_{\wp} F ℘ a ≤ 4 a \le4 a ≤ 4 # F ℘ \#\mathbb{F}_{\wp} # F ℘ p p p p p p E ( K ) E(K) E ( K ) m n ∣ a ≤ 4 mn \mid a \le4 mn ∣ a ≤ 4 m n > 4 mn > 4 mn > 4 □ \square □
Lemma 2.3 ([11 Lemma 5.3] ). Let p ∤ n p \nmid n p ∤ n K ⊇ Q ( ζ m ) K \supseteq\mathbb{Q}(\zeta_m) K ⊇ Q ( ζ m ) ℘ \wp ℘ K K K p p p E ′ E' E ′ F ℘ \mathbb{F}_{\wp} F ℘ Z / m Z × Z / n Z ⊆ E ′ ( F ℘ ) \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \subseteq E'(\mathbb{F}_{\wp}) Z / m Z × Z / n Z ⊆ E ′ ( F ℘ ) E / K E/K E / K Z / m Z × Z / n Z ⊆ E ( K ) \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \subseteq E(K) Z / m Z × Z / n Z ⊆ E ( K ) K K K p p p
Whenever we invoke Lemma 2.3 , the hypothesis follows from the Hasse bound, from Waterhouse’s description [Wat69, Theorem 4.1] of the possible traces of Frobenius, or from an explicit enumeration of all elliptic curves over the relevant finite fields. For example, over F 64 \mathbb{F}_{64} F 64 t t t ∣ t ∣ ≤ 16 |t| \le16 ∣ t ∣ ≤ 16 t ∈ { 0 , ± 8 , ± 16 } t \in\{0,\pm8,\pm16\} t ∈ { 0 , ± 8 , ± 16 } F 64 \mathbb{F}_{64} F 64 # E ( F 64 ) = 81 \#E(\mathbb{F}_{64}) = 81 # E ( F 64 ) = 81 π \pi π ( π + 8 ) 2 = 0 (\pi+ 8)^2 = 0 ( π + 8 ) 2 = 0 π = − 8 \pi= -8 π = − 8 End ( E ) \operatorname{End}(E) End ( E ) E ( F 64 ) = ker ( 1 − π ) = E [ 9 ] E(\mathbb{F}_{64}) = \ker(1-\pi) = E[9] E ( F 64 ) = ker ( 1 − π ) = E [ 9 ] # E ( F 64 ) = 49 \#E(\mathbb{F}_{64}) = 49 # E ( F 64 ) = 49 π = 8 \pi= 8 π = 8 E ( F 64 ) = E [ 7 ] E(\mathbb{F}_{64}) = E[7] E ( F 64 ) = E [ 7 ]
We will often use the following simple lemma. Its first part is the argument at the end of the proof of [36 Proposition 7.2] .
Lemma 2.4. Let H ⊆ Δ ( n ) H \subseteq\Delta(n) H ⊆ Δ ( n ) D D D X 1 ( n ) X_1(n) X 1 ( n ) h ≠ 1 h \ne1 h = 1 H H H D D D h D = D hD = D h D = D h ∈ H h \in H h ∈ H D D D H H H # H \#H # H D = ∑ O m O ∑ x ∈ O x D = \sum_O m_O \sum_{x \in O} x D = ∑ O m O ∑ x ∈ O x O O O # H ∣ deg D \#H \mid\deg D # H ∣ deg D
Proof. If ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ x = ( E , P ) x = (E, P) x = ( E , P ) D D D ( E , a P ) ≃ ( E , P ) (E, aP) \simeq(E, P) ( E , a P ) ≃ ( E , P ) u u u E E E u ( P ) = a P u(P) = aP u ( P ) = a P E E E Aut ( E ) = { ± 1 } \operatorname{Aut}(E) = \{\pm1\} Aut ( E ) = { ± 1 } a P = ± P aP = \pm P a P = ± P P P P n n n a ≡ ± 1 ( m o d n ) a \equiv\pm1 \pmod n a ≡ ± 1 ( mod n ) ⟨ a ⟩ = 1 \langle a\rangle= 1 ⟨ a ⟩ = 1 Δ ( n ) \Delta(n) Δ ( n ) h D = D hD = D h D = D h ∈ H h \in H h ∈ H D D D x x x h x hx h x D D D H H H D D D # H \#H # H
The groups that have to be considered
Call a finite abelian group G G G sextic torsion group if G ≃ E ( K ) t o r s G \simeq E(K)_{\mathrm{tors}} G ≃ E ( K ) tors E E E K K K G G G Z / m Z × Z / n Z \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} Z / m Z × Z / n Z m ∣ n m \mid n m ∣ n
B : = Φ ∞ ( 6 ) ∪ { Z / 25 Z , Z / 37 Z , Z / 4 Z × Z / 12 Z , Z / 2 Z × Z / 24 Z } . \mathcal{B} := \Phi^\infty(6) \cup\{\mathbb{Z}/25\mathbb{Z}, \mathbb{Z}/37\mathbb{Z}, \mathbb{Z}/4\mathbb{Z} \times\mathbb{Z}/12\mathbb{Z}, \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/24\mathbb{Z}\}. B := Φ ∞ ( 6 ) ∪ { Z /25 Z , Z /37 Z , Z /4 Z × Z /12 Z , Z /2 Z × Z /24 Z } . We will show in § 10 that all the groups in B \mathcal{B} B B \mathcal{B} B
Lemma 3.1. Let G \mathcal{G} G G = Z / m Z × Z / n Z G = \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} G = Z / m Z × Z / n Z m ∣ n m \mid n m ∣ n
G ∉ B G \notin\mathcal{B} G ∈ / B H ∈ B H \in\mathcal{B} H ∈ B H ⊊ G H \subsetneq G H ⊊ G
φ ( m ) ∣ 6 \varphi(m) \mid6 φ ( m ) ∣ 6
every prime divisor of n n n S ( 6 ) = { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 37 } S(6) = \{2, 3, 5, 7, 11, 13, 17, 19, 37\} S ( 6 ) = { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 37 }
Then G \mathcal{G} G
Z / n Z for n ∈ { 32 , 33 , 34 , 35 , 36 , 38 , 39 , 40 , 42 , 44 , 45 , 48 , 49 , 50 , 51 , 52 , 54 , 55 , 56 , 57 , 60 , 63 , 65 , 75 , 77 , 81 , 85 , 91 , 95 , 119 , 121 , 125 , 133 , 143 , 169 , 187 , 209 , 221 , 247 , 289 , 323 , 361 } , Z / 37 n Z for n ∈ { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 37 } , Z / 2 Z × Z / 2 n Z for n ∈ { 11 , 13 , 14 , 15 } , Z / 3 Z × Z / 3 n Z for n ∈ { 5 , 6 , 7 , 8 , 9 } , Z / 4 Z × Z / 4 n Z for n ∈ { 4 , 5 , 6 } , Z / 6 Z × Z / 12 Z , Z / 7 Z × Z / 7 Z , Z / 9 Z × Z / 9 Z . \begin{aligned}
&\mathbb{Z}/n\mathbb{Z}\quad\text{for }n \in\{32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50,\\
&\qquad51, 52, 54, 55, 56, 57, 60, 63, 65, 75, 77, 81, 85, 91, 95,\\
&\qquad119, 121, 125, 133, 143, 169, 187, 209, 221, 247, 289, 323, 361\},\\
&\mathbb{Z}/37n\mathbb{Z}\quad\text{for }n \in\{2, 3, 5, 7, 11, 13, 17, 19, 37\},\\
&\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2n\mathbb{Z}\quad\text{for }n \in\{11, 13, 14, 15\},\\
&\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/3n\mathbb{Z}\quad\text{for }n \in\{5, 6, 7, 8, 9\},\\
&\mathbb{Z}/4\mathbb{Z} \times\mathbb{Z}/4n\mathbb{Z}\quad\text{for }n \in\{4, 5, 6\},\\
&\mathbb{Z}/6\mathbb{Z} \times\mathbb{Z}/12\mathbb{Z},\quad\mathbb{Z}/7\mathbb{Z} \times\mathbb{Z}/7\mathbb{Z},\quad\mathbb{Z}/9\mathbb{Z} \times\mathbb{Z}/9\mathbb{Z}.
\end{aligned} Z / n Z for n ∈ { 32 , 33 , 34 , 35 , 36 , 38 , 39 , 40 , 42 , 44 , 45 , 48 , 49 , 50 , 51 , 52 , 54 , 55 , 56 , 57 , 60 , 63 , 65 , 75 , 77 , 81 , 85 , 91 , 95 , 119 , 121 , 125 , 133 , 143 , 169 , 187 , 209 , 221 , 247 , 289 , 323 , 361 } , Z /37 n Z for n ∈ { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 37 } , Z /2 Z × Z /2 n Z for n ∈ { 11 , 13 , 14 , 15 } , Z /3 Z × Z /3 n Z for n ∈ { 5 , 6 , 7 , 8 , 9 } , Z /4 Z × Z /4 n Z for n ∈ { 4 , 5 , 6 } , Z /6 Z × Z /12 Z , Z /7 Z × Z /7 Z , Z /9 Z × Z /9 Z . Moreover every sextic torsion group not in B \mathcal{B} B G \mathcal{G} G
Proof. Conditions (2) and (3) are necessary for G G G E ( K ) E(K) E ( K ) K K K Q ( ζ m ) ⊆ K \mathbb{Q}(\zeta_m) \subseteq K Q ( ζ m ) ⊆ K φ ( m ) ∣ 6 \varphi(m) \mid6 φ ( m ) ∣ 6 p ∣ n p \mid n p ∣ n p p p p ∈ S ( 6 ) p \in S(6) p ∈ S ( 6 ) [10 Theorem 1.2] . Since B \mathcal{B} B B \mathcal{B} B B \mathcal{B} B G \mathcal{G} G G = Z / n Z G = \mathbb{Z}/n\mathbb{Z} G = Z / n Z Z / ( n / p ) Z ∈ B \mathbb{Z}/(n/p)\mathbb{Z} \in\mathcal{B} Z / ( n / p ) Z ∈ B p ∣ n p \mid n p ∣ n n / p ≤ 37 n/p \le37 n / p ≤ 37 n ≤ 37 2 n \le37^2 n ≤ 3 7 2 m > 1 m > 1 m > 1 Z / n Z ∈ B \mathbb{Z}/n\mathbb{Z} \in\mathcal{B} Z / n Z ∈ B n ≤ 37 n \le37 n ≤ 37
Let G = Z / m Z × Z / n Z ∈ G G = \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \in\mathcal{G} G = Z / m Z × Z / n Z ∈ G E E E K K K G ↪ E ( K ) G \hookrightarrow E(K) G ↪ E ( K ) x ∈ Y 1 ( m , n ) ( K ) x \in Y_1(m,n)(K) x ∈ Y 1 ( m , n ) ( K ) x ( d m ) x^{(d_m)} x ( d m ) § 2 . Conversely, a point of Y 1 ( m , n ) Y_1(m,n) Y 1 ( m , n ) ( K , E ) (K,E) ( K , E ) X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) G G G X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) d m d_m d m Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) G G G Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) d m d_m d m X 1 ( m , n ) X_1(m,n) X 1 ( m , n )
Several members of G \mathcal{G} G
Lemma 3.2. Let k ≥ 2 k \ge2 k ≥ 2 Q \mathbb{Q} Q X 1 ( 4 k ) → X 1 ( 2 , 2 k ) X_1(4k) \to X_1(2,2k) X 1 ( 4 k ) → X 1 ( 2 , 2 k ) ( E , P ) ↦ ( E / ⟨ 2 k P ⟩ , Q , P m o d ⟨ 2 k P ⟩ ) (E,P) \mapsto(E/\langle2kP\rangle,Q,P\bmod\langle2kP\rangle) ( E , P ) ↦ ( E / ⟨ 2 k P ⟩ , Q , P mod ⟨ 2 k P ⟩) Q Q Q E [ 2 ] / ⟨ 2 k P ⟩ E[2]/\langle2kP\rangle E [ 2 ] / ⟨ 2 k P ⟩ Z / 2 Z × Z / 2 k Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2k\mathbb{Z} Z /2 Z × Z /2 k Z Z / 4 k Z \mathbb{Z}/4k\mathbb{Z} Z /4 k Z
Proof. The morphism is the one described in [11 §2] (see also [14 §2] ). It maps non-cuspidal points to non-cuspidal points, and the image of a point defined over K K K K K K
We will apply Lemma 3.2 with k = 11 , 13 , 14 , 15 k = 11, 13, 14, 15 k = 11 , 13 , 14 , 15 Z / 44 Z \mathbb{Z}/44\mathbb{Z} Z /44 Z Z / 52 Z \mathbb{Z}/52\mathbb{Z} Z /52 Z Z / 56 Z \mathbb{Z}/56\mathbb{Z} Z /56 Z Z / 60 Z \mathbb{Z}/60\mathbb{Z} Z /60 Z k = 12 k = 12 k = 12 Z / 2 Z × Z / 24 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/24\mathbb{Z} Z /2 Z × Z /24 Z Z / 48 Z \mathbb{Z}/48\mathbb{Z} Z /48 Z Proposition 10.4 .
The rest of the paper treats the 66 groups of Lemma 3.1 one family at a time. Table 1 shows where each of them is dealt with.
Table 1. Treatment of the 66 groups of Lemma 3.1 .
The global method and diamond operators
In this section we eliminate the seventeen groups Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n ≥ 77 n \ge77 n ≥ 77 37 ∤ n 37 \nmid n 37 ∤ n n ≠ 81 n \ne81 n = 81 [36 §7] . If ψ ∈ Z [ Δ ( n ) ] \psi\in\mathbb{Z}[\Delta(n)] ψ ∈ Z [ Δ ( n )] A [ f ] A_{[f]} A [ f ] J 1 ( n ) J_1(n) J 1 ( n ) L ( f , 1 ) = 0 L(f,1) = 0 L ( f , 1 ) = 0 ψ ( J 1 ( n ) ( Q ) ) \psi(J_1(n)(\mathbb{Q})) ψ ( J 1 ( n ) ( Q )) J 1 ( n ) J_1(n) J 1 ( n ) ⨁ n ′ ∣ n ⨁ f A [ f ] σ 0 ( n / n ′ ) \bigoplus_{n' \mid n}\bigoplus_{f} A_{[f]}^{\sigma_0(n/n')} ⨁ n ′ ∣ n ⨁ f A [ f ] σ 0 ( n / n ′ ) n ′ n' n ′ ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ A [ f ] A_{[f]} A [ f ] χ f ( a ) \chi_f(a) χ f ( a ) χ f \chi_f χ f f f f f f f L ( f , 1 ) ≠ 0 L(f,1) \ne0 L ( f , 1 ) = 0 A [ f ] ( Q ) A_{[f]}(\mathbb{Q}) A [ f ] ( Q ) ψ = ∑ a c a ⟨ a ⟩ \psi= \sum_a c_a\langle a\rangle ψ = ∑ a c a ⟨ a ⟩ ∑ a c a χ f ( a ) = 0 \sum_a c_a\chi_f(a) = 0 ∑ a c a χ f ( a ) = 0 f f f n n n L ( f , 1 ) = 0 L(f,1) = 0 L ( f , 1 ) = 0 [30 ] , where they are recorded as proved, and we check them and the character values below.
The global method
The main tool is [11 Proposition 4.3] , due to Derickx. We use it in the form of [36 Proposition 4.1] : let n n n 3 ∤ n 3 \nmid n 3 ∤ n a ∈ ( Z / n Z ) × a \in(\mathbb{Z}/n\mathbb{Z})^\times a ∈ ( Z / n Z ) × ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle- 1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) k 3 : = 7 k_3 := 7 k 3 := 7 a ∈ { 3 , 3 − 1 } a \in\{3,3^{-1}\} a ∈ { 3 , 3 − 1 } k 3 : = 8 k_3 := 8 k 3 := 8
6 k 3 < 325 2 15 [ PSL 2 ( Z ) : Γ ‾ 1 ( n ) ] , ord ⟨ a ⟩ ∤ 6 , d C M X 1 ( n ) > 6 , (2) 6k_3 < \frac{325}{2^{15}}[\operatorname{PSL}_2(\mathbb{Z}):\overline{\Gamma}_1(n)], \qquad\operatorname{ord}\langle a\rangle\nmid6, \qquad d_{\mathrm{CM}}X_1(n)>6, \tag*{(2)} 6 k 3 < 2 15 325 [ PSL 2 ( Z ) : Γ 1 ( n )] , ord ⟨ a ⟩ ∤ 6 , d CM X 1 ( n ) > 6 , (2) then X 1 ( n ) X_1(n) X 1 ( n ) [36 Proposition 4.1] applies without change with d = 6 d = 6 d = 6 d = 5 d = 5 d = 5 [11 Proposition 4.3] shows that an effective Q \mathbb{Q} Q X 1 ( n ) X_1(n) X 1 ( n ) ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ Lemma 2.4 each orbit has length ord ⟨ a ⟩ \operatorname{ord}\langle a\rangle ord ⟨ a ⟩
Proposition 4.1. The group Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z
n ∈ { 119 , 121 , 125 , 143 , 169 , 187 , 209 , 221 , 289 , 323 , 361 } . n \in\{119,121,125,143,169,187,209,221,289,323,361\}. n ∈ { 119 , 121 , 125 , 143 , 169 , 187 , 209 , 221 , 289 , 323 , 361 } . Proof. We use the elements a a a [36 Table 2] , which are listed in Table 2 . For these a a a ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle- 1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) n ∈ { 119 , 121 , 125 , 169 , 221 , 289 , 361 } n \in\{119,121,125,169,221,289,361\} n ∈ { 119 , 121 , 125 , 169 , 221 , 289 , 361 } n n n n n n a a a (2) are checked in Table 2 . □
Table 2. Data for Proposition 4.1. Here i ( n ) = [ PSL 2 ( Z ) : Γ ‾ 1 ( n ) ] i(n) = [\operatorname{PSL}_2(\mathbb{Z}):\overline{\Gamma}_1(n)] i ( n ) = [ PSL 2 ( Z ) : Γ 1 ( n )] gon C X 1 ( n ) \operatorname{gon}_{\mathbb{C}}X_1(n) gon C X 1 ( n ) 6 k 3 ≤ 48 6k_3 \le48 6 k 3 ≤ 48 ord ⟨ a ⟩ \operatorname{ord}\langle a\rangle ord ⟨ a ⟩ Δ ( n ) \Delta(n) Δ ( n )
Several diamond operators
For the levels n = 77 , 85 , 91 , 95 , 133 , 247 n = 77,85,91,95,133,247 n = 77 , 85 , 91 , 95 , 133 , 247 ord ⟨ a ⟩ ∣ 6 \operatorname{ord}\langle a\rangle\mid6 ord ⟨ a ⟩ ∣ 6 a a a ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle- 1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) [36 ] the levels 91, 95, 133 and 247 were handled by the global method, with a = 3 , 11 , 64 , 144 a = 3,11,64,144 a = 3 , 11 , 64 , 144 1 + ⟨ 31 ⟩ 1 + \langle31\rangle 1 + ⟨ 31 ⟩ ( ⟨ 6 ⟩ − 1 ) ( ⟨ 3 ⟩ − 1 ) (\langle6\rangle- 1)(\langle3\rangle- 1) (⟨ 6 ⟩ − 1 ) (⟨ 3 ⟩ − 1 ) X 0 ( 91 ) X_0(91) X 0 ( 91 ) ψ = ∑ a c a ⟨ a ⟩ ∈ Z [ Δ ( n ) ] \psi= \sum_a c_a\langle a\rangle\in\mathbb{Z}[\Delta(n)] ψ = ∑ a c a ⟨ a ⟩ ∈ Z [ Δ ( n )] ∥ ψ ∥ : = ∑ a ∣ c a ∣ \|\psi\| := \sum_a |c_a| ∥ ψ ∥ := ∑ a ∣ c a ∣
Proposition 4.2. Let n n n ψ ∈ Z [ Δ ( n ) ] \psi\in\mathbb{Z}[\Delta(n)] ψ ∈ Z [ Δ ( n )] ψ ( J 1 ( n ) ( Q ) ) \psi(J_1(n)(\mathbb{Q})) ψ ( J 1 ( n ) ( Q )) gon C X 1 ( n ) > 12 ∥ ψ ∥ \operatorname{gon}_{\mathbb{C}} X_1(n) > 12\|\psi\| gon C X 1 ( n ) > 12∥ ψ ∥ K K K E / K E/K E / K n n n E E E K K K D = x ( 6 ) D = x^{(6)} D = x ( 6 ) ψ ( D ) = 0 \psi(D) = 0 ψ ( D ) = 0
Proof. By Lemma 2.2 the reduction of E E E K K K ℘ \wp ℘ L : = Q ( ζ n ) L := \mathbb{Q}(\zeta_n) L := Q ( ζ n ) D D D ℘ \wp ℘ Z [ 1 n ] \mathbb{Z}[\frac{1}{n}] Z [ n 1 ] L L L C C C X 1 ( n ) L X_1(n)_L X 1 ( n ) L
ψ [ D − C ] = ψ [ D − 6 c 0 ] + ψ [ 6 c 0 − C ] . \psi[D-C] = \psi[D-6c_0] + \psi[6c_0-C]. ψ [ D − C ] = ψ [ D − 6 c 0 ] + ψ [ 6 c 0 − C ] . The first summand lies in ψ ( J 1 ( n ) ( Q ) ) \psi(J_1(n)(\mathbb{Q})) ψ ( J 1 ( n ) ( Q )) ψ [ D − C ] ∈ J 1 ( n ) ( L ) t o r s \psi[D-C] \in J_1(n)(L)_{\mathrm{tors}} ψ [ D − C ] ∈ J 1 ( n ) ( L ) tors ℘ \wp ℘ L L L 2 ψ [ D − C ] = 0 2\psi[D-C] = 0 2 ψ [ D − C ] = 0 2 ψ ( D − C ) = div ( f ) 2\psi(D-C) = \operatorname{div}(f) 2 ψ ( D − C ) = div ( f ) f f f X 1 ( n ) L X_1(n)_L X 1 ( n ) L ψ = ψ + − ψ − \psi= \psi^+ - \psi^- ψ = ψ + − ψ − ψ ± \psi^\pm ψ ± f f f 2 ψ + ( C ) + 2 ψ − ( D ) 2\psi^+(C) + 2\psi^-(D) 2 ψ + ( C ) + 2 ψ − ( D ) 12 ∥ ψ ∥ 12\|\psi\| 12∥ ψ ∥ f f f ψ ( D ) = ψ ( C ) \psi(D) = \psi(C) ψ ( D ) = ψ ( C ) D D D ψ ( D ) = 0 \psi(D) = 0 ψ ( D ) = 0
We will apply Proposition 4.2 with ψ = ⟨ a ⟩ − 1 \psi= \langle a\rangle- 1 ψ = ⟨ a ⟩ − 1 D D D ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ ψ = ⟨ a ⟩ − 1 \psi= \langle a\rangle- 1 ψ = ⟨ a ⟩ − 1 [36 Proposition 7.2] . If the support of D D D D D D H ⊆ Δ ( n ) H \subseteq\Delta(n) H ⊆ Δ ( n ) Lemma 2.4 it is a sum of H H H # H \#H # H
Proposition 4.3. The group Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n ∈ { 77 , 85 , 91 , 95 , 133 , 247 } n \in\{77,85,91,95,133,247\} n ∈ { 77 , 85 , 91 , 95 , 133 , 247 } Proof. Suppose E E E K K K n n n E E E K K K Lemma 2.3 it suffices to show that no elliptic curve over F 2 f \mathbb{F}_{2^f} F 2 f f ≤ 6 f \le6 f ≤ 6 n n n f ≤ 5 f \le5 f ≤ 5 ( 2 5 / 2 + 1 ) 2 < 45 (2^{5/2}+1)^2 < 45 ( 2 5/2 + 1 ) 2 < 45 f = 6 f=6 f = 6 # E ′ ( F 64 ) ≤ 81 \#E'(\mathbb{F}_{64}) \le81 # E ′ ( F 64 ) ≤ 81 n ≠ 77 n \ne77 n = 77 n = 77 n=77 n = 77 F 64 \mathbb{F}_{64} F 64 − 12 -12 − 12 − 12 -12 − 12 0 , ± 8 , ± 16 0,\pm8,\pm16 0 , ± 8 , ± 16
Hence Proposition 4.2 applies to D = x ( 6 ) D=x^{(6)} D = x ( 6 ) ψ \psi ψ Table 3 . In every case d C M X 1 ( n ) > 6 d_{\mathrm{CM}X_1(n)}>6 d CM X 1 ( n ) > 6 D D D Lemma 2.4 , D D D # H \#H # H H H H Δ ( n ) \Delta(n) Δ ( n )
For n = 77 n=77 n = 77 ψ = ⟨ 3 ⟩ − 1 \psi=\langle3\rangle-1 ψ = ⟨ 3 ⟩ − 1 ψ ( J 1 ( 77 ) ( Q ) ) \psi(J_1(77)(\mathbb{Q})) ψ ( J 1 ( 77 ) ( Q )) 12 ∥ ψ ∥ = 24 < 28.56 12\lVert\psi\rVert=24<28.56 12 ∥ ψ ∥ = 24 < 28.56 ⟨ 3 ⟩ D = D \langle3\rangle D=D ⟨ 3 ⟩ D = D ord ⟨ 3 ⟩ = 30 \operatorname{ord}\langle3\rangle=30 ord ⟨ 3 ⟩ = 30
For n = 85 n=85 n = 85 ψ = ⟨ 2 ⟩ − 1 \psi=\langle2\rangle-1 ψ = ⟨ 2 ⟩ − 1 85. j 85.j 85. j ord ⟨ 2 ⟩ = 8 ∤ 6 \operatorname{ord}\langle2\rangle=8\nmid6 ord ⟨ 2 ⟩ = 8 ∤ 6 n = 77 n=77 n = 77
For n = 91 n=91 n = 91 ψ = ⟨ 3 ⟩ − 1 \psi=\langle3\rangle-1 ψ = ⟨ 3 ⟩ − 1 91. g 91.g 91. g 91. u 91.u 91. u ⟨ 3 ⟩ D = D \langle3\rangle D=D ⟨ 3 ⟩ D = D ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ D D D ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ x x x x x x X 0 ( 91 ) = X 1 ( 91 ) / Δ ( 91 ) X_0(91)=X_1(91)/\Delta(91) X 0 ( 91 ) = X 1 ( 91 ) /Δ ( 91 ) X 0 ( 91 ) X_0(91) X 0 ( 91 ) [26 ] , which is a contradiction.
For n = 95 n=95 n = 95 ψ = 1 + ⟨ 31 ⟩ \psi=1+\langle31\rangle ψ = 1 + ⟨ 31 ⟩ χ \chi χ χ ( 31 ) = − 1 \chi(31)=-1 χ ( 31 ) = − 1 ψ \psi ψ 12 ∥ ψ ∥ = 24 < 42.84 12\lVert\psi\rVert=24<42.84 12 ∥ ψ ∥ = 24 < 42.84 D + ⟨ 31 ⟩ D = 0 D+\langle31\rangle D=0 D + ⟨ 31 ⟩ D = 0
For n = 133 n=133 n = 133 χ f \chi_f χ f χ s \chi_s χ s 133. f 133.f 133. f 133. s 133.s 133. s χ f ( 6 ) = 1 \chi_f(6)=1 χ f ( 6 ) = 1 χ s ( 3 ) = 1 \chi_s(3)=1 χ s ( 3 ) = 1
ψ = ( ⟨ 6 ⟩ − 1 ) ( ⟨ 3 ⟩ − 1 ) , ∥ ψ ∥ = 4 , 12 ∥ ψ ∥ = 48 < 85.69 , \psi=(\langle6\rangle-1)(\langle3\rangle-1),\qquad\lVert\psi\rVert=4,\qquad12\lVert\psi\rVert=48<85.69, ψ = (⟨ 6 ⟩ − 1 ) (⟨ 3 ⟩ − 1 ) , ∥ ψ ∥ = 4 , 12 ∥ ψ ∥ = 48 < 85.69 , annihilates every factor of positive analytic rank, and Proposition 4.2 gives ψ ( D ) = 0 \psi(D)=0 ψ ( D ) = 0 D ′ = ⟨ 3 ⟩ D − D D'=\langle3\rangle D-D D ′ = ⟨ 3 ⟩ D − D ⟨ 6 ⟩ D ′ = D ′ \langle6\rangle D'=D' ⟨ 6 ⟩ D ′ = D ′ Lemma 2.4 its geometric points form a union of ⟨ 6 ⟩ \langle6\rangle ⟨ 6 ⟩ ord ⟨ 6 ⟩ = 18 \operatorname{ord}\langle6\rangle=18 ord ⟨ 6 ⟩ = 18 D D D ⟨ 3 ⟩ D \langle3\rangle D ⟨ 3 ⟩ D D ′ = 0 D'=0 D ′ = 0 ⟨ 3 ⟩ D = D \langle3\rangle D=D ⟨ 3 ⟩ D = D ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ 9 ∣ deg D = 6 9\mid\deg D=6 9 ∣ deg D = 6
For n = 247 n=247 n = 247 247. b e 247.be 247. b e ⟨ 12 ⟩ D = ⟨ 56 ⟩ D = D \langle12\rangle D=\langle56\rangle D=D ⟨ 12 ⟩ D = ⟨ 56 ⟩ D = D ⟨ 12 ⟩ \langle12\rangle ⟨ 12 ⟩ ⟨ 56 ⟩ \langle56\rangle ⟨ 56 ⟩ Δ ( 247 ) \Delta(247) Δ ( 247 ) Lemma 2.4 , D D D 9 ∣ deg D = 6 9\mid\deg D=6 9 ∣ deg D = 6
Table 3. The newforms of weight 2 and level dividing n n n i ( n ) i(n) i ( n ) Table 2 . Their Conrey labels are 85.4, 91.9, 91.30, 95.8, 133.39, 133.31 and 247.8.
The rank zero criterion
We recall [11 Proposition 5.2] in the form in which we use it.
Proposition 5.1 ([11 Proposition 5.2] ). Let m ∣ n m\mid n m ∣ n m n > 4 mn>4 mn > 4 p ∣ m n p\mid mn p ∣ mn d ≥ 1 d\ge1 d ≥ 1 rk Q ( ζ m ) J 1 ( m , n ) ( Q ( ζ m ) ) = 0 \operatorname{rk}_{\mathbb{Q}(\zeta_m)}J_1(m,n)(\mathbb{Q}(\zeta_m))=0 rk Q ( ζ m ) J 1 ( m , n ) ( Q ( ζ m )) = 0 p ≥ 2 p\ge2 p ≥ 2 gon Q ( ζ m ) X 1 ( m , n ) > d \operatorname{gon}_{\mathbb{Q}(\zeta_m)}X_1(m,n)>d gon Q ( ζ m ) X 1 ( m , n ) > d p ≥ 2 p\ge2 p ≥ 2 gon Q ( ζ m ) X 1 ( m , n ) > 2 d \operatorname{gon}_{\mathbb{Q}(\zeta_m)}X_1(m,n)>2d gon Q ( ζ m ) X 1 ( m , n ) > 2 d
Then every elliptic curve E E E d d d K K K Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) Z / m Z × Z / n Z ⊆ E ( K ) \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \subseteq E(K) Z / m Z × Z / n Z ⊆ E ( K ) p \mathfrak{p} p Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) p p p K K K p \mathfrak{p} p
In [11 Proposition 5.2] the conclusion is stated for the primes of K K K p p p p \mathfrak{p} p Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) p p p E E E K K K p \mathfrak{p} p Z / 7 Z × Z / 7 Z \mathbb{Z}/7\mathbb{Z} \times\mathbb{Z}/7\mathbb{Z} Z /7 Z × Z /7 Z Z / 9 Z × Z / 9 Z \mathbb{Z}/9\mathbb{Z} \times\mathbb{Z}/9\mathbb{Z} Z /9 Z × Z /9 Z [14 Theorem 4.1] .
Lemma 5.2. For m = 7 m = 7 m = 7 m = 9 m = 9 m = 9 rk J 1 ( m , m ) ( Q ( ζ m ) ) = 0 \operatorname{rk} J_1(m,m)(\mathbb{Q}(\zeta_m)) = 0 rk J 1 ( m , m ) ( Q ( ζ m )) = 0
Proof. Put F = Q ( ζ m ) F = \mathbb{Q}(\zeta_m) F = Q ( ζ m ) [14 Theorem 4.1] , where F F F L ( g , 1 ) ≠ 0 L(g,1) \ne0 L ( g , 1 ) = 0 g g g F F F X 1 ( m , m ) X_1(m,m) X 1 ( m , m ) X H X_H X H X 1 ( m 2 ) X_1(m^2) X 1 ( m 2 ) 1 1 1 m m m [14 §2] , so it suffices to show that B ( F ) B(F) B ( F ) B B B X H X_H X H Q \mathbb{Q} Q 2 2 2 X H X_H X H
V m : = ⨁ ε S 2 ( Γ 0 ( m 2 ) , ε ) , V_m := \bigoplus_{\varepsilon} S_2(\Gamma_0(m^2),\varepsilon), V m := ε ⨁ S 2 ( Γ 0 ( m 2 ) , ε ) , the sum being over the even Dirichlet characters ε \varepsilon ε m m m Q ℓ \mathbb{Q}_{\ell} Q ℓ ℓ \ell ℓ B B B ρ f \rho_f ρ f f f f V m V_m V m B ( F ) = R ( Q ) B(F) = R(\mathbb{Q}) B ( F ) = R ( Q ) R : = Res F / Q B F R := \operatorname{Res}_{F/\mathbb{Q}} B_F R := Res F / Q B F ℓ \ell ℓ B B B Gal ( Q ‾ / F ) \operatorname{Gal}(\overline{\mathbb{Q}}/F) Gal ( Q / F ) Q ℓ \mathbb{Q}_{\ell} Q ℓ ρ f ⊗ χ \rho_f \otimes\chi ρ f ⊗ χ χ \chi χ Gal ( F / Q ) ≃ ( Z / m Z ) × \operatorname{Gal}(F/\mathbb{Q}) \simeq(\mathbb{Z}/m\mathbb{Z})^{\times} Gal ( F / Q ) ≃ ( Z / m Z ) × m 2 m^2 m 2 ε \varepsilon ε m m m χ \chi χ m m m m 2 m^2 m 2 ε χ 2 \varepsilon\chi^2 ε χ 2 m m m ρ f ⊗ χ ≃ ρ g \rho_f \otimes\chi\simeq\rho_g ρ f ⊗ χ ≃ ρ g g g g V m V_m V m g g g Aut ( C ) \operatorname{Aut}(\mathbb{C}) Aut ( C ) [16 ] the abelian variety R R R Q \mathbb{Q} Q A [ g ] A_{[g]} A [ g ] g g g V m V_m V m L ( g , 1 ) ≠ 0 L(g,1) \ne0 L ( g , 1 ) = 0 g g g V m V_m V m V m V_m V m 49 49 49 m = 7 m = 7 m = 7 27 27 27 81 81 81 m = 9 m = 9 m = 9
Proposition 5.3. The groups Z / 55 Z \mathbb{Z}/55\mathbb{Z} Z /55 Z Z / 75 Z \mathbb{Z}/75\mathbb{Z} Z /75 Z Z / 81 Z \mathbb{Z}/81\mathbb{Z} Z /81 Z Z / 3 Z × Z / 27 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/27\mathbb{Z} Z /3 Z × Z /27 Z Z / 4 Z × Z / 24 Z \mathbb{Z}/4\mathbb{Z} \times\mathbb{Z}/24\mathbb{Z} Z /4 Z × Z /24 Z Z / 7 Z × Z / 7 Z \mathbb{Z}/7\mathbb{Z} \times\mathbb{Z}/7\mathbb{Z} Z /7 Z × Z /7 Z Z / 9 Z × Z / 9 Z \mathbb{Z}/9\mathbb{Z} \times\mathbb{Z}/9\mathbb{Z} Z /9 Z × Z /9 Z
Proof. In each case suppose that E E E K K K d = d m = [ K : Q ( ζ m ) ] d = d_m = [K : \mathbb{Q}(\zeta_m)] d = d m = [ K : Q ( ζ m )] p p p E E E K K K p p p Lemma 2.3 , which is a contradiction.
For n = 55 , 75 , 81 n = 55,75,81 n = 55 , 75 , 81 rk J 1 ( n ) ( Q ) = 0 \operatorname{rk} J_1(n)(\mathbb{Q}) = 0 rk J 1 ( n ) ( Q ) = 0 [7 Theorem 3.1(2)] , and Abramovich’s bound gives gon Q X 1 ( n ) > 14.28 , 23.80 , 28.92 \operatorname{gon}_{\mathbb{Q}} X_1(n) > 14.28,23.80,28.92 gon Q X 1 ( n ) > 14.28 , 23.80 , 28.92 12 12 12 p = 2 p = 2 p = 2 d = 6 d = 6 d = 6 F 2 f \mathbb{F}_{2^f} F 2 f f ≤ 5 f \le5 f ≤ 5 n n n F 64 \mathbb{F}_{64} F 64 55 55 55 # E ( F 64 ) = 55 \#E(\mathbb{F}_{64}) = 55 # E ( F 64 ) = 55 75 75 75 81 81 81 E ( F 64 ) = E [ 9 ] E(\mathbb{F}_{64}) = E[9] E ( F 64 ) = E [ 9 ] 81 81 81 Lemma 2.3 ).
For Z / 3 Z × Z / 27 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/27\mathbb{Z} Z /3 Z × Z /27 Z d = 3 d = 3 d = 3 [14 Theorem 4.1] , rk J 1 ( 3 , 27 ) ( Q ( ζ 3 ) ) = 0 \operatorname{rk} J_1(3, 27)(\mathbb{Q}(\zeta_3)) = 0 rk J 1 ( 3 , 27 ) ( Q ( ζ 3 )) = 0 [ PSL 2 ( Z ) : Γ 1 ( 27 ) ∩ Γ ( 3 ) ] = 972 [\operatorname{PSL}_2(\mathbb{Z}) : \Gamma_1(27) \cap\Gamma(3)] = 972 [ PSL 2 ( Z ) : Γ 1 ( 27 ) ∩ Γ ( 3 )] = 972 gon C X 1 ( 3 , 27 ) > 9.64 > 6 \operatorname{gon}_{\mathbb{C}} X_1(3, 27) > 9.64 > 6 gon C X 1 ( 3 , 27 ) > 9.64 > 6 p = 2 p = 2 p = 2 2 2 2 Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) K K K 2 2 2 F 4 \mathbb{F}_4 F 4 F 16 \mathbb{F}_{16} F 16 F 64 \mathbb{F}_{64} F 64 81 81 81 F 64 \mathbb{F}_{64} F 64 Z / 3 Z × Z / 27 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/27\mathbb{Z} Z /3 Z × Z /27 Z E ( F 64 ) = E [ 9 ] ≃ ( Z / 9 Z ) 2 E(\mathbb{F}_{64}) = E[9] \simeq(\mathbb{Z}/9\mathbb{Z})^2 E ( F 64 ) = E [ 9 ] ≃ ( Z /9 Z ) 2 Lemma 2.3 ), which has no point of order 27 27 27
For Z / 4 Z × Z / 24 Z \mathbb{Z}/4\mathbb{Z} \times\mathbb{Z}/24\mathbb{Z} Z /4 Z × Z /24 Z d = 3 d = 3 d = 3 [14 Theorem 4.1] , rk J 1 ( 4 , 24 ) ( Q ( i ) ) = 0 \operatorname{rk} J_1(4, 24)(\mathbb{Q}(i)) = 0 rk J 1 ( 4 , 24 ) ( Q ( i )) = 0 [ PSL 2 ( Z ) : Γ 1 ( 24 ) ∩ Γ ( 4 ) ] = 768 [\operatorname{PSL}_2(\mathbb{Z}) : \Gamma_1(24) \cap\Gamma(4)] = 768 [ PSL 2 ( Z ) : Γ 1 ( 24 ) ∩ Γ ( 4 )] = 768 gon C X 1 ( 4 , 24 ) > 7.61 > 3 \operatorname{gon}_{\mathbb{C}} X_1(4, 24) > 7.61 > 3 gon C X 1 ( 4 , 24 ) > 7.61 > 3 p = 5 p = 5 p = 5 5 5 5 Q ( i ) \mathbb{Q}(i) Q ( i ) K K K 5 5 5 F 5 \mathbb{F}_5 F 5 F 25 \mathbb{F}_{25} F 25 F 125 \mathbb{F}_{125} F 125 96 96 96 ( 25 + 1 ) 2 < 96 (\sqrt{25} + 1)^2 < 96 ( 25 + 1 ) 2 < 96 96 < ( 125 − 1 ) 2 < ( 125 + 1 ) 2 < 192 96 < (\sqrt{125} - 1)^2 < (\sqrt{125} + 1)^2 < 192 96 < ( 125 − 1 ) 2 < ( 125 + 1 ) 2 < 192
For Z / 7 Z × Z / 7 Z \mathbb{Z}/7\mathbb{Z} \times\mathbb{Z}/7\mathbb{Z} Z /7 Z × Z /7 Z Z / 9 Z × Z / 9 Z \mathbb{Z}/9\mathbb{Z} \times\mathbb{Z}/9\mathbb{Z} Z /9 Z × Z /9 Z K = Q ( ζ m ) K = \mathbb{Q}(\zeta_m) K = Q ( ζ m ) d = 1 d = 1 d = 1 X 1 ( 7 , 7 ) X_1(7, 7) X 1 ( 7 , 7 ) X 1 ( 9 , 9 ) X_1(9, 9) X 1 ( 9 , 9 ) 3 3 3 10 10 10 gon Q ( ζ m ) X 1 ( m , m ) > 1 \operatorname{gon}_{\mathbb{Q}(\zeta_m)} X_1(m, m) > 1 gon Q ( ζ m ) X 1 ( m , m ) > 1 Lemma 5.2 gives rk J 1 ( m , m ) ( Q ( ζ m ) ) = 0 \operatorname{rk} J_1(m, m)(\mathbb{Q}(\zeta_m)) = 0 rk J 1 ( m , m ) ( Q ( ζ m )) = 0 p = 29 p = 29 p = 29 m = 7 m = 7 m = 7 p = 19 p = 19 p = 19 m = 9 m = 9 m = 9 p ≡ 1 ( m o d m ) p \equiv1 \pmod{m} p ≡ 1 ( mod m ) K K K p p p F p \mathbb{F}_p F p ( p + 1 ) 2 < m 2 (\sqrt{p} + 1)^2 < m^2 ( p + 1 ) 2 < m 2 F p \mathbb{F}_p F p m m m
A criterion using twice a cuspidal divisor
When p = 2 p = 2 p = 2 gon Q ( ζ m ) X 1 ( m , n ) > 2 d \operatorname{gon}_{\mathbb{Q}(\zeta_m)} X_1(m, n) > 2d gon Q ( ζ m ) X 1 ( m , n ) > 2 d 2 2 2 X 1 ( 45 ) X_1(45) X 1 ( 45 ) X 1 ( 3 , 21 ) X_1(3, 21) X 1 ( 3 , 21 )
Lemma 5.4. Let m ∣ n m \mid n m ∣ n n n n m n > 4 mn > 4 mn > 4 p \mathfrak{p} p Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) 2 2 2 k k k d ≥ 1 d \ge1 d ≥ 1 X = X 1 ( m , n ) X = X_1(m, n) X = X 1 ( m , n ) rk J ( Q ( ζ m ) ) = 0 \operatorname{rk} J(\mathbb{Q}(\zeta_m)) = 0 rk J ( Q ( ζ m )) = 0
dim L ( 2 C ‾ ) = 1 \dim L(2\overline{C}) = 1 dim L ( 2 C ) = 1 for every effective cuspidal divisor C ‾ \overline{C} C d d d X k X_k X k E E E d d d K K K Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) Z / m Z × Z / n Z ⊆ E ( K ) \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \subseteq E(K) Z / m Z × Z / n Z ⊆ E ( K ) K K K p \mathfrak{p} p
Proof. Suppose the opposite, and let D D D Lemma 2.2 the reduction of E E E K K K p \mathfrak{p} p D D D C ‾ \overline{C} C d d d X k X_k X k M = Q ( ζ n ) M = \mathbb{Q}(\zeta_n) M = Q ( ζ n ) P \mathfrak{P} P M M M p \mathfrak{p} p F ⊆ M F \subseteq M F ⊆ M Gal ( M / Q ( ζ m ) ) \operatorname{Gal}(M/\mathbb{Q}(\zeta_m)) Gal ( M / Q ( ζ m )) P ∩ F \mathfrak{P} \cap F P ∩ F F F F k k k F F F 2 2 2 n n n M M M 2 2 2 C ‾ \overline{C} C Gal ( M / F ) \operatorname{Gal}(M/F) Gal ( M / F ) C ‾ \overline{C} C F F F C C C d d d [ D − C ] = [ D − d c 0 ] + [ d c 0 − C ] [D - C] = [D - dc_0] + [dc_0 - C] [ D − C ] = [ D − d c 0 ] + [ d c 0 − C ] J ( F ) J(F) J ( F ) 0 0 0 P ∩ F \mathfrak{P} \cap F P ∩ F 2 [ D − C ] = 0 2[D - C] = 0 2 [ D − C ] = 0 2 D 2D 2 D 2 C 2C 2 C [19 Theorem III.12.8] , the dimension of the Riemann–Roch space of 2 C 2C 2 C X F X_F X F
dim L ( 2 C ) ≤ dim L ( 2 C ‾ ) = 1 , \dim L(2C) \le\dim L(2\overline{C}) = 1, dim L ( 2 C ) ≤ dim L ( 2 C ) = 1 , so 2 C 2C 2 C 2 D = 2 C 2D = 2C 2 D = 2 C D D D C C C
We check the condition of Lemma 5.4 using differentials. Let g g g X k X_k X k P P P X k X_k X k j ≥ 0 j \ge0 j ≥ 0 Ω ( j P ) \Omega(jP) Ω ( j P ) X k X_k X k j j j P P P g g g H 0 ( X k , Ω 1 ) H^0(X_k,\Omega^1) H 0 ( X k , Ω 1 ) j j j P P P P P P Ω ( j P ) \Omega(jP) Ω ( j P ) j deg P j\deg P j deg P C ‾ = ∑ P m P P \overline{C} = \sum_P m_P P C = ∑ P m P P Ω ( C ‾ ) \Omega(\overline{C}) Ω ( C ) Ω ( m P P ) \Omega(m_P P) Ω ( m P P ) deg C ‾ \deg\overline{C} deg C
dim L ( 2 C ‾ ) = 2 d + 1 − g + dim Ω ( 2 C ‾ ) = 1 + 2 d − codim Ω ( 2 C ‾ ) . \dim L(2\overline{C}) = 2d + 1 - g + \dim\Omega(2\overline{C}) = 1 + 2d - \operatorname{codim} \Omega(2\overline{C}). dim L ( 2 C ) = 2 d + 1 − g + dim Ω ( 2 C ) = 1 + 2 d − codim Ω ( 2 C ) . As Ω ( 2 C ‾ ) \Omega(2\overline{C}) Ω ( 2 C ) ∑ P 2 m P deg P = 2 d \sum_P 2m_P \deg P = 2d ∑ P 2 m P deg P = 2 d k k k 2 m P 2m_P 2 m P P P P dim L ( 2 C ‾ ) = 1 \dim L(2\overline{C}) = 1 dim L ( 2 C ) = 1
This is much more efficient than computing each Riemann–Roch space L ( 2 C ‾ ) L(2\overline{C}) L ( 2 C ) Ω ( 2 m P ) \Omega(2mP) Ω ( 2 m P ) P P P m ≤ d / deg P m \le d/\deg P m ≤ d / deg P H 0 ( X k , Ω 1 ) H^0(X_k,\Omega^1) H 0 ( X k , Ω 1 ) P P P X 1 ( 45 ) F 2 X_1(45)_{\mathbb{F}_2} X 1 ( 45 ) F 2 C ‾ \overline{C} C C ‾ \overline{C} C H 0 ( X k , Ω 1 ) H^0(X_k,\Omega^1) H 0 ( X k , Ω 1 ) k k k
Proposition 5.5. The groups Z / 45 Z \mathbb{Z}/45\mathbb{Z} Z /45 Z Z / 3 Z × Z / 21 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/21\mathbb{Z} Z /3 Z × Z /21 Z Z / 3 Z × Z / 15 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/15\mathbb{Z} Z /3 Z × Z /15 Z
Proof. Let E E E K K K E E E K K K Z / 45 Z \mathbb{Z}/45\mathbb{Z} Z /45 Z F 2 f \mathbb{F}_{2^f} F 2 f f ≤ 6 f \le6 f ≤ 6 Z / 3 Z × Z / 21 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/21\mathbb{Z} Z /3 Z × Z /21 Z Z / 3 Z × Z / 15 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/15\mathbb{Z} Z /3 Z × Z /15 Z F 4 \mathbb{F}_4 F 4 F 16 \mathbb{F}_{16} F 16 F 64 \mathbb{F}_{64} F 64 F 64 \mathbb{F}_{64} F 64 E E E
For Z / 45 Z \mathbb{Z}/45\mathbb{Z} Z /45 Z rk J 1 ( 45 ) ( Q ) = 0 \operatorname{rk} J_1(45)(\mathbb{Q}) = 0 rk J 1 ( 45 ) ( Q ) = 0 [7 Theorem 3.1(2)] , and we apply Lemma 5.4 with m = 1 m = 1 m = 1 d = 6 d = 6 d = 6 k = F 2 k = \mathbb{F}_2 k = F 2 X 1 ( 45 ) F 2 X_1(45)_{\mathbb{F}_2} X 1 ( 45 ) F 2 1 12 , 2 4 , 4 5 , 6 2 1^{12}, 2^4, 4^5, 6^2 1 12 , 2 4 , 4 5 , 6 2 a r a^r a r r r r a a a dim L ( 2 C ‾ ) = 1 \dim L(2\overline{C}) = 1 dim L ( 2 C ) = 1
For Z / 3 Z × Z / 21 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/21\mathbb{Z} Z /3 Z × Z /21 Z rk J 1 ( 3 , 21 ) ( Q ( ζ 3 ) ) = 0 \operatorname{rk} J_1(3,21)(\mathbb{Q}(\zeta_3)) = 0 rk J 1 ( 3 , 21 ) ( Q ( ζ 3 )) = 0 [14 Theorem 4.1] , and we apply Lemma 5.4 with m = 3 m = 3 m = 3 d = 3 d = 3 d = 3 k = F 4 k = \mathbb{F}_4 k = F 4 X 1 ( 3 , 21 ) F 4 X_1(3,21)_{\mathbb{F}_4} X 1 ( 3 , 21 ) F 4 1 24 , 3 8 1^{24}, 3^8 1 24 , 3 8 ( 26 3 ) + 8 = 2608 \binom{26}{3} + 8 = 2608 ( 3 26 ) + 8 = 2608 dim L ( 2 C ‾ ) = 1 \dim L(2\overline{C}) = 1 dim L ( 2 C ) = 1
For Z / 3 Z × Z / 15 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/15\mathbb{Z} Z /3 Z × Z /15 Z X = X 1 ( 3 , 15 ) X = X_1(3,15) X = X 1 ( 3 , 15 ) Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) rk J ( Q ( ζ 3 ) ) = 0 \operatorname{rk} J(\mathbb{Q}(\zeta_3)) = 0 rk J ( Q ( ζ 3 )) = 0 [14 Theorem 4.1] , and let p \mathfrak{p} p Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) F 4 \mathbb{F}_4 F 4 X F 4 X_{\mathbb{F}_4} X F 4 ( 18 3 ) + 16 ⋅ 8 = 944 \binom{18}{3} + 16 \cdot8 = 944 ( 3 18 ) + 16 ⋅ 8 = 944 dim L ( 2 C ‾ ) = 1 \dim L(2\overline{C}) = 1 dim L ( 2 C ) = 1 D D D dim L ( C ‾ ) = 1 \dim L(\overline{C}) = 1 dim L ( C ) = 1 X F 4 X_{\mathbb{F}_4} X F 4 X X X Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) a ∈ ( Z / 15 Z ) × a \in(\mathbb{Z}/15\mathbb{Z})^\times a ∈ ( Z /15 Z ) × σ a \sigma_a σ a Q ( ζ 15 ) \mathbb{Q}(\zeta_{15}) Q ( ζ 15 ) σ a ( ζ 15 ) = ζ 15 a \sigma_a(\zeta_{15}) = \zeta_{15}^a σ a ( ζ 15 ) = ζ 15 a Gal ( Q ( ζ 15 ) / Q ( ζ 3 ) ) \operatorname{Gal}(\mathbb{Q}(\zeta_{15})/\mathbb{Q}(\zeta_3)) Gal ( Q ( ζ 15 ) / Q ( ζ 3 )) σ 7 \sigma_7 σ 7 p \mathfrak{p} p σ 4 = σ 7 2 \sigma_4 = \sigma_7^2 σ 4 = σ 7 2 X X X σ 4 \sigma_4 σ 4 Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) σ 7 \sigma_7 σ 7 σ 7 \sigma_7 σ 7 σ 4 \sigma_4 σ 4 X X X σ 4 = σ 7 2 \sigma_4=\sigma_7^2 σ 4 = σ 7 2 C ‾ \overline{C} C Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) C C C [ D − C ] ∈ J ( Q ( ζ 3 ) ) [D-C]\in J(\mathbb{Q}(\zeta_3)) [ D − C ] ∈ J ( Q ( ζ 3 ))
We claim that reduction modulo p \mathfrak{p} p J ( Q ( ζ 3 ) ) = J ( Q ( ζ 3 ) ) t o r s J(\mathbb{Q}(\zeta_3))=J(\mathbb{Q}(\zeta_3))_{\mathrm{tors}} J ( Q ( ζ 3 )) = J ( Q ( ζ 3 ) ) tors G ⊆ J ( Q ( ζ 3 ) ) G\subseteq J(\mathbb{Q}(\zeta_3)) G ⊆ J ( Q ( ζ 3 )) Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) G G G p \mathfrak{p} p [ 3 , 3 , 24 , 96 , 96 ] [3,3,24,96,96] [ 3 , 3 , 24 , 96 , 96 ] 2 13 2^{13} 2 13 J ( F 7 ) J(\mathbb{F}_7) J ( F 7 ) Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) [ 3 , 3 , 3 , 24 , 480 , 480 ] [3,3,3,24,480,480] [ 3 , 3 , 3 , 24 , 480 , 480 ] 2 13 2^{13} 2 13 J ( Q ( ζ 3 ) ) t o r s J(\mathbb{Q}(\zeta_3))_{\mathrm{tors}} J ( Q ( ζ 3 ) ) tors J ( Q ( ζ 3 ) ) J(\mathbb{Q}(\zeta_3)) J ( Q ( ζ 3 )) 2 13 2^{13} 2 13 G G G 2 13 2^{13} 2 13 J ( Q ( ζ 3 ) ) [ 2 ∞ ] = G [ 2 ∞ ] J(\mathbb{Q}(\zeta_3))[2^\infty]=G[2^\infty] J ( Q ( ζ 3 )) [ 2 ∞ ] = G [ 2 ∞ ] p \mathfrak{p} p D ∼ C D\sim C D ∼ C [19 Theorem III.12.8] dim L ( C ) ≤ dim L ( C ‾ ) = 1 \dim L(C)\leq\dim L(\overline{C})=1 dim L ( C ) ≤ dim L ( C ) = 1 D = C D=C D = C
The Hecke sieve in degree six
Let X = X 1 ( m , n ) X=X_1(m,n) X = X 1 ( m , n ) rk J ( Q ( ζ m ) ) = 0 \operatorname{rk}J(\mathbb{Q}(\zeta_m))=0 rk J ( Q ( ζ m )) = 0 p \mathfrak{p} p Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) p ∤ n p\nmid n p ∤ n k k k K ⊇ Q ( ζ m ) K\supseteq\mathbb{Q}(\zeta_m) K ⊇ Q ( ζ m ) x ∈ Y 1 ( m , n ) ( K ) x\in Y_1(m,n)(K) x ∈ Y 1 ( m , n ) ( K ) D x : = x ( d m ) D_x:=x^{(d_m)} D x := x ( d m ) p \mathfrak{p} p
( D x ) k = D + C , (D_x)_k=D+C, ( D x ) k = D + C , where D D D d ′ d' d ′ C C C d m − d ′ d_m-d' d m − d ′ d ′ ≥ 1 d'\geq1 d ′ ≥ 1 x x x K K K p \mathfrak{p} p [36 §6] we call the multiset of degrees of the closed points occurring in D D D d ′ d' d ′ [11 Proposition 5.5] says that if [ A q ( D ) ] ≠ 0 [A_q(D)]\neq0 [ A q ( D )] = 0 [ A q ( C ) ] = 0 [A_q(C)]=0 [ A q ( C )] = 0 J ( k ) J(k) J ( k ) q ∤ 2 p n q\nmid2pn q ∤ 2 p n q ≡ 1 ( m o d m ) q\equiv1\pmod m q ≡ 1 ( mod m ) [ A q ( D x ) ] = 0 [A_q(D_x)]=0 [ A q ( D x )] = 0 [ A q ( D ) ] + [ A q ( C ) ] [A_q(D)]+[A_q(C)] [ A q ( D )] + [ A q ( C )] A q A_q A q D D D
The condition [ A q ( C ) ] = 0 [A_q(C)]=0 [ A q ( C )] = 0 [36 Lemma 6.1] , whose proof applies to X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) m m m M : = Q ( ζ n ) M:=\mathbb{Q}(\zeta_n) M := Q ( ζ n ) p \mathfrak{p} p p f p^f p f Gal ( M / Q ( ζ m ) ) ⊆ ( Z / n Z ) × \operatorname{Gal}(M/\mathbb{Q}(\zeta_m))\subseteq(\mathbb{Z}/n\mathbb{Z})^\times Gal ( M / Q ( ζ m )) ⊆ ( Z / n Z ) × k = F p f k=\mathbb{F}_{p^f} k = F p f C i C_i C i X k X_k X k H i ⊆ Gal ( M / Q ( ζ m ) ) H_i\subseteq\operatorname{Gal}(M/\mathbb{Q}(\zeta_m)) H i ⊆ Gal ( M / Q ( ζ m )) [ A q ( C ) ] = 0 [A_q(C)]=0 [ A q ( C )] = 0 C C C C i C_i C i q m o d n ∈ H i q\bmod n\in H_i q mod n ∈ H i q ≡ p f j ( m o d n ) q\equiv p^{fj}\pmod n q ≡ p f j ( mod n ) j ≥ 0 j\geq0 j ≥ 0 C i C_i C i Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) H i = Gal ( M / Q ( ζ m ) ) H_i=\operatorname{Gal}(M/\mathbb{Q}(\zeta_m)) H i = Gal ( M / Q ( ζ m )) q q q
The curves over Q \mathbb{Q} Q
Proposition 6.1. The groups Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n ∈ { 33 , 34 , 35 , 38 , 39 , 40 , 42 , 50 } n\in\{33,34,35,38,39,40,42,50\} n ∈ { 33 , 34 , 35 , 38 , 39 , 40 , 42 , 50 } Z / 2 Z × Z / 22 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/22\mathbb{Z} Z /2 Z × Z /22 Z Z / 2 Z × Z / 28 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/28\mathbb{Z} Z /2 Z × Z /28 Z Z / 44 Z \mathbb{Z}/44\mathbb{Z} Z /44 Z Z / 56 Z \mathbb{Z}/56\mathbb{Z} Z /56 Z
Proof. Let X = X 1 ( m , n ) X=X_1(m,n) X = X 1 ( m , n ) p p p Table 4 . In every case rk J ( Q ) = 0 \operatorname{rk}J(\mathbb{Q})=0 rk J ( Q ) = 0 m = 1 m=1 m = 1 [7 Theorem 3.1(2)] , and for m = 2 m=2 m = 2 [7 Theorem 3.1(3)] (or [14 Theorem 4.1] ).
Table 4. The Hecke sieve computations of Propositions 6.1 , 6.3 , 7.2 and 7.3 . The fourth column lists the number of non-cuspidal closed points of X F p X_{\mathbb{F}_p} X F p 1 , … , 6 1,\ldots,6 1 , … , 6 [ 3 , 3 ] [3,3] [ 3 , 3 ] [ 4 , 2 ] [4,2] [ 4 , 2 ] X 1 ( 32 ) X_1(32) X 1 ( 32 ) X 1 ( 2 , 26 ) X_1(2,26) X 1 ( 2 , 26 ) § 7 . In the digital version of the paper, the name of each curve is a link to the script that runs the sieve on it. For X 1 ( 33 ) X_1(33) X 1 ( 33 ) X 1 ( 42 ) X_1(42) X 1 ( 42 ) X 1 ( 49 ) X_1(49) X 1 ( 49 ) X 1 ( 2 , 22 ) X_1(2,22) X 1 ( 2 , 22 )
We first exclude the totally cuspidal reduction using Proposition 5.1 with d = 6 d = 6 d = 6 gon Q X > 6 \operatorname{gon}_{\mathbb{Q}} X > 6 gon Q X > 6 p > 2 p > 2 p > 2 gon Q X > 12 \operatorname{gon}_{\mathbb{Q}} X > 12 gon Q X > 12 p = 2 p = 2 p = 2 n = 39 n = 39 n = 39 [15 Theorem 4 and Table 1] we have gon Q X 1 ( n ) = 10 , 10 , 12 , 12 , 14 , 12 \operatorname{gon}_{\mathbb{Q}} X_1(n) = 10, 10, 12, 12, 14, 12 gon Q X 1 ( n ) = 10 , 10 , 12 , 12 , 14 , 12 n = 33 , 34 , 35 , 38 , 39 , 40 n = 33, 34, 35, 38, 39, 40 n = 33 , 34 , 35 , 38 , 39 , 40 gon Q X 1 ( 42 ) > 8 \operatorname{gon}_{\mathbb{Q}} X_1(42) > 8 gon Q X 1 ( 42 ) > 8 [15 proof of Proposition 6] . Abramovich’s bound gives gon Q X 1 ( 50 ) > 8.92 \operatorname{gon}_{\mathbb{Q}} X_1(50) > 8.92 gon Q X 1 ( 50 ) > 8.92 X 1 ( 2 , 22 ) X_1(2, 22) X 1 ( 2 , 22 ) X 1 ( 2 , 28 ) X_1(2, 28) X 1 ( 2 , 28 ) gon F 3 X 1 ( 2 , 2 k ) > 6 \operatorname{gon}_{\mathbb{F}_3} X_1(2, 2k) > 6 gon F 3 X 1 ( 2 , 2 k ) > 6 [14 proof of Proposition 5.5] , so gon Q X 1 ( 2 , 2 k ) > 6 \operatorname{gon}_{\mathbb{Q}} X_1(2, 2k) > 6 gon Q X 1 ( 2 , 2 k ) > 6
It remains to exclude every decomposition ( D x ) F p = D + C (D_x)_{\mathbb{F}_p} = D + C ( D x ) F p = D + C deg D = d ′ ≥ 1 \deg D = d' \ge1 deg D = d ′ ≥ 1 X F p X_{\mathbb{F}_p} X F p Table 4 . For one representative D D D C C C Table 4 , that [ A q ( D ) ] ≠ 0 [A_q(D)] \ne0 [ A q ( D )] = 0 q q q [ A q ( C ) ] = 0 [A_q(C)] = 0 [ A q ( C )] = 0
For example, X 1 ( 39 ) F 2 X_1(39)_{\mathbb{F}_2} X 1 ( 39 ) F 2 [ 6 ] [6] [ 6 ] C = 0 C = 0 C = 0 Δ ( 39 ) \Delta(39) Δ ( 39 ) D D D [ A q ( D ) ] ≠ 0 [A_q(D)] \ne0 [ A q ( D )] = 0 q = 5 q = 5 q = 5 q = 7 q = 7 q = 7
Remark 6.2. The cuspidal complements impose the following conditions on q q q ( n , p ) = ( 34 , 3 ) , ( 38 , 3 ) (n,p) = (34,3), (38,3) ( n , p ) = ( 34 , 3 ) , ( 38 , 3 ) p p p n n n X 1 ( 35 ) X_1(35) X 1 ( 35 ) X 1 ( 39 ) X_1(39) X 1 ( 39 ) X 1 ( 50 ) X_1(50) X 1 ( 50 ) Q \mathbb{Q} Q p p p q q q X 1 ( 33 ) X_1(33) X 1 ( 33 ) Y 1 ( 33 ) ( F 5 ) = ∅ Y_1(33)(\mathbb{F}_5)=\varnothing Y 1 ( 33 ) ( F 5 ) = ∅ X 1 ( 32 ) X_1(32) X 1 ( 32 ) q = 11 ≡ 3 7 ( m o d 32 ) q=11\equiv3^7\pmod{32} q = 11 ≡ 3 7 ( mod 32 ) X 1 ( 40 ) X_1(40) X 1 ( 40 ) q m o d 40 ∈ { 1 , 3 , 9 , 11 , 17 , 19 , 27 , 33 } q\bmod40\in\{1,3,9,11,17,19,27,33\} q mod 40 ∈ { 1 , 3 , 9 , 11 , 17 , 19 , 27 , 33 } q = 11 q=11 q = 11 X 1 ( 42 ) X_1(42) X 1 ( 42 ) q = 17 ≡ 5 5 ( m o d 42 ) q=17\equiv5^5\pmod{42} q = 17 ≡ 5 5 ( mod 42 ) X 1 ( 2 , 22 ) X_1(2,22) X 1 ( 2 , 22 ) q = 5 ≡ 3 8 ( m o d 44 ) q=5\equiv3^8\pmod{44} q = 5 ≡ 3 8 ( mod 44 ) X 1 ( 2 , 26 ) X_1(2,26) X 1 ( 2 , 26 ) X 1 ( 2 , 28 ) X_1(2,28) X 1 ( 2 , 28 ) F 3 \mathbb{F}_3 F 3 X 1 ( n ) X_1(n) X 1 ( n ) [7 Lemma 2.8] . For X 1 ( 2 , 26 ) X_1(2,26) X 1 ( 2 , 26 ) X 1 ( 2 , 28 ) X_1(2,28) X 1 ( 2 , 28 ) X 1 ( 2 , 22 ) X_1(2,22) X 1 ( 2 , 22 ) 5 ≡ 3 8 ( m o d 44 ) 5\equiv3^8\pmod{44} 5 ≡ 3 8 ( mod 44 )
Proposition 6.3. The group Z / 49 Z \mathbb{Z}/49\mathbb{Z} Z /49 Z
Proof. We have rk J 1 ( 49 ) ( Q ) = 0 \operatorname{rk} J_1(49)(\mathbb{Q})=0 rk J 1 ( 49 ) ( Q ) = 0 [7 Theorem 3.1(2)] , and Abramovich’s bound gives gon Q X 1 ( 49 ) > 11.66 > 6 \operatorname{gon}_{\mathbb{Q}} X_1(49)>11.66>6 gon Q X 1 ( 49 ) > 11.66 > 6 p = 3 p=3 p = 3 Q \mathbb{Q} Q 1 21 1^{21} 1 21 6 3 6^3 6 3 21 21 21 Table 4 the non-cuspidal part D D D [ A 5 ( D ) ] ≠ 0 [A_5(D)]\ne0 [ A 5 ( D )] = 0 [ A 5 ( D ) ] ≠ 0 [A_5(D)]\ne0 [ A 5 ( D )] = 0
Remark 6.4. As an independent check of the case of degree 6 in the proof of Proposition 6.3 , we also used the operator A 2 = T 2 − 2 ⟨ 2 ⟩ − 1 A_2=T_2-2\langle2\rangle-1 A 2 = T 2 − 2 ⟨ 2 ⟩ − 1 q = 2 q=2 q = 2 § 11 .
The curves over Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m )
For Z / 3 Z × Z / 24 Z \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} Z /3 Z × Z /24 Z Z / 6 Z × Z / 12 Z \mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z} Z /6 Z × Z /12 Z Z / 4 Z × Z / 16 Z \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/16\mathbb{Z} Z /4 Z × Z /16 Z Z / 4 Z × Z / 20 Z \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/20\mathbb{Z} Z /4 Z × Z /20 Z Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) d = d m = 3 d=d_m=3 d = d m = 3 p \mathfrak{p} p Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) p = 7 , 3 , 3 , 3 p=7,3,3,3 p = 7 , 3 , 3 , 3 m = 6 m=6 m = 6 ⟨ 13 ⟩ = 1 \langle13\rangle=1 ⟨ 13 ⟩ = 1 X 1 ( 6 , 12 ) X_1(6,12) X 1 ( 6 , 12 ) A 13 = T 13 − 14 A_{13}=T_{13}-14 A 13 = T 13 − 14
Lemma 6.5. We have gon C X 1 ( 6 , 12 ) > 3 \operatorname{gon}_{\mathbb{C}} X_1(6,12)>3 gon C X 1 ( 6 , 12 ) > 3
Proof. The forgetful map X 1 ( 6 , 12 ) → X 1 ( 3 , 12 ) X_1(6,12)\to X_1(3,12) X 1 ( 6 , 12 ) → X 1 ( 3 , 12 ) ( E , P , Q ) ↦ ( E , 2 P , Q ) (E,P,Q)\mapsto(E,2P,Q) ( E , P , Q ) ↦ ( E , 2 P , Q ) X 1 ( 6 , 12 ) → P 1 X_1(6,12)\to\mathbb{P}^1 X 1 ( 6 , 12 ) → P 1 d ≤ 3 d\le3 d ≤ 3 X 1 ( 3 , 12 ) X_1(3,12) X 1 ( 3 , 12 ) [42 Theorem 3.11.3] gives 9 ≤ 2 ⋅ 3 + ( d − 1 ) ≤ 8 9\le2\cdot3+(d-1)\le8 9 ≤ 2 ⋅ 3 + ( d − 1 ) ≤ 8
Proposition 6.6. The groups Z / 3 Z × Z / 24 Z \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} Z /3 Z × Z /24 Z Z / 6 Z × Z / 12 Z \mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z} Z /6 Z × Z /12 Z Z / 4 Z × Z / 16 Z \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/16\mathbb{Z} Z /4 Z × Z /16 Z Z / 4 Z × Z / 20 Z \mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/20\mathbb{Z} Z /4 Z × Z /20 Z
Proof. Let E E E K K K X = X 1 ( m , n ) X=X_1(m,n) X = X 1 ( m , n ) rk J ( Q ( ζ m ) ) = 0 \operatorname{rk}J(\mathbb{Q}(\zeta_m))=0 rk J ( Q ( ζ m )) = 0 [14 Theorem 4.1] . The indices [ PSL 2 ( Z ) : Γ 1 ( n ) ∩ Γ ( m ) ‾ ] = 576 [\operatorname{PSL}_2(\mathbb{Z}):\overline{\Gamma_1(n)\cap\Gamma(m)}]=576 [ PSL 2 ( Z ) : Γ 1 ( n ) ∩ Γ ( m ) ] = 576 X 1 ( 3 , 24 ) X_1(3,24) X 1 ( 3 , 24 ) X 1 ( 4 , 16 ) X_1(4,16) X 1 ( 4 , 16 ) X 1 ( 4 , 20 ) X_1(4,20) X 1 ( 4 , 20 ) gon C X > 5.71 , 3.80 , 5.71 \operatorname{gon}_{\mathbb{C}} X > 5.71, 3.80, 5.71 gon C X > 5.71 , 3.80 , 5.71 gon C X 1 ( 6 , 12 ) > 3 \operatorname{gon}_{\mathbb{C}} X_1(6,12) > 3 gon C X 1 ( 6 , 12 ) > 3 Lemma 6.5 . Hence Proposition 5.1(a) with d = 3 d = 3 d = 3 E E E K K K p \mathfrak{p} p D D D
The residue field k k k p \mathfrak{p} p F 7 \mathbb{F}_7 F 7 m = 3 , 6 m = 3, 6 m = 3 , 6 F 9 \mathbb{F}_9 F 9 m = 4 m = 4 m = 4 K K K p \mathfrak{p} p k k k Z / 3 Z × Z / 24 Z \mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/24\mathbb{Z} Z /3 Z × Z /24 Z Z / 6 Z × Z / 12 Z \mathbb{Z}/6\mathbb{Z} \times\mathbb{Z}/12\mathbb{Z} Z /6 Z × Z /12 Z F 7 \mathbb{F}_7 F 7 F 49 \mathbb{F}_{49} F 49 Z / 4 Z × Z / 16 Z \mathbb{Z}/4\mathbb{Z} \times\mathbb{Z}/16\mathbb{Z} Z /4 Z × Z /16 Z F 9 \mathbb{F}_9 F 9 F 81 \mathbb{F}_{81} F 81 E ( F 81 ) = E [ 8 ] E(\mathbb{F}_{81}) = E[8] E ( F 81 ) = E [ 8 ] D D D Z / 4 Z × Z / 20 Z \mathbb{Z}/4\mathbb{Z} \times\mathbb{Z}/20\mathbb{Z} Z /4 Z × Z /20 Z
The numbers of the non-cuspidal closed points of these degrees and of their diamond orbits are listed in Table 5 , together with the primes q q q q ≡ 1 ( m o d m ) q \equiv1 \pmod{m} q ≡ 1 ( mod m ) Q ( ζ m ) \mathbb{Q}(\zeta_m) Q ( ζ m ) c 0 c_0 c 0 A q ( c 0 ) = 0 A_q(c_0) = 0 A q ( c 0 ) = 0 Z / 4 Z × Z / 20 Z \mathbb{Z}/4\mathbb{Z} \times\mathbb{Z}/20\mathbb{Z} Z /4 Z × Z /20 Z D D D q = 29 q = 29 q = 29 29 ≡ 9 = 3 2 ( m o d 20 ) 29 \equiv9 = 3^2 \pmod{20} 29 ≡ 9 = 3 2 ( mod 20 ) C C C [ A 29 ( C ) ] = 0 [A_{29}(C)] = 0 [ A 29 ( C )] = 0 § 6 . For one representative D D D Table 5 , that [ A q ( D ) ] ≠ 0 [A_q(D)] \ne0 [ A q ( D )] = 0
Table 5. The Hecke sieve computations of Proposition 6.6 . For X 1 ( 4 , 20 ) X_1(4,20) X 1 ( 4 , 20 ) q = 29 q = 29 q = 29 q = 13 q = 13 q = 13 Table 4 , the name of each curve is a link to the script that runs the sieve on it.
Remark 6.7. In Proposition 6.6 we check the completeness of the enumeration of the closed points by an independent count of the elliptic curves over the extensions of k k k independent_m3_counts.m, independent_m6_counts.m, independent_m4_p3_counts.m and independent_m4_20_p3_counts.m). For instance, # Y 1 ( 4 , 16 ) ( F 9 d ) = 0 , 0 , 960 \#Y_1(4,16)(\mathbb{F}_{9^d}) = 0, 0, 960 # Y 1 ( 4 , 16 ) ( F 9 d ) = 0 , 0 , 960 d = 1 , 2 , 3 d = 1, 2, 3 d = 1 , 2 , 3 Table 4 (see § 11 ).
Arguments with cuspidal divisor classes
The Hecke sieve uses only the fact that A q A_q A q J ( Q ( ζ m ) ) J(\mathbb{Q}(\zeta_m)) J ( Q ( ζ m )) J ( Q ( ζ m ) ) J(\mathbb{Q}(\zeta_m)) J ( Q ( ζ m )) J ( Q ( ζ m ) ) J(\mathbb{Q}(\zeta_m)) J ( Q ( ζ m )) [7 §5.2] and [36 §8] , which we use in the following form.
Lemma 7.1. Let X X X F F F c ∗ c_* c ∗ p \mathfrak{p} p X X X k k k G ⊆ J ( k ) G \subseteq J(k) G ⊆ J ( k )
the reduction of J ( F ) J(F) J ( F ) D ∈ X ( d ) ( F ) D \in X^{(d)}(F) D ∈ X ( d ) ( F ) p p p Z + C Z+C Z + C Z Z Z e e e C C C
[ Z + C − d c ∗ ] ∈ G . [Z+C-dc_*]\in G. [ Z + C − d c ∗ ] ∈ G . Proof. This is the reduction of the class [ D − d c ∗ ] ∈ J ( F ) [D-dc_*]\in J(F) [ D − d c ∗ ] ∈ J ( F )
So an effective divisor Z Z Z e e e D D D [ Z + C − d c ∗ ] ∈ G [Z+C-dc_*]\in G [ Z + C − d c ∗ ] ∈ G C C C d − e d-e d − e X k X_k X k Z Z Z j ≥ 0 j\geq0 j ≥ 0 S j ⊆ J ( k ) / G S_j\subseteq J(k)/G S j ⊆ J ( k ) / G [ j c ∗ − C ] [jc_*-C] [ j c ∗ − C ] C C C j j j X k X_k X k [ Z − e c ∗ ] [Z-ec_*] [ Z − e c ∗ ] J ( k ) / G J(k)/G J ( k ) / G S d − e S_{d-e} S d − e
In practice G G G J ( F ) J(F) J ( F ) J ( F ) J(F) J ( F ) [ c ˉ − deg ( c ˉ ) c ∗ ] [\bar{c}-\deg(\bar{c})c_*] [ c ˉ − deg ( c ˉ ) c ∗ ] c ˉ \bar{c} c ˉ X k X_k X k J ( F ) J(F) J ( F ) G G G J ( F ) J(F) J ( F ) 0 0 0 X k X_k X k S j S_j S j [ Z − e c ∗ ] ∈ G [Z-ec_*]\in G [ Z − e c ∗ ] ∈ G
Usually we test only one representative Z Z Z Z Z Z
[ δ Z − e c ∗ ] = δ [ Z − e c ∗ ] + e [ δ c ∗ − c ∗ ] ( δ ∈ Δ ( n ) ) . [\delta Z-ec_*]=\delta[Z-ec_*]+e[\delta c_*-c_*]\qquad(\delta\in\Delta(n)). [ δ Z − e c ∗ ] = δ [ Z − e c ∗ ] + e [ δ c ∗ − c ∗ ] ( δ ∈ Δ ( n )) . In both cases above G G G [ δ c ∗ − c ∗ ] [\delta c_*-c_*] [ δ c ∗ − c ∗ ] δ c ∗ \delta c_* δ c ∗ F F F δ \delta δ Z Z Z δ Z \delta Z δ Z
The curves X 1 ( 32 ) X_1(32) X 1 ( 32 ) X 1 ( 2 , 26 ) X_1(2,26) X 1 ( 2 , 26 ) X 1 ( 36 ) X_1(36) X 1 ( 36 )
Proposition 7.2. The group Z / 32 Z \mathbb{Z}/32\mathbb{Z} Z /32 Z
Proof. We have rk J 1 ( 32 ) ( Q ) = 0 \operatorname{rk} J_1(32)(\mathbb{Q})=0 rk J 1 ( 32 ) ( Q ) = 0 J 1 ( 32 ) ( Q ) = Cl Q c X 1 ( 32 ) ≃ ( Z / 2 Z ) 3 × Z / 4 Z × Z / 120 Z × Z / 11640 Z J_1(32)(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^c X_1(32)\simeq(\mathbb{Z}/2\mathbb{Z})^3\times\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/120\mathbb{Z}\times\mathbb{Z}/11640\mathbb{Z} J 1 ( 32 ) ( Q ) = Cl Q c X 1 ( 32 ) ≃ ( Z /2 Z ) 3 × Z /4 Z × Z /120 Z × Z /11640 Z [7 Theorem 3.1(2), Corollary 4.14 and Table 2] , and gon Q X 1 ( 32 ) = 8 \operatorname{gon}_{\mathbb{Q}}X_1(32)=8 gon Q X 1 ( 32 ) = 8 [15 Theorem 4 and Table 1] . As in the proof of Proposition 6.1 , Proposition 5.1 with p = 3 p=3 p = 3 p = 3 p=3 p = 3 Table 4 and Remark 6.2) excludes every decomposition except four, three of type [ 6 ] [6] [ 6 ] [ 3 , 3 ] [3,3] [ 3 , 3 ] G G G J 1 ( 32 ) ( F 3 ) J_1(32)(\mathbb{F}_3) J 1 ( 32 ) ( F 3 ) 0 0 0 G ≃ ( Z / 2 Z ) 3 × Z / 4 Z × Z / 120 Z × Z / 11640 Z G\simeq(\mathbb{Z}/2\mathbb{Z})^3\times\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/120\mathbb{Z}\times\mathbb{Z}/11640\mathbb{Z} G ≃ ( Z /2 Z ) 3 × Z /4 Z × Z /120 Z × Z /11640 Z G G G J 1 ( 32 ) ( Q ) J_1(32)(\mathbb{Q}) J 1 ( 32 ) ( Q ) [ Z − 6 c ∗ ] [Z-6c_*] [ Z − 6 c ∗ ] G G G Lemma 7.1 none of them is the reduction of a trace divisor.
Proposition 7.3. The group Z / 2 Z × Z / 26 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/26\mathbb{Z} Z /2 Z × Z /26 Z Lemma 3.2 , neither is Z / 52 Z \mathbb{Z}/52\mathbb{Z} Z /52 Z
Proof. Let X = X 1 ( 2 , 26 ) X=X_1(2,26) X = X 1 ( 2 , 26 ) 25 25 25 0 0 0 Q \mathbb{Q} Q J ( Q ) = Cl Q c X ≃ Z / 2 Z × Z / 14 Z × Z / 266 Z × Z / 3990 Z × Z / 11970 Z J(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^c X\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/14\mathbb{Z}\times\mathbb{Z}/266\mathbb{Z}\times\mathbb{Z}/3990\mathbb{Z}\times\mathbb{Z}/11970\mathbb{Z} J ( Q ) = Cl Q c X ≃ Z /2 Z × Z /14 Z × Z /266 Z × Z /3990 Z × Z /11970 Z [7 Theorem 3.1(3), Proposition 4.16 and Table 3] , and gon Q X > 6 \operatorname{gon}_{\mathbb{Q}}X>6 gon Q X > 6 [14 proof of Proposition 5.5] . As in the proof of Proposition 6.1 , the totally cuspidal reduction is excluded by Proposition 5.1 with p = 3 p=3 p = 3 p = 3 p=3 p = 3 [ 6 ] [6] [ 6 ] G G G J ( F 3 ) J(\mathbb{F}_3) J ( F 3 ) [ 2 , 14 , 266 , 3990 , 11970 ] [2,14,266,3990,11970] [ 2 , 14 , 266 , 3990 , 11970 ] J ( Q ) J(\mathbb{Q}) J ( Q ) [ Z − 6 c ˉ ∗ ] [Z-6\bar{c}_{*}] [ Z − 6 c ˉ ∗ ] G G G G G G Lemma 7.1 completes the proof. □
Proposition 7.4. The group Z / 36 Z \mathbb{Z}/36\mathbb{Z} Z /36 Z
Proof. We have J 1 ( 36 ) ( Q ) = Cl Q c X 1 ( 36 ) ≃ Z / 12 Z × Z / 252 Z × Z / 7812 Z J_{1}(36)(\mathbb{Q})=\operatorname{Cl}^{c}_{\mathbb{Q}}X_{1}(36)\simeq\mathbb{Z}/12\mathbb{Z}\times\mathbb{Z}/252\mathbb{Z}\times\mathbb{Z}/7812\mathbb{Z} J 1 ( 36 ) ( Q ) = Cl Q c X 1 ( 36 ) ≃ Z /12 Z × Z /252 Z × Z /7812 Z [7 Theorem 3.1(2), Corollary 4.14 and Table 2] and gon Q X 1 ( 36 ) = 8 \operatorname{gon}_{\mathbb{Q}}X_{1}(36)=8 gon Q X 1 ( 36 ) = 8 [15 Theorem 4 and Table 1] , so Proposition 5.1 with p = 5 p=5 p = 5 G ⊆ J 1 ( 36 ) ( F 5 ) G\subseteq J_{1}(36)(\mathbb{F}_{5}) G ⊆ J 1 ( 36 ) ( F 5 ) [ c ˉ − deg ( c ˉ ) c ˉ ∗ ] [\bar{c}-\deg(\bar{c})\bar{c}_{*}] [ c ˉ − deg ( c ˉ ) c ˉ ∗ ] c ˉ \bar{c} c ˉ X 1 ( 36 ) F 5 X_{1}(36)_{\mathbb{F}_{5}} X 1 ( 36 ) F 5 J 1 ( 36 ) ( Q ) J_{1}(36)(\mathbb{Q}) J 1 ( 36 ) ( Q )
J 1 ( 36 ) ( F 5 ) ≃ ( Z / 4 Z ) 2 × ( Z / 12 Z ) 2 × Z / 36 Z × ( Z / 7812 Z ) 2 , J_{1}(36)(\mathbb{F}_{5})\simeq(\mathbb{Z}/4\mathbb{Z})^{2}\times(\mathbb{Z}/12\mathbb{Z})^{2}\times\mathbb{Z}/36\mathbb{Z}\times(\mathbb{Z}/7812\mathbb{Z})^{2}, J 1 ( 36 ) ( F 5 ) ≃ ( Z /4 Z ) 2 × ( Z /12 Z ) 2 × Z /36 Z × ( Z /7812 Z ) 2 , G ≃ Z / 3 Z × Z / 36 Z × ( Z / 7812 Z ) 2 , G\simeq\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/36\mathbb{Z}\times(\mathbb{Z}/7812\mathbb{Z})^{2}, G ≃ Z /3 Z × Z /36 Z × ( Z /7812 Z ) 2 , so G G G 768 768 768 X 1 ( 36 ) F 5 X_{1}(36)_{\mathbb{F}_{5}} X 1 ( 36 ) F 5 1 , … , 6 1,\ldots,6 1 , … , 6 0 , 0 , 48 , 144 , 660 , 2856 0,0,48,144,660,2856 0 , 0 , 48 , 144 , 660 , 2856 Z Z Z 3 , 4 , 5 3,4,5 3 , 4 , 5 6 6 6 3 3 3 [ Z − ( deg Z ) c ˉ ∗ ] ∉ G [Z-(\deg Z)\bar{c}_{*}]\notin G [ Z − ( deg Z ) c ˉ ∗ ] ∈ / G 48 + 144 + 660 + 2856 + 1176 48+144+660+2856+1176 48 + 144 + 660 + 2856 + 1176 Z Z Z G G G 0 0 0 Lemma 7.1 shows that none of them occurs. □
The sextic points on X 1 ( 2 , 24 ) X_{1}(2,24) X 1 ( 2 , 24 )
On X 1 ( 2 , 24 ) X_{1}(2,24) X 1 ( 2 , 24 ) Proposition 10.3 we show, with an explicit elliptic curve, that this bound is attained.
Proposition 7.5. Let X = X 1 ( 2 , 24 ) X=X_{1}(2,24) X = X 1 ( 2 , 24 ) Q \mathbb{Q} Q L 1 , L 2 L_{1},L_{2} L 1 , L 2 6 6 6 X X X D D D Q \mathbb{Q} Q 6 6 6 X X X δ D ∼ L i \delta D\sim L_{i} δ D ∼ L i δ ∈ Δ ( 24 ) \delta\in\Delta(24) δ ∈ Δ ( 24 ) i ∈ { 1 , 2 } i\in\{1,2\} i ∈ { 1 , 2 } ∣ L 1 ∣ \lvert L_{1}\rvert ∣ L 1 ∣ ∣ L 2 ∣ \lvert L_{2}\rvert ∣ L 2 ∣
Proof. The curve X X X 13 13 13 0 0 0 J ( Q ) = Cl Q c X ≃ Z / 2 Z × Z / 4 Z × Z / 4 Z × Z / 120 Z × Z / 240 Z J(\mathbb{Q})=\operatorname{Cl}^{c}_{\mathbb{Q}}X\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/120\mathbb{Z}\times\mathbb{Z}/240\mathbb{Z} J ( Q ) = Cl Q c X ≃ Z /2 Z × Z /4 Z × Z /4 Z × Z /120 Z × Z /240 Z [7 Theorem 3.1(3), Proposition 4.16 and Table 3] , and gon F 5 X > 6 \operatorname{gon}_{\mathbb{F}_{5}}X>6 gon F 5 X > 6 [14 proof of Proposition 5.5] , so gon Q X > 6 \operatorname{gon}_{\mathbb{Q}}X>6 gon Q X > 6 p = 5 p=5 p = 5
Let D D D Q \mathbb{Q} Q 6 6 6 X X X D D D 3 3 3 Z / 2 Z × Z / 24 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} Z /2 Z × Z /24 Z D D D 6 6 6 Y 1 ( 2 , 24 ) ( K ) Y_{1}(2,24)(K) Y 1 ( 2 , 24 ) ( K ) K K K D D D
We use the normalized universal curve y 2 = x 3 + c x 2 + d x y^{2}=x^{3}+cx^{2}+dx y 2 = x 3 + c x 2 + d x P = ( 0 , 0 ) P=(0,0) P = ( 0 , 0 ) 2 2 2 Q = ( b , b ) Q=(b,b) Q = ( b , b ) 24 24 24 d = b ( 1 − b − c ) d=b(1-b-c) d = b ( 1 − b − c ) b b b d d d c 2 − 4 d c^{2}-4d c 2 − 4 d x ( j Q ) x(jQ) x ( j Q ) 1 ≤ j ≤ 12 1\leq j\leq12 1 ≤ j ≤ 12 Y 1 ( 2 , 24 ) Z [ 1 / 6 ] Y_{1}(2,24)_{\mathbb{Z}[1/6]} Y 1 ( 2 , 24 ) Z [ 1/6 ] j Q jQ j Q j ≤ 12 j\leq12 j ≤ 12 O O O P P P 5 5 5 19 19 19 B 1 , … , B 19 B_{1},\ldots,B_{19} B 1 , … , B 19 Q \mathbb{Q} Q ord b \operatorname{ord} b ord b ord d \operatorname{ord} d ord d ord ( c 2 − 4 d ) \operatorname{ord}(c^{2}-4d) ord ( c 2 − 4 d ) ord x ( Q ) , … , ord x ( 12 Q ) \operatorname{ord} x(Q),\ldots,\operatorname{ord} x(12Q) ord x ( Q ) , … , ord x ( 12 Q ) B 1 B_{1} B 1 b b b − 22 -22 − 22 c ∗ : = B 1 c_{*}:=B_{1} c ∗ := B 1 Lemma 7.1 , and we put
L 1 = 15 B 1 − B 7 − B 8 + B 11 − B 14 + B 17 , L 2 = 3 B 1 + B 3 + B 4 + B 13 − B 16 + B 19 . \begin{aligned}
L_{1} &= 15B_{1}-B_{7}-B_{8}+B_{11}-B_{14}+B_{17},\\
L_{2} &= 3B_{1}+B_{3}+B_{4}+B_{13}-B_{16}+B_{19}.
\end{aligned} L 1 L 2 = 15 B 1 − B 7 − B 8 + B 11 − B 14 + B 17 , = 3 B 1 + B 3 + B 4 + B 13 − B 16 + B 19 . Let G ⊆ J ( F 5 ) G\subseteq J(\mathbb{F}_{5}) G ⊆ J ( F 5 ) [ B ‾ j − deg ( B j ) c ‾ ∗ ] [\overline{B}_{j}-\deg(B_{j})\overline{c}_{*}] [ B j − deg ( B j ) c ∗ ] G G G [ 2 , 4 , 4 , 120 , 240 ] [2,4,4,120,240] [ 2 , 4 , 4 , 120 , 240 ] G G G J ( Q ) J(\mathbb{Q}) J ( Q ) J ( F 5 ) / G ≅ ( Z / 2 Z ) 2 × Z / 24 Z × Z / 48 Z J(\mathbb{F}_{5})/G\cong(\mathbb{Z}/2\mathbb{Z})^{2}\times\mathbb{Z}/24\mathbb{Z}\times\mathbb{Z}/48\mathbb{Z} J ( F 5 ) / G ≅ ( Z /2 Z ) 2 × Z /24 Z × Z /48 Z S j S_{j} S j Lemma 7.1 have 1 , 5 , 23 , 75 1,5,23,75 1 , 5 , 23 , 75 j = 0 , 1 , 2 , 3 j=0,1,2,3 j = 0 , 1 , 2 , 3
Over F 5 \mathbb{F}_{5} F 5 1 , … , 6 1,\ldots,6 1 , … , 6 0 , 0 , 16 , 96 , 672 , 2712 0,0,16,96,672,2712 0 , 0 , 16 , 96 , 672 , 2712 Z Z Z D D D [ 3 ] [3] [ 3 ] [ 4 ] [4] [ 4 ] [ 5 ] [5] [ 5 ] [ 6 ] [6] [ 6 ] [ 3 , 3 ] [3,3] [ 3 , 3 ] 4 4 4 32 32 32 168 168 168 684 684 684 64 64 64 Z Z Z [ Z − ( deg Z ) c ‾ ∗ ] m o d G ∈ S 6 − deg Z [Z-(\deg Z)\overline{c}_{*}]\bmod G\in S_{6-\deg Z} [ Z − ( deg Z ) c ∗ ] mod G ∈ S 6 − d e g Z [ 6 ] [6] [ 6 ] L 1 L_{1} L 1 L 2 L_{2} L 2
So, after applying a diamond operator, D D D [ D − L i ] [D-L_{i}] [ D − L i ] D ∼ L i D\sim L_{i} D ∼ L i gon Q X > 6 \operatorname{gon}_{\mathbb{Q}}X>6 gon Q X > 6 ∣ L 1 ∣ |L_{1}| ∣ L 1 ∣ ∣ L 2 ∣ |L_{2}| ∣ L 2 ∣
The group Z/3Z × Z/18Z
For Z / 3 Z × Z / 18 Z \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z} Z /3 Z × Z /18 Z J 1 ( 3 , 18 ) ( Q ( ζ 3 ) ) J_{1}(3,18)(\mathbb{Q}(\zeta_{3})) J 1 ( 3 , 18 ) ( Q ( ζ 3 ))
Lemma 7.6. Let X X X F F F J ( F ) J(F) J ( F ) c ∗ c_{*} c ∗ p 1 , p 2 \mathfrak{p}_{1},\mathfrak{p}_{2} p 1 , p 2 k 1 , k 2 k_{1},k_{2} k 1 , k 2 p 1 \mathfrak{p}_{1} p 1 J ( F ) J(F) J ( F ) H i ⊆ J ( k i ) H_{i}\subseteq J(k_{i}) H i ⊆ J ( k i ) Cl F c ∗ X \operatorname{Cl}^{c_{*}}_{F}X Cl F c ∗ X p i \mathfrak{p}_{i} p i ψ : H 1 → H 2 \psi:H_{1}\to H_{2} ψ : H 1 → H 2 Cl F c ∗ X → H 1 \operatorname{Cl}^{c_{*}}_{F}X\to H_{1} Cl F c ∗ X → H 1 Cl F c ∗ X → H 2 \operatorname{Cl}^{c_{*}}_{F}X\to H_{2} Cl F c ∗ X → H 2 J ( k 1 ) / H 1 J(k_{1})/H_{1} J ( k 1 ) / H 1 e e e D ∈ X ( d ) ( F ) D\in X^{(d)}(F) D ∈ X ( d ) ( F ) D ‾ i \overline{D}_{i} D i c ‾ ∗ , i \overline{c}_{*,i} c ∗ , i D D D c ∗ c_{*} c ∗ p i \mathfrak{p}_{i} p i
ψ ( e [ D ‾ 1 − d c ‾ ∗ , 1 ] ) = e [ D ‾ 2 − d c ‾ ∗ , 2 ] . \psi\left(e[\overline{D}_{1}-d\overline{c}_{*,1}]\right)=e[\overline{D}_{2}-d\overline{c}_{*,2}]. ψ ( e [ D 1 − d c ∗ , 1 ] ) = e [ D 2 − d c ∗ , 2 ] . Proof. Let y = [ D − d c ∗ ] ∈ J ( F ) y=[D-dc_{*}]\in J(F) y = [ D − d c ∗ ] ∈ J ( F ) e y ey ey p 1 \mathfrak{p}_{1} p 1 H 1 H_{1} H 1 c ∈ Cl F c ∗ X c\in\operatorname{Cl}^{c_{*}}_{F}X c ∈ Cl F c ∗ X e y − c ey-c ey − c p 1 \mathfrak{p}_{1} p 1 e y = c ey=c ey = c p 2 \mathfrak{p}_{2} p 2
Proposition 7.7. The group Z / 3 Z × Z / 18 Z \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z} Z /3 Z × Z /18 Z
Proof. Let X = X 1 ( 3 , 18 ) X=X_{1}(3,18) X = X 1 ( 3 , 18 ) Q ( ζ 3 ) \mathbb{Q}(\zeta_{3}) Q ( ζ 3 ) 10 10 10 rk J ( Q ( ζ 3 ) ) = 0 \operatorname{rk}J(\mathbb{Q}(\zeta_{3}))=0 rk J ( Q ( ζ 3 )) = 0 [14 Theorem 4.1] , and the index [ PSL 2 ( Z ) : Γ 1 ( 18 ) ∩ Γ ( 3 ) ] = 324 [\operatorname{PSL}_{2}(\mathbb{Z}):\Gamma_{1}(18)\cap\Gamma(3)]=324 [ PSL 2 ( Z ) : Γ 1 ( 18 ) ∩ Γ ( 3 )] = 324 gon Q X > 3.21 > 3 \operatorname{gon}_{\mathbb{Q}}X>3.21>3 gon Q X > 3.21 > 3 p 1 \mathfrak{p}_{1} p 1 p 2 \mathfrak{p}_{2} p 2 Q ( ζ 3 ) \mathbb{Q}(\zeta_{3}) Q ( ζ 3 ) 7 7 7 13 13 13 F 7 \mathbb{F}_{7} F 7 F 13 \mathbb{F}_{13} F 13 p 1 \mathfrak{p}_{1} p 1 p 2 \mathfrak{p}_{2} p 2 D D D p i \mathfrak{p}_{i} p i 1 , 2 , 3 1,2,3 1 , 2 , 3 0 , 27 , 144 0,27,144 0 , 27 , 144 F 7 \mathbb{F}_{7} F 7 0 , 54 , 846 0,54,846 0 , 54 , 846 F 13 \mathbb{F}_{13} F 13 3 3 3 2 2 2 1 1 1 Q ( ζ 3 ) \mathbb{Q}(\zeta_{3}) Q ( ζ 3 ) 3 3 3 p 1 \mathfrak{p}_{1} p 1 A 13 A_{13} A 13 9 9 9 2 2 2 36 36 36 48 48 48 3 3 3 p 2 \mathfrak{p}_{2} p 2 A 7 A_{7} A 7 18 18 18
To compare the two primes we need to identify the reductions of the cuspidal divisor classes. We use the model of X X X [33 ] , on which the universal curve is y 2 + a 1 x y + a 3 y = x 3 y^2+a_1xy+a_3y=x^3 y 2 + a 1 x y + a 3 y = x 3 P = ( 0 , 0 ) P=(0,0) P = ( 0 , 0 ) y ( Q ) = x ( Q ) 2 y(Q)=x(Q)^2 y ( Q ) = x ( Q ) 2 Q Q Q x = 0 x=0 x = 0 ± P \pm P ± P y = 0 y=0 y = 0 P P P x ( Q ) x(Q) x ( Q ) y ( Q ) y(Q) y ( Q ) Y 1 ( 3 , 18 ) Y_1(3,18) Y 1 ( 3 , 18 ) a 1 a_1 a 1 a 3 a_3 a 3 Y 1 ( 3 , 18 ) Y_1(3,18) Y 1 ( 3 , 18 ) Z [ ζ 3 , 1 6 ] \mathbb{Z}[\zeta_3,\frac{1}{6}] Z [ ζ 3 , 6 1 ]
a 3 , a 1 3 − 27 a 3 , x ( k Q + j P ) , y ( k Q + j P ) ( 1 ≤ k ≤ 9 , 0 ≤ j ≤ 2 ) , a_3,\quad a_1^3-27a_3,\quad x(kQ+jP),\quad y(kQ+jP)\quad(1\leq k\leq9,\ 0\leq j\leq2), a 3 , a 1 3 − 27 a 3 , x ( k Q + j P ) , y ( k Q + j P ) ( 1 ≤ k ≤ 9 , 0 ≤ j ≤ 2 ) , x ( k Q ) − x ( k Q + P ) ( 1 ≤ k ≤ 9 ) , x ( k Q + P ) − x ( k Q + 2 P ) ( 1 ≤ k ≤ 8 ) x(kQ)-x(kQ+P)\quad(1\leq k\leq9),\qquad x(kQ+P)-x(kQ+2P)\quad(1\leq k\leq8) x ( k Q ) − x ( k Q + P ) ( 1 ≤ k ≤ 9 ) , x ( k Q + P ) − x ( k Q + 2 P ) ( 1 ≤ k ≤ 8 ) are units on Y 1 ( 3 , 18 ) Y_1(3,18) Y 1 ( 3 , 18 ) Z [ ζ 3 , 1 6 ] \mathbb{Z}[\zeta_3,\frac{1}{6}] Z [ ζ 3 , 6 1 ] a 3 3 ( a 1 3 − 27 a 3 ) a_3^3(a_1^3-27a_3) a 3 3 ( a 1 3 − 27 a 3 ) k Q + j P kQ+jP k Q + j P O O O ± P \pm P ± P k Q + P ≠ ± k Q kQ+P\neq\pm kQ k Q + P = ± k Q k Q + 2 P ≠ ± ( k Q + P ) kQ+2P\neq\pm(kQ+P) k Q + 2 P = ± ( k Q + P ) k k k 1 18 1^ {18} 1 18 3 6 3^6 3 6 Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) H 1 , H 2 H_1,H_2 H 1 , H 2 Cl Q ( ζ 3 ) c X \operatorname{Cl}_{\mathbb{Q}(\zeta_3)}^c X Cl Q ( ζ 3 ) c X [ 3 , 3 , 3 , 3 , 63 , 63 , 63 ] [3,3,3,3,63,63,63] [ 3 , 3 , 3 , 3 , 63 , 63 , 63 ] J ( F 7 ) / H 1 ≃ Z / 3 Z × Z / 3 Z × Z / 9 Z J(\mathbb{F}_7)/H_1\simeq\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/9\mathbb{Z} J ( F 7 ) / H 1 ≃ Z /3 Z × Z /3 Z × Z /9 Z Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) X X X p 1 \mathfrak p_1 p 1 J ( Q ( ζ 3 ) ) = J ( Q ( ζ 3 ) ) t o r s J(\mathbb{Q}(\zeta_3))=J(\mathbb{Q}(\zeta_3))_{\mathrm{tors}} J ( Q ( ζ 3 )) = J ( Q ( ζ 3 ) ) tors ψ \psi ψ Lemma 7.6 sends the reduction modulo p 1 \mathfrak p_1 p 1 p 2 \mathfrak p_2 p 2 Lemma 7.6 with e = 9 e=9 e = 9 ( D ‾ 1 , D ‾ 2 ) (\overline{D}_1,\overline{D}_2) ( D 1 , D 2 ) ψ ( 9 [ D ‾ 1 − 3 c ‾ ∗ , 1 ] ) = 9 [ D ‾ 2 − 3 c ‾ ∗ , 2 ] \psi(9[\overline{D}_1-3\overline{c}_{*,1}])=9[\overline{D}_2-3\overline{c}_{*,2}] ψ ( 9 [ D 1 − 3 c ∗ , 1 ]) = 9 [ D 2 − 3 c ∗ , 2 ] 36 ⋅ 36 36\cdot36 36 ⋅ 36
The group Z/2Z × Z/30Z
For X = X 1 ( 2 , 30 ) X=X_1(2,30) X = X 1 ( 2 , 30 ) Y = X 1 ( 30 ) Y=X_1(30) Y = X 1 ( 30 )
π 1 ( E , P , Q ) = ( E , Q ) , π 2 ( E , P , Q ) = ( E , P + 16 Q ) , π 3 ( E , P , Q ) = ( E , P + Q ) \pi_1(E,P,Q)=(E,Q),\qquad\pi_2(E,P,Q)=(E,P+16Q),\qquad\pi_3(E,P,Q)=(E,P+Q) π 1 ( E , P , Q ) = ( E , Q ) , π 2 ( E , P , Q ) = ( E , P + 16 Q ) , π 3 ( E , P , Q ) = ( E , P + Q ) from X X X Y Y Y Q \mathbb{Q} Q 15 Q 15Q 15 Q P P P P + 15 Q P+15Q P + 15 Q E [ 2 ] E[2] E [ 2 ] 16 Q 16Q 16 Q W W W X X X ( E , P , Q ) ↦ ( E , 15 Q , P + Q ) (E,P,Q)\mapsto(E,15Q,P+Q) ( E , P , Q ) ↦ ( E , 15 Q , P + Q ) E [ 2 ] E[2] E [ 2 ] W W W J ( X ) J(X) J ( X ) Y Y Y X 1 ( 15 ) X_1(15) X 1 ( 15 ) X 1 ( 15 ) X_1(15) X 1 ( 15 ) X 1 ( 15 ) X_1(15) X 1 ( 15 )
Proposition 7.8. The group Z / 2 Z × Z / 30 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/30\mathbb{Z} Z /2 Z × Z /30 Z Lemma 3.2 , neither is Z / 60 Z \mathbb{Z}/60\mathbb{Z} Z /60 Z
Proof. We have rk J ( Q ) = 0 \operatorname{rk}J(\mathbb{Q})=0 rk J ( Q ) = 0 [14 Theorem 4.1] , and gon X > 6 \operatorname{gon}X>6 gon X > 6 [14 Lemma 5.6 and Proposition 5.7] , so Proposition 5.1 with p = 7 p=7 p = 7 Z ′ + C Z'+C Z ′ + C Z ′ Z' Z ′ e > 0 e>0 e > 0
By [7 Theorem 3.1(2), Corollary 4.14 and Table 2] , J ( Y ) ( Q ) = Cl Q c Y ≃ Z / 4 Z × Z / 8160 Z J(Y)(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^cY\simeq\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/8160\mathbb{Z} J ( Y ) ( Q ) = Cl Q c Y ≃ Z /4 Z × Z /8160 Z J ( Y ) ( F 7 ) ≃ ( Z / 4 Z ) 3 × Z / 136 Z × Z / 16320 Z J(Y)(\mathbb{F}_7)\simeq(\mathbb{Z}/4\mathbb{Z})^3\times\mathbb{Z}/136\mathbb{Z}\times\mathbb{Z}/16320\mathbb{Z} J ( Y ) ( F 7 ) ≃ ( Z /4 Z ) 3 × Z /136 Z × Z /16320 Z G G G J ( Y ) ( Q ) J(Y)(\mathbb{Q}) J ( Y ) ( Q ) Lemma 7.1 on Y Y Y π i ∗ \pi_{i*} π i ∗ Y Y Y
[ π i ∗ Z ′ − e c ˉ ∗ ] m o d G ∈ S 6 − e ( i = 1 , 2 , 3 ) , [\pi_{i*}Z' - e\bar{c}_*] \bmod G \in S_{6-e} \qquad(i=1,2,3), [ π i ∗ Z ′ − e c ˉ ∗ ] mod G ∈ S 6 − e ( i = 1 , 2 , 3 ) , where the sets S 0 , … , S 4 S_0,\ldots,S_4 S 0 , … , S 4 X F 7 X_{\mathbb{F}_7} X F 7 … \ldots … Z ′ Z' Z ′ π 1 \pi_1 π 1
To exclude them we use the quotient W W W [7 Proposition 4.16] , every element of J ( Q ) J(\mathbb{Q}) J ( Q ) Q \mathbb{Q} Q X X X Q \mathbb{Q} Q W W W W W W X X X W W W H ⊆ J ( W ) ( F 7 ) H \subseteq J(W)(\mathbb{F}_7) H ⊆ J ( W ) ( F 7 ) W W W
J ( W ) ( F 7 ) ≃ ( Z / 2 Z ) 6 × Z / 102 Z × Z / 10608 Z , H ≃ ( Z / 2 Z ) 5 × Z / 8 Z , J(W)(\mathbb{F}_7) \simeq(\mathbb{Z}/2\mathbb{Z})^6 \times\mathbb{Z}/102\mathbb{Z} \times\mathbb{Z}/10608\mathbb{Z}, \qquad H \simeq(\mathbb{Z}/2\mathbb{Z})^5 \times\mathbb{Z}/8\mathbb{Z}, J ( W ) ( F 7 ) ≃ ( Z /2 Z ) 6 × Z /102 Z × Z /10608 Z , H ≃ ( Z /2 Z ) 5 × Z /8 Z , and that the classes of the push-forwards to W W W 6 c ∗ 6c_* 6 c ∗ c ∗ c_* c ∗ W W W H H H
Remark 7.9. The covering X → X 1 ( 15 ) X \to X_1(15) X → X 1 ( 15 ) ( E , P , Q ) ↦ ( E , 2 Q ) (E,P,Q) \mapsto(E,2Q) ( E , P , Q ) ↦ ( E , 2 Q ) SL 2 ( F 2 ) ≃ S 3 \operatorname{SL}_2(\mathbb{F}_2) \simeq S_3 SL 2 ( F 2 ) ≃ S 3 ( P , 15 Q ) (P,15Q) ( P , 15 Q ) E [ 2 ] E[2] E [ 2 ] W W W A 3 A_3 A 3 H 0 ( X , Ω 1 ) H^0(X,\Omega^1) H 0 ( X , Ω 1 ) S 3 S_3 S 3 X 1 ( 15 ) X_1(15) X 1 ( 15 ) W W W Y Y Y X X X π i \pi_i π i S 3 S_3 S 3 Y Y Y W W W A 3 A_3 A 3 W W W J ( X ) J(X) J ( X ) S 3 S_3 S 3 Y Y Y
The group Z/54Z
Let X = X 1 ( 54 ) X=X_1(54) X = X 1 ( 54 ) Y = X 1 ( 27 ) Y=X_1(27) Y = X 1 ( 27 ) π : X → Y \pi:X\to Y π : X → Y ( E , P ) ↦ ( E , 2 P ) (E,P)\mapsto(E,2P) ( E , P ) ↦ ( E , 2 P ) Q \mathbb{Q} Q z z z Y F 5 Y_{\mathbb{F}_5} Y F 5 E z E_z E z z z z X F 5 X_{\mathbb{F}_5} X F 5 z z z E z E_z E z x x x deg z \deg z deg z 2 deg z 2\deg z 2 deg z E z E_z E z z z z 3 deg z 3\deg z 3 deg z π ∗ x = ( deg x / deg z ) z \pi_*x=(\deg x/\deg z)z π ∗ x = ( deg x / deg z ) z
Proposition 7.10. The group Z / 54 Z \mathbb{Z}/54\mathbb{Z} Z /54 Z
Proof. Suppose that E / K E/K E / K P P P k J 1 ( 54 ) ( Q ) = 0 kJ_1(54)(\mathbb{Q})=0 k J 1 ( 54 ) ( Q ) = 0 [7 Theorem 3.1(2)] , and the index [ PSL 2 ( Z ) : Γ 1 ( 54 ) ] = 972 [\operatorname{PSL}_2(\mathbb{Z}):\Gamma_1(54)]=972 [ PSL 2 ( Z ) : Γ 1 ( 54 )] = 972 gon Q X ≥ gon C X > 9.64 > 6 \operatorname{gon}_{\mathbb{Q}}X \ge\operatorname{gon}_{\mathbb{C}}X > 9.64 > 6 gon Q X ≥ gon C X > 9.64 > 6 p = 5 p=5 p = 5 D D D ( E , P ) (E,P) ( E , P ) Z ′ + C Z'+C Z ′ + C Z ′ Z' Z ′ e > 0 e>0 e > 0
By [7 Corollary 4.14 and Table 2] , J 1 ( 27 ) ( Q ) = C Q J ≃ Z / 3 Z × Z / 3 Z × Z / 5249477 Z J_1(27)(\mathbb{Q})=C_{\mathbb{Q}}J \simeq\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/5249477\mathbb{Z} J 1 ( 27 ) ( Q ) = C Q J ≃ Z /3 Z × Z /3 Z × Z /5249477 Z G ⊆ J 1 ( 27 ) ( F 5 ) G\subseteq J_1(27)(\mathbb{F}_5) G ⊆ J 1 ( 27 ) ( F 5 ) [ c ˉ − deg ( c ˉ ) c ∗ ] [\bar{c}-\deg(\bar{c})c_*] [ c ˉ − deg ( c ˉ ) c ∗ ] c ˉ \bar{c} c ˉ Y F 5 Y_{\mathbb{F}_5} Y F 5 J 1 ( 27 ) ( Q ) J_1(27)(\mathbb{Q}) J 1 ( 27 ) ( Q )
J 1 ( 27 ) ( F 5 ) ≃ Z / 3 Z × Z / 3 Z × Z / 342175446 Z , G ≃ Z / 3 Z × Z / 3 Z × Z / 52497 Z , J_1(27)(\mathbb{F}_5)\simeq\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/342175446\mathbb{Z}, \qquad G\simeq\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/52497\mathbb{Z}, J 1 ( 27 ) ( F 5 ) ≃ Z /3 Z × Z /3 Z × Z /342175446 Z , G ≃ Z /3 Z × Z /3 Z × Z /52497 Z , so G G G π \pi π Q \mathbb{Q} Q π ∗ D \pi_*D π ∗ D Y Y Y Q \mathbb{Q} Q π ∗ Z ′ + π ∗ C \pi_*Z' + \pi_*C π ∗ Z ′ + π ∗ C π ∗ Z ′ \pi_*Z' π ∗ Z ′ e e e π ∗ C \pi_*C π ∗ C Lemma 7.1 on Y Y Y [ π ∗ Z ′ + π ∗ C − 6 c ∗ ] ∈ G [\pi_*Z' + \pi_*C - 6c_*] \in G [ π ∗ Z ′ + π ∗ C − 6 c ∗ ] ∈ G G G G [ π ∗ Z ′ − e c ∗ ] ∈ G [\pi_*Z' - ec_*] \in G [ π ∗ Z ′ − e c ∗ ] ∈ G
We also compute that the numbers of non-cuspidal closed points of Y F 5 Y_{\mathbb{F}_5} Y F 5 1 , … , 6 1,\ldots,6 1 , … , 6 0 , 9 , 57 , 171 , 612 , 2700 0,9,57,171,612,2700 0 , 9 , 57 , 171 , 612 , 2700 0 , 0 , 30 , 171 , 396 , 1890 0,0,30,171,396,1890 0 , 0 , 30 , 171 , 396 , 1890 e ≤ 6 e \leq6 e ≤ 6 π ∗ Z ′ \pi_*Z' π ∗ Z ′ 3 z 3z 3 z z z z z z z z + z ′ z+z' z + z ′ z z z z ′ z' z ′ [ π ∗ Z ′ − e c ∗ ] ∈ G [\pi_*Z' - ec_*] \in G [ π ∗ Z ′ − e c ∗ ] ∈ G 9 + 30 + 171 + 396 + 1890 + 465 9+30+171+396+1890+465 9 + 30 + 171 + 396 + 1890 + 465 z i + z j z_i+z_j z i + z j 1 ≤ i ≤ j ≤ 3 1 \leq i \leq j \leq3 1 ≤ i ≤ j ≤ 3 z 1 , z 2 , z 3 z_1,z_2,z_3 z 1 , z 2 , z 3 Y F 5 Y_{\mathbb{F}_5} Y F 5 j j j ℘ \wp ℘ K K K ( E , P ) (E,P) ( E , P ) ℘ \wp ℘ X X X F ℘ \mathbb{F}_\wp F ℘ X F 5 X_{\mathbb{F}_5} X F 5 E E E ℘ \wp ℘ D D D Z ′ Z' Z ′ ℘ \wp ℘ E E E e > 0 e>0 e > 0 π \pi π z 1 , z 2 , z 3 z_1,z_2,z_3 z 1 , z 2 , z 3 E E E ℘ \wp ℘ j j j
Let ℘ \wp ℘ E ′ = E / ⟨ 27 P ⟩ E'=E/\langle27P\rangle E ′ = E / ⟨ 27 P ⟩ K K K ϕ : E → E ′ \phi:E\to E' ϕ : E → E ′ ϕ ( 2 P ) \phi(2P) ϕ ( 2 P ) ker ϕ ^ = ϕ ( E [ 2 ] ) \ker\hat{\phi}=\phi(E[2]) ker ϕ ^ = ϕ ( E [ 2 ]) K K K E ′ E' E ′ E ′ E' E ′ ℘ \wp ℘ ℘ \wp ℘ j j j E E E y 2 = x 3 + x + 1 y^2=x^3+x+1 y 2 = x 3 + x + 1 j j j 1728 ⋅ 4 / 31 = 2 1728\cdot4/31=2 1728 ⋅ 4/31 = 2 F 5 \mathbb{F}_5 F 5 E E E ( r , 0 ) (r,0) ( r , 0 ) r 3 + r + 1 = 0 r^3+r+1=0 r 3 + r + 1 = 0 x 3 + x + 1 x^3+x+1 x 3 + x + 1 F 5 \mathbb{F}_5 F 5 r r r F 5 \mathbb{F}_5 F 5 ( r , 0 ) (r,0) ( r , 0 ) ( 0 , 0 ) (0,0) ( 0 , 0 ) y 2 = x ( x 2 + 3 r x + 3 r 2 + 1 ) y^2=x(x^2+3rx+3r^2+1) y 2 = x ( x 2 + 3 r x + 3 r 2 + 1 ) y 2 = x ( x 2 + a x + b ) y^2=x(x^2+ax+b) y 2 = x ( x 2 + a x + b ) ⟨ ( 0 , 0 ) ⟩ \langle(0,0)\rangle ⟨( 0 , 0 )⟩ y 2 = x ( x 2 − 2 a x + a 2 − 4 b ) y^2=x(x^2-2ax+a^2-4b) y 2 = x ( x 2 − 2 a x + a 2 − 4 b ) ⟨ ( r , 0 ) ⟩ \langle(r,0)\rangle ⟨( r , 0 )⟩ y 2 = x ( x 2 + 4 r x + 2 r 2 + 1 ) y^2=x(x^2+4rx+2r^2+1) y 2 = x ( x 2 + 4 r x + 2 r 2 + 1 ) j j j
1 r 2 + r + 1 = 2 r 2 + r + 4 ∉ F 5 , \frac{1}{r^2+r+1}=2r^2+r+4\notin\mathbb{F}_5, r 2 + r + 1 1 = 2 r 2 + r + 4 ∈ / F 5 , which is a contradiction.
Formal immersion at the prime 2
For n = 51 , 63 n=51,63 n = 51 , 63 n n n n = 65 n=65 n = 65 p = 2 p=2 p = 2 n = 51 n=51 n = 51 gon C X 1 ( 51 ) > 11.42 \operatorname{gon}_{\mathbb{C}} X_1(51)>11.42 gon C X 1 ( 51 ) > 11.42 n = 63 , 65 n=63,65 n = 63 , 65 n n n J 1 ( n ) ( Q ) J_1(n)(\mathbb{Q}) J 1 ( n ) ( Q ) ψ = ⟨ a ⟩ − 1 \psi=\langle a\rangle-1 ψ = ⟨ a ⟩ − 1 gon C X 1 ( n ) > 11.42 , 17.13 , 19.99 \operatorname{gon}_{\mathbb{C}} X_1(n)>11.42,17.13,19.99 gon C X 1 ( n ) > 11.42 , 17.13 , 19.99
The difficulty at p = 2 p=2 p = 2 μ 2 = { ± 1 } \mu_2=\{\pm1\} μ 2 = { ± 1 } Z 2 \mathbb{Z}_2 Z 2 − 1 -1 − 1
Lemma 8.1. Let A A A Z 2 \mathbb{Z}_2 Z 2 T T T δ \delta δ A A A T = F + δ V T = F + \delta V T = F + δ V A F 2 A_{\mathbb{F}_2} A F 2 F F F V V V A F 2 A_{\mathbb{F}_2} A F 2 z ∈ A ( Q 2 ) z \in A(\mathbb{Q}_2) z ∈ A ( Q 2 ) 2 z = 0 2z = 0 2 z = 0 0 0 0 2 2 2 ( T − δ ) z = 0 (T-\delta)z = 0 ( T − δ ) z = 0
Proof. Choose coordinates X = ( X 1 , … , X g ) X = (X_1,\ldots,X_g) X = ( X 1 , … , X g ) A A A A F 2 A_{\mathbb{F}_2} A F 2 A ( Q 2 ) A(\mathbb{Q}_2) A ( Q 2 ) 0 0 0 2 Z 2 2\mathbb{Z}_2 2 Z 2 ′ ' ′ [24 Appendix] ). Write
[ 2 ] ( X ) = 2 X + Q ( X ) + R ( X ) , [2](X) = 2X + Q(X) + R(X), [ 2 ] ( X ) = 2 X + Q ( X ) + R ( X ) , where the coordinates of Q Q Q 2 2 2 R R R 3 3 3 Z 2 \mathbb{Z}_2 Z 2 M M M Z 2 \mathbb{Z}_2 Z 2 M ‾ \overline{M} M V V V [ 2 ] = V ∘ F [2] = V \circ F [ 2 ] = V ∘ F A F 2 A_{\mathbb{F}_2} A F 2 F F F Q Q Q Q ‾ ( X ) = M ‾ ( X 1 2 , … , X g 2 ) t \overline{Q}(X) = \overline{M}(X_1^2,\ldots,X_g^2)^t Q ( X ) = M ( X 1 2 , … , X g 2 ) t
Write the coordinates of z z z 2 u 2u 2 u u ∈ Z 2 g u \in\mathbb{Z}_2^g u ∈ Z 2 g 2 z = 0 2z = 0 2 z = 0
0 = [ 2 ] ( 2 u ) ≡ 4 u + 4 Q ( u ) ( m o d 8 ) , 0 = [2](2u) \equiv4u + 4Q(u) \pmod{8}, 0 = [ 2 ] ( 2 u ) ≡ 4 u + 4 Q ( u ) ( mod 8 ) , so u ‾ + M ‾ ( u ‾ 1 2 , … , u ‾ g 2 ) t = 0 \overline{u}+\overline{M}(\overline{u}_1^2,\ldots,\overline{u}_g^2)^t = 0 u + M ( u 1 2 , … , u g 2 ) t = 0 a 2 = a a^2 = a a 2 = a a ∈ F 2 a \in\mathbb{F}_2 a ∈ F 2 ( M ‾ − 1 ) u ‾ = 0 (\overline{M}-1)\overline{u}=0 ( M − 1 ) u = 0 α = T − δ \alpha= T-\delta α = T − δ F F F 0 0 0 α \alpha α A F 2 A_{\mathbb{F}_2} A F 2 N ‾ ( M ‾ − 1 ) \overline{N}(\overline{M}-1) N ( M − 1 ) N ‾ \overline{N} N δ \delta δ α \alpha α 2 u 2u 2 u 4 4 4 α z \alpha z α z 4 Z 2 4\mathbb{Z}_2 4 Z 2
Finally, a non-zero point v v v 4 Z 2 4\mathbb{Z}_2 4 Z 2 2 2 2 s ≥ 2 s \geq2 s ≥ 2 v v v [ 2 ] ( v ) [2](v) [ 2 ] ( v ) s + 1 s+1 s + 1 2 s > s + 1 2s>s+1 2 s > s + 1 2 α z = α ( 2 z ) = 0 2\alpha z=\alpha(2z)=0 2 α z = α ( 2 z ) = 0 α z = 0 \alpha z=0 α z = 0
:::
We apply this to quotients of X 1 ( n ) X_1(n) X 1 ( n ) n n n H ⊆ Δ ( n ) H \subseteq\Delta(n) H ⊆ Δ ( n ) X 1 ( n ) X_1(n) X 1 ( n ) 0 0 0 2 2 2 Y : = X 1 ( n ) / H Y := X_1(n)/H Y := X 1 ( n ) / H Z ( 2 ) \mathbb{Z}_{(2)} Z ( 2 ) 2 2 2 T 2 T_2 T 2 Y Y Y T 2 = F + ⟨ 2 ⟩ V T_2 = F+\langle2\rangle V T 2 = F + ⟨ 2 ⟩ V J ( Y ) F 2 J(Y)_{\mathbb{F}_2} J ( Y ) F 2 c 0 c_0 c 0 C C C Y F 2 Y_{\mathbb{F}_2} Y F 2 x x x ( f 2 + g 2 x ) d x (f^2+g^2x)\,dx ( f 2 + g 2 x ) d x f , g f,g f , g Y F 2 Y_{\mathbb{F}_2} Y F 2 C ( ( f 2 + g 2 x ) d x ) = g d x C((f^2+g^2x)\,dx)=g\,dx C (( f 2 + g 2 x ) d x ) = g d x t t t C ( ∑ n a n t n d t ) = ∑ n a 2 n + 1 1 / 2 t n d t C(\sum_n a_nt^n\,dt)=\sum_n a_{2n+1}^{1/2}t^n\,dt C ( ∑ n a n t n d t ) = ∑ n a 2 n + 1 1/2 t n d t C C C H 0 ( Y F 2 , Ω 1 ) H^0(Y_{\mathbb{F}_2},\Omega^1) H 0 ( Y F 2 , Ω 1 ) F 2 \mathbb{F}_2 F 2 q q q q q q ∑ n a n q n d q / q ↦ ∑ n a 2 n 1 / 2 q n d q / q \sum_n a_nq^n\,dq/q\mapsto\sum_n a_{2n}^{1/2}q^n\,dq/q ∑ n a n q n d q / q ↦ ∑ n a 2 n 1/2 q n d q / q a n a_n a n F 2 \mathbb{F}_2 F 2 U 2 U_2 U 2 ∑ n a n q n ↦ ∑ n a 2 n q n \sum_n a_nq^n\mapsto\sum_n a_{2n}q^n ∑ n a n q n ↦ ∑ n a 2 n q n ∑ n a n q n \sum_n a_nq^n ∑ n a n q n
Every endomorphism ψ \psi ψ J ( Y ) J(Y) J ( Y ) J ( Y ) J(Y) J ( Y ) Z 2 \mathbb{Z}_2 Z 2 ψ ∗ \psi^* ψ ∗ H 0 ( Y F 2 , Ω 1 ) H^0(Y_{\mathbb{F}_2},\Omega^1) H 0 ( Y F 2 , Ω 1 ) 0 0 0 C C C [38 ] , the Eichler–Shimura relation gives
( T 2 − ⟨ 2 ⟩ ) ∗ = ( C − 1 ) ⟨ 2 ⟩ ∗ on H 0 ( Y F 2 , Ω 1 ) . (3) (T_2-\langle2\rangle)^*=(C-1)\langle2\rangle^* \quad\text{on } H^0(Y_{\mathbb{F}_2},\Omega^1). \tag*{(3)} ( T 2 − ⟨ 2 ⟩ ) ∗ = ( C − 1 ) ⟨ 2 ⟩ ∗ on H 0 ( Y F 2 , Ω 1 ) . (3) The following criterion combines three ingredients. The elements ψ i \psi_i ψ i J ( Y ) ( Q ) J(Y)(\mathbb{Q}) J ( Y ) ( Q ) Lemma 8.1 shows that T 2 − ⟨ 2 ⟩ T_2-\langle2\rangle T 2 − ⟨ 2 ⟩ C ‾ \overline{C} C Y ( d ) → J ( Y ) r Y^{(d)} \to J(Y)^r Y ( d ) → J ( Y ) r C ‾ \overline{C} C U U U U U U C ‾ \overline{C} C J ( Y ) J(Y) J ( Y ) H 0 ( Y F 2 , Ω 1 ) H^0(Y_{\mathbb{F}_2}, \Omega^1) H 0 ( Y F 2 , Ω 1 )
Proposition 8.2. Let Y = X 1 ( n ) / H Y = X_1(n)/H Y = X 1 ( n ) / H ψ 1 , … , ψ r ∈ Z [ Δ ( n ) / H ] \psi_1,\ldots,\psi_r \in\mathbb{Z}[\Delta(n)/H] ψ 1 , … , ψ r ∈ Z [ Δ ( n ) / H ] ψ i ( J ( Y ) ( Q ) ) \psi_i(J(Y)(\mathbb{Q})) ψ i ( J ( Y ) ( Q ))
U : = ( C − 1 ) ( ∑ i = 1 r ψ i ∗ H 0 ( Y F 2 , Ω 1 ) ) ⊆ H 0 ( Y F 2 , Ω 1 ) . U := (C-1)\left(\sum_{i=1}^{r}\psi_i^*H^0(Y_{\mathbb{F}_2},\Omega^1)\right) \subseteq H^0(Y_{\mathbb{F}_2},\Omega^1). U := ( C − 1 ) ( i = 1 ∑ r ψ i ∗ H 0 ( Y F 2 , Ω 1 ) ) ⊆ H 0 ( Y F 2 , Ω 1 ) . Let C ‾ \overline{C} C d d d Y F 2 Y_{\mathbb{F}_2} Y F 2 U → H 0 ( C ‾ , Ω 1 ∣ C ‾ ) U \to H^0(\overline{C},\Omega^1|_{\overline{C}}) U → H 0 ( C , Ω 1 ∣ C ) Q \mathbb{Q} Q D D D d d d Y Y Y C ‾ \overline{C} C
Proof. Suppose that D D D C ‾ \overline{C} C Y Y Y C ‾ \overline{C} C C C C L ⊆ Q ( ζ n ) L \subseteq\mathbb{Q}(\zeta_n) L ⊆ Q ( ζ n ) λ \lambda λ L λ = Q 2 L_\lambda= \mathbb{Q}_2 L λ = Q 2 i i i ψ i [ D − C ] = ψ i [ D − d c 0 ] − ψ i [ C − d c 0 ] \psi_i[D-C] = \psi_i[D-dc_0]-\psi_i[C-dc_0] ψ i [ D − C ] = ψ i [ D − d c 0 ] − ψ i [ C − d c 0 ] ψ i \psi_i ψ i λ \lambda λ J ( Y ) J(Y) J ( Y ) Z 2 \mathbb{Z}_2 Z 2 T = T 2 T=T_2 T = T 2 δ = ⟨ 2 ⟩ \delta=\langle2\rangle δ = ⟨ 2 ⟩ ( T 2 − ⟨ 2 ⟩ ) ψ i [ D − C ] = 0 (T_2-\langle2\rangle)\psi_i[D-C]=0 ( T 2 − ⟨ 2 ⟩) ψ i [ D − C ] = 0
Consider the morphism f : Y ( d ) → J ( Y ) r f:Y^{(d)}\to J(Y)^r f : Y ( d ) → J ( Y ) r Z 2 \mathbb{Z}_2 Z 2 f ( B ) = ( ( T 2 − ⟨ 2 ⟩ ) ψ i [ B − d c 0 ] ) i f(B)=((T_2-\langle2\rangle)\psi_i[B-dc_0])_i f ( B ) = (( T 2 − ⟨ 2 ⟩) ψ i [ B − d c 0 ] ) i f ( D ) = f ( C ) f(D)=f(C) f ( D ) = f ( C ) D D D C C C C ‾ \overline{C} C (3) , and since ⟨ 2 ⟩ ∗ \langle2\rangle^* ⟨ 2 ⟩ ∗ C C C ψ i ∗ \psi_i^* ψ i ∗ f f f C ‾ \overline{C} C U U U C ‾ \overline{C} C Y ( d ) → J ( Y ) Y^{(d)}\to J(Y) Y ( d ) → J ( Y ) B ↦ [ B − d c 0 ] B\mapsto[B-dc_0] B ↦ [ B − d c 0 ] C ‾ \overline{C} C H 0 ( Y F 2 , Ω 1 ) → H 0 ( C ‾ , Ω 1 ∣ C ‾ ) H^0(Y_{\mathbb{F}_2},\Omega^1)\to H^0(\overline{C},\Omega^1|_{\overline{C}}) H 0 ( Y F 2 , Ω 1 ) → H 0 ( C , Ω 1 ∣ C ) f f f C ‾ \overline{C} C [10 Lemma 4.2] (which is due to Parent [39 Lemme 4.13] ) f f f C ‾ \overline{C} C D = C D=C D = C D D D
As in § 5 , we check the surjectivity of the restriction maps by computing, for each closed cusp P P P Y F 2 Y_{\mathbb{F}_2} Y F 2 j j j Ω ( j P ) \Omega(jP) Ω ( j P ) H 0 ( Y F 2 , Ω 1 ) H^0(Y_{\mathbb{F}_2},\Omega^1) H 0 ( Y F 2 , Ω 1 ) C ‾ \overline{C} C dim U − dim ( U ∩ Ω ( C ‾ ) ) \dim U-\dim(U\cap\Omega(\overline{C})) dim U − dim ( U ∩ Ω ( C )) U U U U ∩ Ω ( C ‾ ) U\cap\Omega(\overline{C}) U ∩ Ω ( C ) J ( Y ) J(Y) J ( Y ) ψ i \psi_i ψ i r = 1 r=1 r = 1 ψ 1 = 1 \psi_1=1 ψ 1 = 1 J ( Y ) ( Q ) J(Y)(\mathbb{Q}) J ( Y ) ( Q )
Remark 8.3. The condition in Proposition 8.2 is the usual formal immersion criterion, written in terms of cotangent spaces. When C ‾ \overline{C} C q q q [23 ] for X 0 ( p ) X_0(p) X 0 ( p ) [40 Théorème 1.10 and Proposition 2.7] (see [10 Proposition 5.3] ), and when C ‾ \overline{C} C [10 Lemma 8.3] . The usual way of dealing with the torsion in the kernel of reduction at 2 is Parent’s trick [10 Corollary 5.2] , which multiplies the Hecke operator by an operator at an odd prime q q q A q A_q A q q q q L L L q q q n n n T 2 − ⟨ 2 ⟩ T_2-\langle2\rangle T 2 − ⟨ 2 ⟩ Y F 2 Y_{\mathbb{F}_2} Y F 2 A q A_q A q T q T_q T q Y Y Y q q q n = 63 n=63 n = 63 q q q Proposition 8.2 is checked in the same way at every effective cuspidal divisor C ‾ \overline{C} C
Lemma 8.4. Let Y 51 : = X 1 ( 51 ) / ⟨ 16 ⟩ Y_{51}:=X_1(51)/\langle16\rangle Y 51 := X 1 ( 51 ) / ⟨ 16 ⟩ Y 63 : = X 1 ( 63 ) / ⟨ 8 ⟩ Y_{63}:=X_1(63)/\langle8\rangle Y 63 := X 1 ( 63 ) / ⟨ 8 ⟩ ⟨ 16 ⟩ \langle16\rangle ⟨ 16 ⟩ ⟨ 8 ⟩ \langle8\rangle ⟨ 8 ⟩ X 1 ( 51 ) X_1(51) X 1 ( 51 ) X 1 ( 63 ) X_1(63) X 1 ( 63 ) Y 51 Y_{51} Y 51 Y 63 Y_{63} Y 63 Y 51 Y_{51} Y 51 X 1 ( 17 ) X_1(17) X 1 ( 17 ) Y 63 Y_{63} Y 63 X 1 ( 9 ) X_1(9) X 1 ( 9 )
Proof. Write ( r , s , a ) = ( 17 , 3 , 16 ) (r,s,a)=(17,3,16) ( r , s , a ) = ( 17 , 3 , 16 ) ( 9 , 7 , 8 ) (9,7,8) ( 9 , 7 , 8 ) a ≡ − 1 ( m o d r ) a\equiv-1\pmod r a ≡ − 1 ( mod r ) a ≡ 1 ( m o d s ) a\equiv1\pmod s a ≡ 1 ( mod s ) r s rs r s ( P r , P s ) (P_r,P_s) ( P r , P s ) r r r s s s ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ ( P r , P s ) (P_r,P_s) ( P r , P s ) ( P r , − P s ) (P_r,-P_s) ( P r , − P s ) Y Y Y ( E , P r ) (E,P_r) ( E , P r ) x x x P s P_s P s ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ u u u u ( P ) = a P u(P)=aP u ( P ) = a P a 2 = 1 a^2=1 a 2 = 1 u 2 u^2 u 2 P P P u 2 = 1 u^2=1 u 2 = 1 u = ± 1 u=\pm1 u = ± 1 P P P ± 1 ≠ a \pm1\ne a ± 1 = a § 2 , a cusp fixed by ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ a a a n n n a ≡ − 1 ( m o d r ) a\equiv-1\pmod r a ≡ − 1 ( mod r ) a ≡ 1 ( m o d s ) a\equiv1\pmod s a ≡ 1 ( mod s ) ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ Z [ ζ n , 1 n ] \mathbb{Z}[\zeta_n,\frac{1}{n}] Z [ ζ n , n 1 ] n n n ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ X 1 ( 51 ) X_1(51) X 1 ( 51 ) X 1 ( 63 ) X_1(63) X 1 ( 63 )
Proposition 8.5. The groups Z / 51 Z \mathbb{Z}/51\mathbb{Z} Z /51 Z Z / 63 Z \mathbb{Z}/63\mathbb{Z} Z /63 Z
Proof. Let n ∈ { 51 , 63 } n\in\{51,63\} n ∈ { 51 , 63 } E E E K K K n n n F 2 f \mathbb{F}_{2^f} F 2 f f ≤ 6 f\le6 f ≤ 6 n n n f ≤ 5 f\le5 f ≤ 5 F 64 \mathbb{F}_{64} F 64 Lemma 2.3 and Lemma 2.2 , E E E x ( 6 ) x^{(6)} x ( 6 ) D D D Y = X 1 ( n ) / H Y=X_1(n)/H Y = X 1 ( n ) / H H H H ⟨ 16 ⟩ \langle16\rangle ⟨ 16 ⟩ n = 51 n=51 n = 51 ⟨ 8 ⟩ \langle8\rangle ⟨ 8 ⟩ n = 63 n=63 n = 63 Lemma 8.4 , H H H X 1 ( n ) X_1(n) X 1 ( n ) Y Y Y Proposition 8.2 . The divisor D D D Q \mathbb{Q} Q
The Jacobian J ( Y ) J(Y) J ( Y ) J 1 ( n ) J_1(n) J 1 ( n ) H H H n n n H H H n = 51 n=51 n = 51 rk J 1 ( 51 ) ( Q ) = 0 \operatorname{rk} J_1(51)(\mathbb{Q})=0 rk J 1 ( 51 ) ( Q ) = 0 [7 Theorem 3.1(2)] . For n = 63 n=63 n = 63 63.2.3. a 63.2.3.a 63.2.3. a χ \chi χ χ ( 8 ) = − 1 \chi(8)=-1 χ ( 8 ) = − 1 J ( Y 63 ) J(Y_{63}) J ( Y 63 ) J ( Y ) ( Q ) J(Y)(\mathbb{Q}) J ( Y ) ( Q ) Proposition 8.2 with r = 1 r=1 r = 1 ψ 1 = 1 \psi_1=1 ψ 1 = 1
For Y 51 Y_{51} Y 51 U = ( C − 1 ) H 0 ( Y F 2 , Ω 1 ) U=(C-1)H^0(Y_{\mathbb{F}_2},\Omega^1) U = ( C − 1 ) H 0 ( Y F 2 , Ω 1 ) Y F 2 Y_{\mathbb{F}_2} Y F 2 1 16 , 4 4 1^{16},4^4 1 16 , 4 4 ( 21 6 ) + 4 ( 17 2 ) = 54808 \binom{21}{6}+4\binom{17}{2}=54808 ( 6 21 ) + 4 ( 2 17 ) = 54808 U U U Y 63 Y_{63} Y 63 1 9 , 2 3 , 3 9 , 6 1^9,2^3,3^9,6 1 9 , 2 3 , 3 9 , 6 U U U U U U Proposition 8.2 no reduction of D D D
The groups Z / 57 Z \mathbb{Z}/57\mathbb{Z} Z /57 Z Z / 65 Z \mathbb{Z}/65\mathbb{Z} Z /65 Z
For n = 57 n=57 n = 57 n = 65 n=65 n = 65 J 1 ( n ) J_1(n) J 1 ( n ) F 64 \mathbb{F}_{64} F 64 n n n
Proposition 9.1. Let n ∈ { 57 , 65 } n\in\{57,65\} n ∈ { 57 , 65 } E E E K K K n n n E E E K K K
Proof. There are no points of order n n n [7 ] , so the point x ∈ Y 1 ( n ) ( K ) x\in Y_1(n)(K) x ∈ Y 1 ( n ) ( K ) D D D E E E ℘ \wp ℘ K K K F ℘ = F 64 \mathbb{F}_{\wp}=\mathbb{F}_{64} F ℘ = F 64 ℘ \wp ℘ K K K E ~ \widetilde{E} E n n n F 64 \mathbb{F}_{64} F 64 # E ~ ( F 64 ) = n \#\widetilde{E}(\mathbb{F}_{64})=n # E ( F 64 ) = n E ~ \widetilde{E} E j ( E ~ ) = 0 j(\widetilde{E})=0 j ( E ) = 0 E ~ \widetilde{E} E [ 2 ] [2] [ 2 ] [41 Theorem V.3.1] , so [ 8 ] [8] [ 8 ] π \pi π E ~ \widetilde{E} E [41 Corollary II.2.12] both are the 64-th power Frobenius morphism followed by an isomorphism, so π = ζ ∘ [ 8 ] \pi=\zeta\circ[8] π = ζ ∘ [ 8 ] ζ \zeta ζ E ~ \widetilde{E} E π \pi π E ~ ( F 64 ) \widetilde{E}(\mathbb{F}_{64}) E ( F 64 ) n n n ζ \zeta ζ 8 − 1 8^{-1} 8 − 1 ζ \zeta ζ 8 P ~ 8\widetilde{P} 8 P P ~ \widetilde{P} P P ~ \widetilde{P} P n n n ( E ~ , 8 P ~ ) ≃ ( E ~ , P ~ ) (\widetilde{E},8\widetilde{P})\simeq(\widetilde{E},\widetilde{P}) ( E , 8 P ) ≃ ( E , P ) ⟨ 8 ⟩ \langle8\rangle ⟨ 8 ⟩ x x x ⟨ 8 ⟩ \langle8\rangle ⟨ 8 ⟩ Q \mathbb{Q} Q D D D ⟨ 8 ⟩ \langle8\rangle ⟨ 8 ⟩
The only newforms of level dividing n n n ( ⟨ 8 ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle8\rangle-1)J_1(n)(\mathbb{Q}) (⟨ 8 ⟩ − 1 ) J 1 ( n ) ( Q ) ( ⟨ 8 ⟩ − 1 ) D (\langle8\rangle-1)D (⟨ 8 ⟩ − 1 ) D
[ ( ⟨ 8 ⟩ − 1 ) D ] = ( ⟨ 8 ⟩ − 1 ) [ D − 6 c 0 ] + 6 [ ⟨ 8 ⟩ c 0 − c 0 ] , [(\langle8\rangle-1)D]=(\langle8\rangle-1)[D-6c_0]+6[\langle8\rangle c_0-c_0], [(⟨ 8 ⟩ − 1 ) D ] = (⟨ 8 ⟩ − 1 ) [ D − 6 c 0 ] + 6 [⟨ 8 ⟩ c 0 − c 0 ] , which is torsion, as the first summand lies in ( ⟨ 8 ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle8\rangle-1)J_1(n)(\mathbb{Q}) (⟨ 8 ⟩ − 1 ) J 1 ( n ) ( Q ) ℘ \wp ℘ ( ⟨ 8 ⟩ − 1 ) (\langle8\rangle-1) (⟨ 8 ⟩ − 1 ) D D D 2 ( ⟨ 8 ⟩ − 1 ) D = div ( f ) 2(\langle8\rangle-1)D=\operatorname{div}(f) 2 (⟨ 8 ⟩ − 1 ) D = div ( f ) f ∈ Q ( X 1 ( n ) ) f\in\mathbb{Q}(X_1(n)) f ∈ Q ( X 1 ( n )) gon C X 1 ( n ) > 14.28 \operatorname{gon}_{\mathbb{C}}X_1(n)>14.28 gon C X 1 ( n ) > 14.28 19.99 19.99 19.99 f f f ⟨ 8 ⟩ D = D \langle8\rangle D=D ⟨ 8 ⟩ D = D X 1 ( n ) X_1(n) X 1 ( n ) D D D Lemma 2.4 , ⟨ 8 ⟩ \langle8\rangle ⟨ 8 ⟩
For n = 57 n=57 n = 57 ⟨ 8 ⟩ \langle8\rangle ⟨ 8 ⟩ Δ ( 57 ) \Delta(57) Δ ( 57 ) x x x X 1 ( 57 ) / ⟨ 8 ⟩ X_1(57)/\langle8\rangle X 1 ( 57 ) / ⟨ 8 ⟩ x x x X 0 ( 57 ) X_0(57) X 0 ( 57 ) X 0 ( 57 ) X_0(57) X 0 ( 57 ) Q ( − 23 ) \mathbb{Q}(\sqrt{-23}) Q ( − 23 ) Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 ) K K K Q ( − 23 ) \mathbb{Q}(\sqrt{-23}) Q ( − 23 ) K K K j j j Q ( 13 ) \mathbb{Q}(\sqrt{13}) Q ( 13 )
3387888351672962316333 13 − 12215205167504087643323 2 and − 17787 13 − 61763 2 , \begin{aligned}
&\frac{3387888351672962316333\sqrt{13}-12215205167504087643323}{2}
\quad\text{and}\quad
\frac{-17787\sqrt{13}-61763}{2},
\end{aligned} 2 3387888351672962316333 13 − 12215205167504087643323 and 2 − 17787 13 − 61763 , and writing them in the basis 1 , t 1,t 1 , t t = ( 1 + 13 ) / 2 t=(1+\sqrt{13})/2 t = ( 1 + 13 ) /2 t t t t 2 + t + 1 ≡ 0 t^2+t+1\equiv0 t 2 + t + 1 ≡ 0 j j j j ( E ~ ) = 0 j(\widetilde{E})=0 j ( E ) = 0
For n = 65 n = 65 n = 65 ⟨ 8 ⟩ \langle8\rangle ⟨ 8 ⟩ x x x X 1 ( 65 ) / ⟨ 8 ⟩ X_1(65)/\langle8\rangle X 1 ( 65 ) / ⟨ 8 ⟩ X 0 ( 65 ) X_0(65) X 0 ( 65 ) X 0 ( 65 ) X_0(65) X 0 ( 65 ) [3 Theorem 1.1 and §5.2.4] all its cubic points are defined over the field Q ( α ) \mathbb{Q}(\alpha) Q ( α ) α 3 − α 2 + α − 2 = 0 \alpha^3-\alpha^2+\alpha-2=0 α 3 − α 2 + α − 2 = 0 K K K − 83 -83 − 83 T ( T 2 + T + 1 ) T(T^2+T+1) T ( T 2 + T + 1 ) Q ( α ) \mathbb{Q}(\alpha) Q ( α )
Lemma 9.2. The diamond operator ⟨ 2 ⟩ \langle2\rangle ⟨ 2 ⟩ J 1 ( 57 ) ( Q ) [ 2 ] J_1(57)(\mathbb{Q})[2] J 1 ( 57 ) ( Q ) [ 2 ]
Proof. Let J = J 1 ( 57 ) J=J_1(57) J = J 1 ( 57 ) J ( Q ) [ 2 ] J(\mathbb{Q})[2] J ( Q ) [ 2 ] J ( Q ) [ 2 ] J(\mathbb{Q})[2] J ( Q ) [ 2 ] A 5 = T 5 − 5 ⟨ 5 ⟩ − 1 A_5=T_5-5\langle5\rangle-1 A 5 = T 5 − 5 ⟨ 5 ⟩ − 1 J [ 2 ] J[2] J [ 2 ] H 1 ( J ( C ) , Z ) / 2 H 1 ( J ( C ) , Z ) ≃ J [ 2 ] H_1(J(\mathbb{C}),\mathbb{Z})/2H_1(J(\mathbb{C}),\mathbb{Z})\simeq J[2] H 1 ( J ( C ) , Z ) /2 H 1 ( J ( C ) , Z ) ≃ J [ 2 ] F 2 \mathbb{F}_2 F 2 A 5 A_5 A 5 ⟨ 2 ⟩ − 1 \langle2\rangle-1 ⟨ 2 ⟩ − 1 Z \mathbb{Z} Z q ⟨ q ⟩ = T q 2 − T q 2 q\langle q\rangle=T_q^2-T_{q^2} q ⟨ q ⟩ = T q 2 − T q 2 q = 2 q=2 q = 2 q = 5 q=5 q = 5
Proposition 9.3. The group Z / 57 Z \mathbb{Z}/57\mathbb{Z} Z /57 Z
Proof. Let E E E K K K D D D Proposition 9.1 and Lemma 2.2 the reduction of D D D X 1 ( 57 ) X_1(57) X 1 ( 57 ) Q \mathbb{Q} Q 1 1 1 2 9 2^9 2 9 18 2 18^2 1 8 2 Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) D D D Q \mathbb{Q} Q C C C
Let g = ⟨ 2 ⟩ g=\langle2\rangle g = ⟨ 2 ⟩ y = ( g − 1 ) [ D − C ] y=(g-1)[D-C] y = ( g − 1 ) [ D − C ] Proposition 9.1 , ( g − 1 ) J 1 ( 57 ) ( Q ) (g-1)J_1(57)(\mathbb{Q}) ( g − 1 ) J 1 ( 57 ) ( Q ) y y y 2 y = 0 2y=0 2 y = 0 Lemma 9.2 we have g y = y gy=y g y = y g g g Δ ( 57 ) \Delta(57) Δ ( 57 )
0 = ( g 9 − 1 ) [ D − C ] = ( 1 + g + ⋯ + g 8 ) y = 9 y = y . 0=(g^9-1)[D-C]=(1+g+\cdots+g^8)y=9y=y. 0 = ( g 9 − 1 ) [ D − C ] = ( 1 + g + ⋯ + g 8 ) y = 9 y = y . So ( g − 1 ) ( D − C ) (g-1)(D-C) ( g − 1 ) ( D − C ) gon C X 1 ( 57 ) > 14.28 \operatorname{gon}_{\mathbb{C}} X_1(57)>14.28 gon C X 1 ( 57 ) > 14.28 g D = D gD=D g D = D D D D Lemma 2.4 gives 9 ∣ deg D = 6 9\mid\deg D=6 9 ∣ deg D = 6
For n = 65 n=65 n = 65 Proposition 8.2 on the quotient Y = X 1 ( 65 ) / ⟨ 4 ⟩ Y=X_1(65)/\langle4\rangle Y = X 1 ( 65 ) / ⟨ 4 ⟩ Y Y Y F 2 \mathbb{F}_2 F 2 Y Y Y ⟨ 51 ⟩ \langle51\rangle ⟨ 51 ⟩ X 1 ( 5 ) X_1(5) X 1 ( 5 ) X 1 ( 13 ) / ⟨ 3 ⟩ X_1(13)/\langle3\rangle X 1 ( 13 ) / ⟨ 3 ⟩ j j j j j j Y Y Y F 2 \mathbb{F}_2 F 2 Y Y Y t t t X 1 ( 5 ) X_1(5) X 1 ( 5 ) b = c = t b=c=t b = c = t z z z X 1 ( 13 ) / ⟨ 3 ⟩ X_1(13)/\langle3\rangle X 1 ( 13 ) / ⟨ 3 ⟩ F 2 \mathbb{F}_2 F 2 j j j
j 5 ( t ) = t 12 + t 8 + t 4 + 1 t 7 + t 6 + t 5 , j 13 ( z ) = z 16 + z 4 + 1 z 28 + z 25 + z 24 + z 21 + z 20 + z 17 + z 16 + z 13 . j_5(t)=\frac{t^{12}+t^8+t^4+1}{t^7+t^6+t^5},\qquad j_{13}(z)=\frac{z^{16}+z^4+1}{z^{28}+z^{25}+z^{24}+z^{21}+z^{20}+z^{17}+z^{16}+z^{13}}. j 5 ( t ) = t 7 + t 6 + t 5 t 12 + t 8 + t 4 + 1 , j 13 ( z ) = z 28 + z 25 + z 24 + z 21 + z 20 + z 17 + z 16 + z 13 z 16 + z 4 + 1 . Over a field of characteristic 2, an ordinary elliptic curve with j j j j ≠ 0 j\ne0 j = 0 y 2 + x y = x 3 + a x 2 + j − 1 y^2+xy=x^3+ax^2+j^{-1} y 2 + x y = x 3 + a x 2 + j − 1 a a a w 2 + w w^2+w w 2 + w X 1 ( 5 ) X_1(5) X 1 ( 5 ) a 5 = t 2 / ( t + 1 ) 3 a_5=t^2/(t+1)^3 a 5 = t 2 / ( t + 1 ) 3 ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ a 13 = ( z 7 + z 6 + z 2 + z ) / ( z 4 + z + 1 ) a_{13} = (z^{7} + z^{6} + z^{2} + z)/(z^{4} + z + 1) a 13 = ( z 7 + z 6 + z 2 + z ) / ( z 4 + z + 1 ) z z z j 13 j_{13} j 13 a 13 a_{13} a 13 X 1 ( 13 ) X_{1}(13) X 1 ( 13 ) F 2 \mathbb{F}_{2} F 2
Lemma 9.4. Let Y = X 1 ( 65 ) / ⟨ 4 ⟩ Y = X_{1}(65)/\langle4\rangle Y = X 1 ( 65 ) / ⟨ 4 ⟩ ⟨ 4 ⟩ \langle4\rangle ⟨ 4 ⟩ X 1 ( 65 ) X_{1}(65) X 1 ( 65 ) Y Y Y Y F 2 Y_{\mathbb{F}_{2}} Y F 2 t t t z z z w w w
j 5 ( t ) = j 13 ( z ) , w 2 + w = a 5 + a 13 , j_{5}(t) = j_{13}(z), \qquad w^{2} + w = a_{5} + a_{13}, j 5 ( t ) = j 13 ( z ) , w 2 + w = a 5 + a 13 , and the residual diamond group Δ ( 65 ) / ⟨ 4 ⟩ \Delta(65)/\langle4\rangle Δ ( 65 ) / ⟨ 4 ⟩
τ T ( t , z , w ) = ( 1 t , z , w + t t + 1 ) , τ U ( t , z , w ) = ( t , z + 1 , w + z ) , \tau_{T}(t,z,w) = \left(\frac{1}{t},z,w+\frac{t}{t+1}\right), \qquad\tau_{U}(t,z,w) = (t,z+1,w+z), τ T ( t , z , w ) = ( t 1 , z , w + t + 1 t ) , τ U ( t , z , w ) = ( t , z + 1 , w + z ) , which have order 4 and commute. More precisely, τ T \tau_{T} τ T ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ ⟨ 3 ⋅ 51 ⟩ \langle3\cdot51\rangle ⟨ 3 ⋅ 51 ⟩ τ U \tau_{U} τ U ⟨ 6 ⟩ \langle6\rangle ⟨ 6 ⟩ ⟨ 6 ⋅ 51 ⟩ \langle6\cdot51\rangle ⟨ 6 ⋅ 51 ⟩ τ T 2 = τ U 2 \tau_{T}^{2}=\tau_{U}^{2} τ T 2 = τ U 2 ⟨ 51 ⟩ \langle51\rangle ⟨ 51 ⟩ ( t , z , w ) ↦ ( t , z , w + 1 ) (t,z,w)\mapsto(t,z,w+1) ( t , z , w ) ↦ ( t , z , w + 1 )
Proof. The element − 4 ≡ 61 ( m o d 65 ) -4 \equiv61 \pmod{65} − 4 ≡ 61 ( mod 65 ) ≡ 1 ( m o d 5 ) \equiv1 \pmod{5} ≡ 1 ( mod 5 ) ⟨ 4 ⟩ \langle4\rangle ⟨ 4 ⟩ u ≠ ± 1 u \ne\pm1 u = ± 1 u 3 = 1 u^{3}=1 u 3 = 1 P 5 P_{5} P 5 3 P 5 = ( u 2 + u + 1 ) P 5 = 0 3P_{5}=(u^{2}+u+1)P_{5}=0 3 P 5 = ( u 2 + u + 1 ) P 5 = 0 61 ≠ ± 1 ( m o d 13 ) 61 \ne\pm1 \pmod{13} 61 = ± 1 ( mod 13 ) § 2 as in Lemma 8.4 that no cusp is fixed either, also in characteristic 2, so the quotient map is étale and Riemann–Hurwitz gives 2 g ( Y ) − 2 = ( 2 ⋅ 121 − 2 ) / 3 2g(Y)-2=(2\cdot121-2)/3 2 g ( Y ) − 2 = ( 2 ⋅ 121 − 2 ) /3 g ( Y ) = 41 g(Y)=41 g ( Y ) = 41 X 1 ( 65 ) / ⟨ 4 , 51 ⟩ X_{1}(65)/\langle4,51\rangle X 1 ( 65 ) / ⟨ 4 , 51 ⟩ 51 ≡ 1 ( m o d 5 ) 51\equiv1\pmod{5} 51 ≡ 1 ( mod 5 ) 51 ≡ − 1 ( m o d 13 ) 51\equiv-1\pmod{13} 51 ≡ − 1 ( mod 13 ) X 1 ( 5 ) X_{1}(5) X 1 ( 5 ) X 1 ( 13 ) / ⟨ 3 ⟩ X_{1}(13)/\langle3\rangle X 1 ( 13 ) / ⟨ 3 ⟩ j j j ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ Y Y Y F 2 \mathbb{F}_{2} F 2 y 2 + x y = x 3 + a 5 x 2 + j − 1 y^{2}+xy=x^{3}+a_{5}x^{2}+j^{-1} y 2 + x y = x 3 + a 5 x 2 + j − 1 y 2 + x y = x 3 + a 13 x 2 + j − 1 y^{2}+xy=x^{3}+a_{13}x^{2}+j^{-1} y 2 + x y = x 3 + a 13 x 2 + j − 1 w 2 + w = a 5 + a 13 w^{2}+w=a_{5}+a_{13} w 2 + w = a 5 + a 13 τ T \tau_{T} τ T τ U \tau_{U} τ U a 5 ( t − 1 ) + a 5 ( t ) = ( t / ( t + 1 ) ) 2 + t / ( t + 1 ) a_{5}(t^{-1})+a_{5}(t)=(t/(t+1))^{2}+t/(t+1) a 5 ( t − 1 ) + a 5 ( t ) = ( t / ( t + 1 ) ) 2 + t / ( t + 1 ) a 13 ( z + 1 ) + a 13 ( z ) = z 2 + z a_{13}(z+1)+a_{13}(z)=z^{2}+z a 13 ( z + 1 ) + a 13 ( z ) = z 2 + z
We verify that the fibre product is irreducible, that Y Y Y τ T \tau_{T} τ T τ U \tau_{U} τ U
It remains to identify τ T \tau_{T} τ T τ U \tau_{U} τ U X 1 ( 65 ) / ⟨ 4 , 51 ⟩ X_{1}(65)/\langle4,51\rangle X 1 ( 65 ) / ⟨ 4 , 51 ⟩ X 1 ( 5 ) X_{1}(5) X 1 ( 5 ) X 1 ( 13 ) / ⟨ 3 ⟩ X_{1}(13)/\langle3\rangle X 1 ( 13 ) / ⟨ 3 ⟩ ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ t t t a m o d 5 a\bmod5 a mod 5 z z z a m o d 13 a\bmod13 a mod 13 X 1 ( 5 ) X_{1}(5) X 1 ( 5 ) ⟨ 2 ⟩ = ⟨ 3 ⟩ \langle2\rangle=\langle3\rangle ⟨ 2 ⟩ = ⟨ 3 ⟩ t ↦ − 1 / t t\mapsto-1/t t ↦ − 1/ t t ↦ 1 / t t\mapsto1/t t ↦ 1/ t F 2 \mathbb{F}_{2} F 2 X 1 ( 13 ) / ⟨ 3 ⟩ X_{1}(13)/\langle3\rangle X 1 ( 13 ) / ⟨ 3 ⟩ F 2 \mathbb{F}_{2} F 2 ⟨ 2 ⟩ \langle2\rangle ⟨ 2 ⟩ z ↦ z + 1 z\mapsto z+1 z ↦ z + 1 3 ≡ − 2 ( m o d 5 ) 3\equiv-2\pmod{5} 3 ≡ − 2 ( mod 5 ) ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ X 1 ( 13 ) / ⟨ 3 ⟩ X_{1}(13)/\langle3\rangle X 1 ( 13 ) / ⟨ 3 ⟩ 6 ≡ 1 ( m o d 5 ) 6\equiv1\pmod{5} 6 ≡ 1 ( mod 5 ) ⟨ 6 ⟩ = ⟨ 2 ⟩ ⟨ 3 ⟩ \langle6\rangle=\langle2\rangle\langle3\rangle ⟨ 6 ⟩ = ⟨ 2 ⟩ ⟨ 3 ⟩ X 1 ( 13 ) X_{1}(13) X 1 ( 13 ) ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ ⟨ 6 ⟩ \langle6\rangle ⟨ 6 ⟩ X 1 ( 65 ) / ⟨ 4 , 51 ⟩ X_{1}(65)/\langle4,51\rangle X 1 ( 65 ) / ⟨ 4 , 51 ⟩ ( t , z ) ↦ ( 1 / t , z ) (t,z)\mapsto(1/t,z) ( t , z ) ↦ ( 1/ t , z ) ( t , z ) ↦ ( t , z + 1 ) (t,z)\mapsto(t,z+1) ( t , z ) ↦ ( t , z + 1 ) Y Y Y X 1 ( 65 ) / ⟨ 4 , 51 ⟩ X_{1}(65)/\langle4,51\rangle X 1 ( 65 ) / ⟨ 4 , 51 ⟩ Y Y Y X 1 ( 65 ) / ⟨ 4 , 51 ⟩ X_{1}(65)/\langle4,51\rangle X 1 ( 65 ) / ⟨ 4 , 51 ⟩ ⟨ 51 ⟩ \langle51\rangle ⟨ 51 ⟩ w ↦ w + 1 w\mapsto w+1 w ↦ w + 1 τ T \tau_{T} τ T ⟨ 3 ⟩ \langle3\rangle ⟨ 3 ⟩ ⟨ 3 ⟩ ⟨ 51 ⟩ \langle3\rangle\langle51\rangle ⟨ 3 ⟩ ⟨ 51 ⟩ τ U \tau_{U} τ U ⟨ 6 ⟩ \langle6\rangle ⟨ 6 ⟩ ⟨ 6 ⟩ ⟨ 51 ⟩ \langle6\rangle\langle51\rangle ⟨ 6 ⟩ ⟨ 51 ⟩ τ T 2 = τ U 2 = ( t , z , w ) ↦ ( t , z , w + 1 ) \tau_{T}^{2}=\tau_{U}^{2}=(t,z,w)\mapsto(t,z,w+1) τ T 2 = τ U 2 = ( t , z , w ) ↦ ( t , z , w + 1 ) τ T \tau_{T} τ T τ U \tau_{U} τ U □ \square □
Proposition 9.5. The group Z / 65 Z \mathbb{Z}/65\mathbb{Z} Z /65 Z
Proof. Let E E E K K K D D D Y = X 1 ( 65 ) / ⟨ 4 ⟩ Y=X_{1}(65)/\langle4\rangle Y = X 1 ( 65 ) / ⟨ 4 ⟩ Proposition 9.1 and Lemma 2.2 , D D D C ‾ \overline{C} C Y F 2 Y_{\mathbb{F}_{2}} Y F 2 Y F 2 Y_{\mathbb{F}_{2}} Y F 2 1 8 1^{8} 1 8 2 4 2^{4} 2 4 4 4 4^{4} 4 4 ψ 1 = τ T − 1 \psi_1=\tau_T-1 ψ 1 = τ T − 1 ψ 2 = τ U − 1 \psi_2=\tau_U-1 ψ 2 = τ U − 1 J ( Y ) ( Q ) J(Y)(\mathbb{Q}) J ( Y ) ( Q ) Lemma 9.4 we view τ T \tau_T τ T τ U \tau_U τ U Y Y Y Lemma 9.4 , that ψ 1 ∗ H 0 ( Y F 2 , Ω 1 ) + ψ 2 ∗ H 0 ( Y F 2 , Ω 1 ) \psi_1^*H^0(Y_{\mathbb{F}_2},\Omega^1)+\psi_2^*H^0(Y_{\mathbb{F}_2},\Omega^1) ψ 1 ∗ H 0 ( Y F 2 , Ω 1 ) + ψ 2 ∗ H 0 ( Y F 2 , Ω 1 ) U U U C − 1 C-1 C − 1 U U U Proposition 8.2 no such C ‾ \overline{C} C D D D
The sporadic groups and the proof of the main theorem
We now describe the four sporadic groups. Recall that for b , c b,c b , c
E ( b , c ) : y 2 + ( 1 − c ) x y − b y = x 3 − b x 2 (4) E(b,c):\quad y^2+(1-c)xy-by=x^3-bx^2 \tag*{(4)} E ( b , c ) : y 2 + ( 1 − c ) x y − b y = x 3 − b x 2 (4) in Tate normal form has the point ( 0 , 0 ) (0,0) ( 0 , 0 )
E ′ ( b , c ) : y 2 = x 3 + c x 2 + ( 1 − b − c ) b x (5) E'(b,c):\quad y^2=x^3+cx^2+(1-b-c)bx \tag*{(5)} E ′ ( b , c ) : y 2 = x 3 + c x 2 + ( 1 − b − c ) b x (5) has the point P = ( 0 , 0 ) P=(0,0) P = ( 0 , 0 ) Q = ( b , b ) Q=(b,b) Q = ( b , b )
Proposition 10.1. Let E 0 : y 2 = x 3 − 10395 x + 444150 E_0:y^2=x^3-10395x+444150 E 0 : y 2 = x 3 − 10395 x + 444150 1225. b 2 1225.b2 1225. b 2 w w w
f 37 = x 6 − 3150 x 5 + 796635 x 4 − 75770100 x 3 + 3111596775 x 2 − 44606598750 x − 85333003875 , f_{37}=x^6-3150x^5+796635x^4-75770100x^3+3111596775x^2-44606598750x-85333003875, f 37 = x 6 − 3150 x 5 + 796635 x 4 − 75770100 x 3 + 3111596775 x 2 − 44606598750 x − 85333003875 , put K 37 : = Q ( w ) K_{37}:=\mathbb{Q}(w) K 37 := Q ( w ) δ : = w 3 − 10395 w + 444150 \delta:=w^3-10395w+444150 δ := w 3 − 10395 w + 444150
E 37 : y 2 = x 3 − 10395 δ 2 x + 444150 δ 3 . E_{37}:\quad y^2=x^3-10395\delta^2x+444150\delta^3. E 37 : y 2 = x 3 − 10395 δ 2 x + 444150 δ 3 . Then K 37 K_{37} K 37 6.6.300125.1 6.6.300125.1 6.6.300125.1 ( δ w , δ 2 ) (\delta w,\delta^2) ( δ w , δ 2 ) E 37 E_{37} E 37 E 37 ( K 37 ) t o r s ≃ Z / 37 Z E_{37}(K_{37})_{\mathrm{tors}}\simeq\mathbb{Z}/37\mathbb{Z} E 37 ( K 37 ) tors ≃ Z /37 Z K K K E / K E/K E / K ( K , E ) (K,E) ( K , E ) ( K 37 , E 37 ) (K_{37},E_{37}) ( K 37 , E 37 ) Z / 37 Z \mathbb{Z}/37\mathbb{Z} Z /37 Z
Proof. Put g ( x ) = x 3 − 10395 x + 444150 g(x)=x^3-10395x+444150 g ( x ) = x 3 − 10395 x + 444150 E 0 : y 2 = g ( x ) E_0:y^2=g(x) E 0 : y 2 = g ( x ) δ = g ( w ) \delta=g(w) δ = g ( w ) ( x , y ) ↦ ( δ x , δ 2 y ) (x,y)\mapsto(\delta x,\delta^2y) ( x , y ) ↦ ( δ x , δ 2 y ) E ∗ : δ y 2 = g ( x ) E^*:\delta y^2=g(x) E ∗ : δ y 2 = g ( x ) E 37 E_{37} E 37 P = ( w , 1 ) P=(w,1) P = ( w , 1 ) ( δ w , δ 2 ) (\delta w,\delta^2) ( δ w , δ 2 ) K 37 K_{37} K 37 6.6.300125.1 6.6.300125.1 6.6.300125.1 P P P Q \mathbb{Q} Q E 0 E_0 E 0
As E ∗ E^* E ∗ E 0 E_0 E 0 x x x x r x_r x r r P rP r P 1 ≤ r ≤ 18 1\le r\le18 1 ≤ r ≤ 18 K 37 K_{37} K 37 Q ( x r ) = K 37 \mathbb{Q}(x_r)=K_{37} Q ( x r ) = K 37 r r r r P = ( x r , y r ) rP=(x_r,y_r) r P = ( x r , y r ) δ y r 2 = g ( x r ) \delta y_r^2=g(x_r) δ y r 2 = g ( x r ) y r ≠ 0 y_r\ne0 y r = 0 r P rP r P
Now let ( K , E ) (K,E) ( K , E ) [10 Proposition 1.4] we have j ( E ) = − 9317 = j ( E 0 ) j(E)=-9317=j(E_0) j ( E ) = − 9317 = j ( E 0 ) E E E E d : d y 2 = g ( x ) E_d:dy^2=g(x) E d : d y 2 = g ( x ) d ∈ K × d\in K^\times d ∈ K × E d E_d E d x x x x ′ ∈ K x'\in K x ′ ∈ K
Finally, we compute that f 37 f_{37} f 37 K 37 K_{37} K 37 E 37 E_{37} E 37 # E 37 ( K 37 ) t o r s \#E_{37}(K_{37})_{\mathrm{tors}} # E 37 ( K 37 ) tors
Proposition 10.2. (1) Let K 25 = Q ( a , b ) K_{25}=\mathbb{Q}(a,b) K 25 = Q ( a , b ) a 3 − a 2 + 1 = 0 a^3-a^2+1=0 a 3 − a 2 + 1 = 0 b 2 + ( a 2 − 2 a − 1 ) b + a = 0 b^2+(a^2-2a-1)b+a=0 b 2 + ( a 2 − 2 a − 1 ) b + a = 0
r = a 2 b − a b + b − 1 a ( a b − 1 ) , s = a b − b + 1 a b . r=\frac{a^2b-ab+b-1}{a(ab-1)},\qquad s=\frac{ab-b+1}{ab}. r = a ( ab − 1 ) a 2 b − ab + b − 1 , s = ab ab − b + 1 . and let E 25 = E ( r s ( r − 1 ) , s ( r − 1 ) ) E_{25}=E(rs(r-1),s(r-1)) E 25 = E ( r s ( r − 1 ) , s ( r − 1 )) (4) . Then ( 0 , 0 ) (0,0) ( 0 , 0 ) E 25 ( K 25 ) t o r s ≃ Z / 25 Z E_{25}(K_{25})_{\mathrm{tors}}\simeq\mathbb{Z}/25\mathbb{Z} E 25 ( K 25 ) tors ≃ Z /25 Z
(2) Let K 4 , 12 = Q ( α , i ) K_{4,12}=\mathbb{Q}(\alpha,i) K 4 , 12 = Q ( α , i ) α 3 − 3 α − 4 = 0 \alpha^3-3\alpha-4=0 α 3 − 3 α − 4 = 0 E 4 , 12 E_{4,12} E 4 , 12 y 2 + x y + y = x 3 − x 2 + 4 x − 1 y^2+xy+y=x^3-x^2+4x-1 y 2 + x y + y = x 3 − x 2 + 4 x − 1 K 4 , 12 K_{4,12} K 4 , 12 E 4 , 12 ( K 4 , 12 ) t o r s ≃ Z / 4 Z × Z / 12 Z E_{4,12}(K_{4,12})_{\mathrm{tors}}\simeq\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z} E 4 , 12 ( K 4 , 12 ) tors ≃ Z /4 Z × Z /12 Z
Proof. The point in (1) is the sextic point on X 1 ( 25 ) X_1(25) X 1 ( 25 ) [46 ] , and the curve in (2) was found by González-Jiménez and Najman [18 ] . We verify both claims directly in Magma [2 ] . □
Proposition 10.3. Let K 2 , 24 = Q ( w ) K_{2,24}=\mathbb{Q}(w) K 2 , 24 = Q ( w ) w 6 − 2 w 4 + 2 w 2 − 3 = 0 w^6-2w^4+2w^2-3=0 w 6 − 2 w 4 + 2 w 2 − 3 = 0 E 2 , 24 = E ′ ( b , c ) E_{2,24}=E'(b,c) E 2 , 24 = E ′ ( b , c ) (5) with
b = 1 32 ( 3 w 5 + 7 w 4 − 24 w 3 − 15 w 2 + 21 w + 18 ) , c = 1 16 ( − 7 w 5 + 5 w 4 + 24 w 3 − 5 w 2 − 17 w − 2 ) . b=\frac{1}{32}(3w^5+7w^4-24w^3-15w^2+21w+18),\qquad c=\frac{1}{16}(-7w^5+5w^4+24w^3-5w^2-17w-2). b = 32 1 ( 3 w 5 + 7 w 4 − 24 w 3 − 15 w 2 + 21 w + 18 ) , c = 16 1 ( − 7 w 5 + 5 w 4 + 24 w 3 − 5 w 2 − 17 w − 2 ) . (1) The point Q = ( b , b ) Q=(b,b) Q = ( b , b ) E 2 , 24 ( K 2 , 24 ) t o r s ≃ Z / 2 Z × Z / 24 Z E_{2,24}(K_{2,24})_{\mathrm{tors}}\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} E 2 , 24 ( K 2 , 24 ) tors ≃ Z /2 Z × Z /24 Z
(2) The curve X 1 ( 2 , 24 ) X_1(2,24) X 1 ( 2 , 24 ) Δ ( 24 ) \Delta(24) Δ ( 24 ) E 2 , 24 E_{2,24} E 2 , 24
(3) If K K K E / K E/K E / K Z / 2 Z × Z / 24 Z ⊆ E ( K ) \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}\subseteq E(K) Z /2 Z × Z /24 Z ⊆ E ( K ) ( K , E ) ≃ ( K 2 , 24 , E 2 , 24 ) (K,E)\simeq(K_{2,24},E_{2,24}) ( K , E ) ≃ ( K 2 , 24 , E 2 , 24 )
The curve E 2 , 24 E_{2,24} E 2 , 24 j j j K 2 , 24 K_{2,24} K 2 , 24 − 139 -139 − 139
Proof. Write K = K 2 , 24 K=K_{2,24} K = K 2 , 24 E = E 2 , 24 E=E_{2,24} E = E 2 , 24 # Aut ( K ) = 2 \#\operatorname{Aut}(K)=2 # Aut ( K ) = 2 w w w h ( x 2 ) h(x^2) h ( x 2 ) h ( t ) = t 3 − 2 t 2 + 2 t − 3 h(t)=t^3-2t^2+2t-3 h ( t ) = t 3 − 2 t 2 + 2 t − 3 3 t 2 − 4 t + 2 3t^2-4t+2 3 t 2 − 4 t + 2 h ( 0 ) < 0 h(0)<0 h ( 0 ) < 0 h h h h ( x 2 ) h(x^2) h ( x 2 ) K K K K K K w w w ± w \pm w ± w w ↦ − w w\mapsto-w w ↦ − w
We prove (2). The group E ( K ) t o r s ≃ Z / 2 Z × Z / 24 Z E(K)_{\mathrm{tors}}\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} E ( K ) tors ≃ Z /2 Z × Z /24 Z Q ′ Q' Q ′ ⟨ Q ′ ⟩ \langle Q'\rangle ⟨ Q ′ ⟩ ( P ′ , Q ′ ) (P',Q') ( P ′ , Q ′ ) E ( K ) t o r s E(K)_{\mathrm{tors}} E ( K ) tors Z / 2 Z × Z / 24 Z ↪ E ( K ) \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}\hookrightarrow E(K) Z /2 Z × Z /24 Z ↪ E ( K ) Y 1 ( 2 , 24 ) Y_1(2,24) Y 1 ( 2 , 24 ) E E E Aut ( E ) = { ± 1 } \operatorname{Aut}(E)=\{\pm1\} Aut ( E ) = { ± 1 } ( P ′ , Q ′ ) (P',Q') ( P ′ , Q ′ ) ( − P ′ , − Q ′ ) = ( P ′ , − Q ′ ) (-P',-Q')=(P',-Q') ( − P ′ , − Q ′ ) = ( P ′ , − Q ′ ) K K K Y 1 ( 2 , 24 ) Y_1(2,24) Y 1 ( 2 , 24 ) Z / 2 Z × Z / 24 Z ⊈ Φ ( d ) \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}\not\subseteq\Phi(d) Z /2 Z × Z /24 Z ⊆ Φ ( d ) d ∈ { 1 , 2 , 3 } d\in\{1,2,3\} d ∈ { 1 , 2 , 3 } K K K # Aut ( K ) = 2 \#\operatorname{Aut}(K)=2 # Aut ( K ) = 2 K K K E E E D D D δ ∈ Δ ( 24 ) \delta\in\Delta(24) δ ∈ Δ ( 24 ) i i i δ D ∼ L i \delta D\sim L_i δ D ∼ L i ∣ L i ∣ |L_i| ∣ L i ∣ 2 ⋅ # Δ ( 24 ) = 8 2\cdot\#\Delta(24)=8 2 ⋅ #Δ ( 24 ) = 8 E E E ∣ L 1 ∣ |L_1| ∣ L 1 ∣ ∣ L 2 ∣ |L_2| ∣ L 2 ∣ K K K
For (3), let ( K ′ , E ′ ) (K', E') ( K ′ , E ′ ) X 1 ( 2 , 24 ) X_1(2, 24) X 1 ( 2 , 24 ) K K K K K K E E E X 1 ( 2 , 24 ) X_1(2, 24) X 1 ( 2 , 24 ) ( K ′ , E ′ ) ≃ ( K , E ) (K', E') \simeq(K, E) ( K ′ , E ′ ) ≃ ( K , E )
Proposition 10.4. The group Z / 48 Z \mathbb{Z}/48\mathbb{Z} Z /48 Z
Proof. Suppose E 1 E_1 E 1 K K K P 1 P_1 P 1 Lemma 3.2 , the curve E ′ = E 1 / ⟨ 24 P 1 ⟩ E' = E_1/\langle24P_1\rangle E ′ = E 1 / ⟨ 24 P 1 ⟩ Z / 2 Z × Z / 24 Z ⊆ E ′ ( K ) \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/24\mathbb{Z} \subseteq E'(K) Z /2 Z × Z /24 Z ⊆ E ′ ( K ) Proposition 10.3 (3) we may assume that K = K 2 , 24 K = K_{2,24} K = K 2 , 24 E ′ = E 2 , 24 E' = E_{2,24} E ′ = E 2 , 24 ϕ : E 1 → E ′ \phi: E_1 \to E' ϕ : E 1 → E ′ ker ϕ ^ = ϕ ( E 1 [ 2 ] ) \ker\hat{\phi} = \phi(E_1[2]) ker ϕ ^ = ϕ ( E 1 [ 2 ]) K K K E 1 E_1 E 1 E 2 , 24 E_{2,24} E 2 , 24 K K K K K K Z / 24 Z \mathbb{Z}/24\mathbb{Z} Z /24 Z Z / 2 Z × Z / 12 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/12\mathbb{Z} Z /2 Z × Z /12 Z Z / 24 Z \mathbb{Z}/24\mathbb{Z} Z /24 Z
Proof of Theorems 1.1 and 1.2. By Propositions 10.1 to 10.3, together with [14 Theorem 1.1] , every group in B \mathcal{B} B G G G G ∉ B G \notin\mathcal{B} G ∈ / B Lemma 3.1 G G G G \mathcal{G} G Table 1 lists, for each of the 66 members of G \mathcal{G} G Φ ( 6 ) = B \Phi(6) = \mathcal{B} Φ ( 6 ) = B Theorem 1.1 . Theorem 1.2 follows from Propositions 10.1 to 10.3. □
Remark 10.5. The sextic points on X 1 ( 37 ) X_1(37) X 1 ( 37 ) X 1 ( 2 , 24 ) X_1(2, 24) X 1 ( 2 , 24 ) X 1 ( 37 ) X_1(37) X 1 ( 37 ) gon Q X 1 ( 37 ) = 18 \operatorname{gon}_{\mathbb{Q}} X_1(37) = 18 gon Q X 1 ( 37 ) = 18 [15 Theorem 1] and Frey’s theorem [17 ] , and for X 1 ( 2 , 24 ) X_1(2, 24) X 1 ( 2 , 24 ) rk J 1 ( 2 , 24 ) ( Q ) = 0 \operatorname{rk} J_1(2, 24)(\mathbb{Q}) = 0 rk J 1 ( 2 , 24 ) ( Q ) = 0 gon Q X 1 ( 2 , 24 ) > 6 \operatorname{gon}_{\mathbb{Q}} X_1(2, 24) > 6 gon Q X 1 ( 2 , 24 ) > 6 [14 Proposition 2.3] . The curve E 2 , 24 E_{2,24} E 2 , 24 Z / 2 Z × Z / 18 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} Z /2 Z × Z /18 Z [36 ] . The points of degree 6 that give the four sporadic groups are listed in [37 ] as 1.25.6.a, 1.37.6.a, 4.12.6.a and 2.24.6.a.
Remark 10.6. We do not know whether E 25 E_{25} E 25 E 4 , 12 E_{4,12} E 4 , 12 Z / 25 Z \mathbb{Z}/25\mathbb{Z} Z /25 Z Z / 4 Z × Z / 12 Z \mathbb{Z}/4\mathbb{Z} \times\mathbb{Z}/12\mathbb{Z} Z /4 Z × Z /12 Z X 1 ( 25 ) X_1(25) X 1 ( 25 ) X 1 ( 4 , 12 ) X_1(4,12) X 1 ( 4 , 12 ) Q ( i ) \mathbb{Q}(i) Q ( i ) Theorem 1.1 : all the groups properly containing these two groups are ruled out directly.
The computations
Our computations were performed in Magma V2.30-1 [2 ] , apart from some auxiliary ones in Python: the list of groups in Lemma 3.1 , the values of Abramovich’s bound and the orders of diamond operators that we quote, the queries to the LMFDB, some checks of orbits of cusps, and the combination and replay of data exported from Magma. They were run on the Mordell server at the Department of Mathematics, University of Zagreb, with an AMD EPYC 9175F CPU and 384 GB of RAM. We used the package mdmagma [33 ] for the moduli-theoretic operations, Sutherland’s optimized models of X 1 ( n ) X_1(n) X 1 ( n ) [44 , 45 ] , the Derickx–Sutherland models of X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) [43 ] for an independent count of points on modular curves over finite fields. The Hecke operators on closed points of X 1 ( m , n ) F p X_1(m,n)_{\mathbb{F}_p} X 1 ( m , n ) F p [36 §10] , extended to m ≥ 3 m \ge3 m ≥ 3 Proposition 9.5 took about 10 hours.
The completeness of the enumeration of closed points of degree at most 6, which every sieve computation assumes, was checked independently of the models by counting the isomorphism classes of elliptic curves with the required level structure over F p d \mathbb{F}_{p^d} F p d d ≤ 6 d \le6 d ≤ 6 # Y 1 ( m , n ) ( F p d ) = ∑ e ∣ d e ⋅ n e \#Y_1(m,n)(\mathbb{F}_{p^d}) = \sum_{e\mid d} e \cdot n_e # Y 1 ( m , n ) ( F p d ) = ∑ e ∣ d e ⋅ n e n e n_e n e e e e independent_counts.m, count_short_sieves.m, count_42_gl2.m and verify_35.m for the curves of Table 4 ).
For the check in Remark 6.4 we need that A 2 A_2 A 2 J 1 ( 49 ) ( Q ) J_1(49)(\mathbb{Q}) J 1 ( 49 ) ( Q ) [7 Theorem 3.1(2), Corollary 4.14 and Table 2] this group is finite with invariant factors [ 7 , 52367710906884085342 ] [7, 52367710906884085342] [ 7 , 52367710906884085342 ] { 0 , z } \{0,z\} { 0 , z } z z z A 2 A_2 A 2 z z z A 5 A_5 A 5 J 1 ( 49 ) [ 2 ] J_1(49)[2] J 1 ( 49 ) [ 2 ] H 1 ( J 1 ( 49 ) ( C ) , Z ) / 2 H 1 ( J 1 ( 49 ) ( C ) , Z ) ≃ J 1 ( 49 ) [ 2 ] H_1(J_1(49)(\mathbb{C}),\mathbb{Z})/2H_1(J_1(49)(\mathbb{C}),\mathbb{Z}) \simeq J_1(49)[2] H 1 ( J 1 ( 49 ) ( C ) , Z ) /2 H 1 ( J 1 ( 49 ) ( C ) , Z ) ≃ J 1 ( 49 ) [ 2 ] T 2 T_2 T 2 T 2 T_2 T 2 ⟨ 2 ⟩ \langle2\rangle ⟨ 2 ⟩ { 0 , z } \{0,z\} { 0 , z } T 2 z = ⟨ 2 ⟩ z = z T_2z = \langle2\rangle z = z T 2 z = ⟨ 2 ⟩ z = z A 2 z = − 2 z = 0 A_2z = -2z = 0 A 2 z = − 2 z = 0 [ A 2 ( D ) ] ≠ 0 [A_2(D)] \ne0 [ A 2 ( D )] = 0 A 5 A_5 A 5
We checked the computations on which the proofs depend most heavily a second time, with code written independently of the first implementation. These are the explicit examples of § 10 (see verify_sporadic_curves.m), the list of Lemma 3.1 (see group_list.py) and the analytic ranks and character values used in § 4 , § 8 and 9, read directly from the LMFDB (see lmfdb_ranks.py). In the folder independent of the repository we recomputed the 2-torsion computations of Remark 6.4 and Lemma 9.2 (in both conventions for the action of the Hecke algebra on homology), and, with models of the curves constructed from scratch, the doubled cuspidal computations of Proposition 5.5 for X 1 ( 45 ) X_1(45) X 1 ( 45 ) X 1 ( 3 , 21 ) X_1(3,21) X 1 ( 3 , 21 ) Proposition 8.5 . For Proposition 9.5 the computation in this folder is the one used in the proof, and the original one (cusp65_genus41_rr_transport_full.m) gives the same dimensions 35 and 28, and rank 6 on the 1792 of these divisors that are compatible with the fields of definition of the cusps of X 1 ( 65 ) X_1(65) X 1 ( 65 )
University of Zagreb Faculty of Science, Department of Mathematics, Bijenička cesta 30, 10000 Zagreb, Croatia
Email address: fnajman@math.hr
