Dedicated with gratitude and admiration to my advisor Andrej Dujella on the occasion of his 60th birthday.
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 [27 ] 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. Determining Φ ( d ) \Phi(d) Φ ( d ) Φ ( 1 ) \Phi(1) Φ ( 1 ) [22 ] , and Φ ( 2 ) \Phi(2) Φ ( 2 ) [17 ] , building on work of Kenku and Momose [21 ] . The set Φ ( 3 ) \Phi(3) Φ ( 3 ) [7 ] , building on work of Jeon, Kim and Schweizer [16 ] , Parent [29 ] and Najman [28 ] , and Φ ( 4 ) \Phi(4) Φ ( 4 ) [9 ] , building on work of Jeon, Kim and Park [15 ] and Bruin and Najman [4 ] . In this paper we determine Φ ( 5 ) \Phi(5) Φ ( 5 )
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 d d [9 §1] , a point of degree d d d X X X X X X
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. at most d d d Φ ∞ ( d ) = Φ ( d ) \Phi^{\infty}(d)=\Phi(d) Φ ∞ ( d ) = Φ ( d ) d = 1 , 2 d=1,2 d = 1 , 2 Z / 21 Z \mathbb{Z}/21\mathbb{Z} Z /21 Z Q ( ζ 9 ) + \mathbb{Q}(\zeta_{9})^{+} Q ( ζ 9 ) + [28 , 7 ] , and there are no sporadic torsion groups in degree 4 [9 ] . The set Φ ∞ ( 5 ) \Phi^{\infty}(5) Φ ∞ ( 5 ) [12 Theorem 1.1] , the cyclic part being due to Derickx and van Hoeij [13 Theorem 3] :
Φ ∞ ( 5 ) = { Z / n Z : 1 ≤ n ≤ 25 , n ≠ 23 } ∪ { Z / 2 Z × Z / 2 n Z : 1 ≤ n ≤ 8 } . (1) \Phi^{\infty}(5)=\{\mathbb{Z}/n\mathbb{Z}:1\le n\le25,\ n\ne23\}\cup\{\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2n\mathbb{Z}:1\le n\le8\}. \tag*{(1)} Φ ∞ ( 5 ) = { Z / n Z : 1 ≤ n ≤ 25 , n = 23 } ∪ { Z /2 Z × Z /2 n Z : 1 ≤ n ≤ 8 } . (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 , 25 , 28 , 30 , Z / 2 Z × Z / 2 n Z , n = 1 , … , 9. \begin{aligned}
\mathbb{Z}/n\mathbb{Z},&\qquad n=1,\ldots,22,\ 24,\ 25,\ 28,\ 30,\\
\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2n\mathbb{Z},&\qquad n=1,\ldots,9.
\end{aligned} Z / n Z , Z /2 Z × Z /2 n Z , n = 1 , … , 22 , 24 , 25 , 28 , 30 , n = 1 , … , 9. Equivalently, Φ ( 5 ) = Φ ∞ ( 5 ) ∪ { Z / 28 Z , Z / 30 Z , Z / 2 Z × Z / 18 Z } \Phi(5)=\Phi^{\infty}(5)\cup\{\mathbb{Z}/28\mathbb{Z},\mathbb{Z}/30\mathbb{Z},\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}\} Φ ( 5 ) = Φ ∞ ( 5 ) ∪ { Z /28 Z , Z /30 Z , Z /2 Z × Z /18 Z } Φ ∞ ( 5 ) \Phi^{\infty}(5) Φ ∞ ( 5 ) ( K , E ) (K,E) ( K , E ) ( K , E ) (K,E) ( K , E ) j j j Q \mathbb{Q} Q Theorem 1.2 .
Thus d = 5 d=5 d = 5 Z / 2 Z × Z / 18 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z} Z /2 Z × Z /18 Z d ≤ 5 d\le5 d ≤ 5 Φ ( d ) \Phi(d) Φ ( d ) § 9 .
Theorem 1.2. Let K K K E / K E/K E / K E ( K ) t o r s E(K)_{\mathrm{tors}} E ( K ) tors Φ ∞ ( 5 ) \Phi^{\infty}(5) Φ ∞ ( 5 ) ( K , E ) (K,E) ( K , E )
(1) E ( K ) t o r s ≅ Z / 28 Z E(K)_{\mathrm{tors}}\cong\mathbb{Z}/28\mathbb{Z} E ( K ) tors ≅ Z /28 Z K = Q ( α ) K=\mathbb{Q}(\alpha) K = Q ( α ) α 5 − α 4 − 2 α 3 − α 2 + 2 α + 2 = 0 \alpha^{5}-\alpha^{4}-2\alpha^{3}-\alpha^{2}+2\alpha+2=0 α 5 − α 4 − 2 α 3 − α 2 + 2 α + 2 = 0 E E E (6) in § 9 .
(2) E ( K ) t o r s ≅ Z / 30 Z E(K)_{\mathrm{tors}}\cong\mathbb{Z}/30\mathbb{Z} E ( K ) tors ≅ Z /30 Z K = Q ( α ) K=\mathbb{Q}(\alpha) K = Q ( α ) α 5 + α 4 − 3 α 3 + 3 α + 1 = 0 \alpha^{5}+\alpha^{4}-3\alpha^{3}+3\alpha+1=0 α 5 + α 4 − 3 α 3 + 3 α + 1 = 0 E E E (7) , (8) in § 9 , which are 2-isogenous over K K K
(3) E ( K ) t o r s ≅ Z / 2 Z × Z / 18 Z E(K)_{\mathrm{tors}}\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z} E ( K ) tors ≅ Z /2 Z × Z /18 Z K = Q ( w ) K=\mathbb{Q}(w) K = Q ( w ) w 5 − 2 w 4 + 2 w 3 − 3 w 2 − w + 4 = 0 w^{5}-2w^{4}+2w^{3}-3w^{2}-w+4=0 w 5 − 2 w 4 + 2 w 3 − 3 w 2 − w + 4 = 0 E E E (9) in § 9 .
In all three cases K K K ( 3 , 1 ) (3,1) ( 3 , 1 ) S 5 S_{5} S 5 j j j Q \mathbb{Q} Q Table 7 .
Table 7. The four exceptional pairs ( K , E ) (K,E) ( K , E ) Theorem 1.2 . Here f E f_E f E E E E N = N K / Q N=N_{K/\mathbb{Q}} N = N K / Q ( 3 , 1 ) (3,1) ( 3 , 1 ) S 5 S_5 S 5 [ Q ( j ) : Q ] = 5 [\mathbb{Q}(j):\mathbb{Q}]=5 [ Q ( j ) : Q ] = 5
The curves in cases (1) and (2) were found by van Hoeij [32 ] . What is new here is that his lists of quintic points on X 1 ( 28 ) X_{1}(28) X 1 ( 28 ) X 1 ( 30 ) X_{1}(30) X 1 ( 30 ) Z / 28 Z \mathbb{Z}/28\mathbb{Z} Z /28 Z Z / 30 Z \mathbb{Z}/30\mathbb{Z} Z /30 Z
Theorem 1.3. (1) X 1 ( 28 ) X_{1}(28) X 1 ( 28 ) Δ ( 28 ) \Delta(28) Δ ( 28 ) { P , − P } \{P,-P\} { P , − P } P P P E E E Theorem 1.2 (1).
(2) X 1 ( 30 ) X_1(30) X 1 ( 30 ) exactly 8 non-cuspidal closed points of degree 5 ; they form two free orbits under Δ ( 30 ) \Delta(30) Δ ( 30 ) { P , − P } \{P,-P\} { P , − P } P P P Theorem 1.2 (2).
(3) X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) exactly 18 non-cuspidal closed points of degree 5 ; they correspond to the 18 equivalence classes, modulo ± 1 \pm1 ± 1 Z / 2 Z × Z / 18 Z ↪ E ( K ) \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} \hookrightarrow E(K) Z /2 Z × Z /18 Z ↪ E ( K ) E E E Theorem 1.2 (3).
The residue field of each of these points is the corresponding quintic field K K K Theorem 1.2 .
Theorem 1.1 also settles, for every level at once, the question of which modular curves of the two families X 1 ( n ) X_1(n) X 1 ( n ) X 1 ( 2 , 2 n ) X_1(2,2n) X 1 ( 2 , 2 n )
Corollary 1.4. (1) X 1 ( n ) X_1(n) X 1 ( n ) n ∈ { 1 , … , 22 , 24 , 25 , 28 , 30 } n \in\{1,\ldots,22,24,25,28,30\} n ∈ { 1 , … , 22 , 24 , 25 , 28 , 30 } n ≤ 25 n \le25 n ≤ 25 n ≠ 23 n \ne23 n = 23 n = 28 n=28 n = 28 n = 30 n=30 n = 30
(2) X 1 ( 2 , 2 n ) X_1(2,2n) X 1 ( 2 , 2 n ) n ≤ 9 n \le9 n ≤ 9 n ≤ 8 n \le8 n ≤ 8 n = 9 n=9 n = 9
(3) In particular, X 1 ( 28 ) X_1(28) X 1 ( 28 ) X 1 ( 30 ) X_1(30) X 1 ( 30 )
The proof of Corollary 1.4 is given in § 9 , together with the reason why the word non-cuspidal cannot be dropped from part (3): the curve X 1 ( 44 ) X_1(44) X 1 ( 44 ) Z / 28 Z \mathbb{Z}/28\mathbb{Z} Z /28 Z Z / 30 Z \mathbb{Z}/30\mathbb{Z} Z /30 Z X 1 ( 28 ) X_1(28) X 1 ( 28 ) X 1 ( 30 ) X_1(30) X 1 ( 30 ) Z / 2 Z × Z / 18 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} Z /2 Z × Z /18 Z gon Q X 1 ( 2 , 18 ) = 4 \operatorname{gon}_{\mathbb{Q}} X_1(2,18)=4 gon Q X 1 ( 2 , 18 ) = 4 [3 Definition 4.1] . The example has a feature not seen in lower degrees: by [15 Theorem 3.6] we have Z / 2 Z × Z / 18 Z ∈ Φ ∞ ( 4 ) \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} \in\Phi^\infty(4) Z /2 Z × Z /18 Z ∈ Φ ∞ ( 4 ) Z / 2 Z × Z / 18 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} Z /2 Z × Z /18 Z ( K , E ) (K,E) ( K , E ) § 9 .
Outline of the proof and of the paper
Outline of the proof and of the paper. As in [9 ] and previous works, the classification reduces to showing that certain modular curves X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) § 3 we recall why 45 groups have to be analyzed. Nineteen of them were eliminated, for quintic fields, in [9 §7] , and four more, namely Z / 4 n Z \mathbb{Z}/4n\mathbb{Z} Z /4 n Z 9 ≤ n ≤ 12 9 \le n \le12 9 ≤ n ≤ 12 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 § 4 , with the numerical conditions checked for d = 5 d=5 d = 5 d = 4 d=4 d = 4 Z / 77 Z \mathbb{Z}/77\mathbb{Z} Z /77 Z Z / 85 Z \mathbb{Z}/85\mathbb{Z} Z /85 Z
The twenty-two groups that [9 ] does not reach are treated here. The group Z / 36 Z \mathbb{Z}/36\mathbb{Z} Z /36 Z Z / 2 Z × Z / 18 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} Z /2 Z × Z /18 Z
the criterion of [9 Proposition 5.2] for curves whose Jacobian has rank 0 (§ 5 );
the Hecke sieve of [9 Proposition 5.5] , together with [9 Lemma 5.10] , which controls the action of A q A_q A q F p \mathbb{F}_p F p Lemma 6.1 in the form needed for our computations and for the twisted operators of § 7 (§ 6 );
a twisted Hecke sieve for the positive analytic rank cases n = 57 , 63 , 65 n = 57, 63, 65 n = 57 , 63 , 65 A q A_q A q A q ( ⟨ a ⟩ − 1 ) A_q(\langle a\rangle- 1) A q (⟨ a ⟩ − 1 ) A q π ∗ π ∗ A_q\pi^*\pi_* A q π ∗ π ∗ H H H p p p § 7 );
a direct analysis of the cuspidal class group, which determines all non-cuspidal closed points of degree 5 on X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) X 1 ( 2 , 20 ) X_1(2,20) X 1 ( 2 , 20 ) X 1 ( 2 , 24 ) X_1(2,24) X 1 ( 2 , 24 ) X 1 ( 28 ) X_1(28) X 1 ( 28 ) X 1 ( 30 ) X_1(30) X 1 ( 30 ) X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) § 8 ).
The computations for n = 57 , 63 , 65 n = 57, 63, 65 n = 57 , 63 , 65 [9 ] and [26 ] . In § 10.1 we describe and prove correct two routines that made them possible: one computes T q T_q T q X 1 ( n ) F p X_1(n)_{\mathbb{F}_p} X 1 ( n ) F p G ⊆ E [ q ] G \subseteq E[q] G ⊆ E [ q ] G G G q q q ( E , a P ) (E,aP) ( E , a P ) X 1 ( 45 ) F 7 X_1(45)_{\mathbb{F}_7} X 1 ( 45 ) F 7 T 11 T_{11} T 11 § 9 assembles the proofs of Theorems 1.1 to 1.3, and § 10 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/quintic-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 use disclosure
AI models were used in the work leading to this paper. Anthropic’s Claude Fable 5.1 and Opus 5 were used for mathematical interactions and exploration, to write, audit and run the Magma and Python code and to organize the resulting data, and in the preparation of the manuscript. In particular, Fable 5.1 suggested Lemmas 10.1 and 10.2 and their proofs. Drafts of the manuscript and its computations were reviewed by OpenAI’s GPT-6 Astra and by Anthropic’s Claude Opus 5.5, which suggested several corrections and expositional improvements. The author takes full responsibility for the content of the paper.
Acknowledgements
I thank Pete Clark and Maarten Derickx for helpful comments.
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 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 Z [ 1 m n ] \mathbb{Z}[\frac{1}{m n}] Z [ mn 1 ] n > 4 n > 4 n > 4 n ≥ 18 n \geq18 n ≥ 18 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 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 = 1 m = 1 m = 1 m = 2 m = 2 m = 2 X X X Q \mathbb{Q} Q d ≥ 1 d \geq1 d ≥ 1 X ( d ) X^{(d)} X ( d ) d d d X X X X ( d ) ( Q ) X^{(d)}(\mathbb{Q}) X ( d ) ( Q ) Q \mathbb{Q} Q d d d X X X Cl Q 0 X \operatorname{Cl}_{\mathbb{Q}}^0 X Cl Q 0 X J ( Q ) J(\mathbb{Q}) J ( Q ) Q \mathbb{Q} Q
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 φ : Z / m Z × Z / n Z → G m × Z / k Z \varphi:\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}\to\mathbb{G}_m\times\mathbb{Z}/k\mathbb{Z} φ : Z / m Z × Z / n Z → G m × Z / k Z [9 §2] ). In particular all cusps are defined over Q ( ζ n ) \mathbb{Q}(\zeta_n) Q ( ζ n ) Gal ( Q ( ζ n ) / Q ) = ( Z / n Z ) × \operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})=(\mathbb{Z}/n\mathbb{Z})^\times Gal ( Q ( ζ n ) / Q ) = ( Z / n Z ) × G m \mathbb{G}_m G m c 0 ∈ X 1 ( m , n ) ( Z [ ζ 1 / n ] ) c_0\in X_1(m,n)(\mathbb{Z}[\zeta_{1/n}]) c 0 ∈ X 1 ( m , n ) ( Z [ ζ 1/ n ]) n n n ( a , b ) ↦ ( ζ m a , b ) (a,b)\mapsto(\zeta_m^a,b) ( a , b ) ↦ ( ζ m a , b ) [9 Lemma 2.8] ), which is Q \mathbb{Q} Q m ≤ 2 m\le2 m ≤ 2
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 ) n > 2 n>2 n > 2 [ PSL 2 ( Z ) : Γ ‾ 1 ( n ) ] = 1 2 [ SL 2 ( Z ) : Γ 1 ( n ) ] [\operatorname{PSL}_2(\mathbb{Z}):\overline{\Gamma}_1(n)]=\frac{1}{2}[\operatorname{SL}_2(\mathbb{Z}):\Gamma_1(n)] [ PSL 2 ( Z ) : Γ 1 ( n )] = 2 1 [ SL 2 ( Z ) : Γ 1 ( 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 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 T q T_q T q ⟨ q ⟩ \langle q\rangle ⟨ q ⟩ A q A_q A q q ∤ 2 n q\nmid2n q ∤ 2 n 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 ) 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. We will use the following facts from [9 §§2 and 5] .
Proposition 2.1. Let m ∈ { 1 , 2 } m\in\{1,2\} m ∈ { 1 , 2 } m ∣ n m\mid n m ∣ n q ∤ 2 n q\nmid2n q ∤ 2 n
(1) (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
(2) If q q q L L L A q ( J ( L ) t o r s ) = 0 A_q(J(L)_{\mathrm{tors}})=0 A q ( J ( L ) tors ) = 0
(3) A q ( c 0 ) = 0 A_q(c_0)=0 A q ( c 0 ) = 0
(4) If rk J ( Q ) = 0 \operatorname{rk}J(\mathbb{Q})=0 rk J ( Q ) = 0 [ A q ( D ) ] = 0 [A_q(D)]=0 [ A q ( D )] = 0 D ∈ X ( d ) ( Q ) D\in X^{(d)}(\mathbb{Q}) D ∈ X ( d ) ( Q ) d d d
Proof. These are, respectively, [9 (2.2)] , [9 Proposition 2.6] , [9 Lemma 2.8] and [9 Lemma 5.4] . □
We will also use the following results.
(Manin–Drinfeld [24 , 10 ] ) the class of a degree 0 cuspidal divisor on X X X J ( Q ) J(\mathbb{Q}) J ( Q )
(Katz [18 Appendix] ) if p > 2 p>2 p > 2 J J J p \mathfrak{p} p L L L p p p L L L p \mathfrak{p} p J ( L ) t o r s J(L)_{\mathrm{tors}} J ( L ) tors
(Parent [29 Proposition 2.4] ) if 2 is unramified in L L L 2 ∤ n 2\nmid n 2 ∤ n L L L J ( L ) t o r s J(L)_{\mathrm{tors}} J ( L ) tors
(Abramovich [1 Theorem 0.1] , with the bound λ 1 ≥ 975 / 4096 \lambda_1\ge975/4096 λ 1 ≥ 975/4096 [20 Appendix 2] ) for a congruence subgroup 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 ] ; (Frey [14 ] , Derickx–Sutherland [12 Propositions 2.2 and 2.3, Corollary 2.4] ) write d ( X ) d(X) d ( X ) d d d X X X d d d d ( X ) ≤ gon Q X ≤ 2 d ( X ) d(X)\le\operatorname{gon}_{\mathbb{Q}}X\le2d(X) d ( X ) ≤ gon Q X ≤ 2 d ( X ) rk J ( Q ) = 0 \operatorname{rk}J(\mathbb{Q})=0 rk J ( Q ) = 0 d ( X ) = gon Q X d(X)=\operatorname{gon}_{\mathbb{Q}}X d ( X ) = gon Q X X X X d d d Q ( X ) \mathbb{Q}(X) Q ( X ) d d d gon Q X ≤ d \operatorname{gon}_{\mathbb{Q}}X\le d gon Q X ≤ d X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 )
(Clark–Corn–Rice–Stankewicz [5 Sections 4.1–4.5] ) the torsion subgroup of a CM elliptic curve over a number field of degree at most 5 has exponent at most 21, so 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 )
for every n ≥ 22 n \ge22 n ≥ 22 d C M X 1 ( n ) d_{\mathrm{CMX1}}(n) d CMX1 ( n ) [6 ] .
Finally, we state the two standard reduction lemmas.
Lemma 2.2. Let m n > 4 mn > 4 mn > 4 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 ) p \mathfrak{p} p K K K p ∤ n p \nmid n p ∤ n E E E p \mathfrak{p} p
Proof. This is contained in the proof of [9 Proposition 5.2] . If E E E p \mathfrak{p} p a ⋅ # F p a \cdot\#\mathbb{F}_{\mathfrak{p}} a ⋅ # F p F p \mathbb{F}_{\mathfrak{p}} F p a ≤ 4 a \le4 a ≤ 4 # F p \#\mathbb{F}_{\mathfrak{p}} # F p p p p p p p E ( K ) E(K) E ( K ) 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 ∣ a ⋅ # F p mn \mid a \cdot\#\mathbb{F}_{\mathfrak{p}} mn ∣ a ⋅ # F p m ∣ n m \mid n m ∣ n p ∤ n p \nmid n p ∤ n gcd ( m n , p ) = 1 \gcd(mn,p)=1 g cd( mn , p ) = 1 m n ∣ a ≤ 4 mn \mid a \le4 mn ∣ a ≤ 4 m n > 4 mn > 4 mn > 4
Lemma 2.3 ([9 Lemma 5.3] ). Let p ∤ n p \nmid n p ∤ n d ′ ≤ d d' \le d d ′ ≤ d E ′ / F p d ′ E'/\mathbb{F}_{p^{d'}} E ′ / F p d ′ Z / m Z × Z / n Z ⊆ E ′ ( F p d ′ ) \mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z} \subseteq E'(\mathbb{F}_{p^{d'}}) Z / m Z × Z / n Z ⊆ E ′ ( F p d ′ ) E E E K K K d d d 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, we have checked the hypothesis by enumerating all elliptic curves over F p d ′ \mathbb{F}_{p^{d'}} F p d ′ j j j § 10 ). In most cases this is also visible from Waterhouse’s description [Wat69, Theorem 4.1] of the possible traces of Frobenius, as in [9 Corollary 4.6] .
The groups that have to be considered
Call a finite abelian group G G G 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 G ≃ Z / m Z × Z / n Z G \simeq\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
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 ∉ Φ ∞ ( 5 ) G \notin\Phi^\infty(5) G ∈ / Φ ∞ ( 5 ) H ∈ Φ ∞ ( 5 ) H \in\Phi^\infty(5) H ∈ Φ ∞ ( 5 ) H ⊊ G H \subsetneq G H ⊊ G
φ ( m ) ∣ 5 \varphi(m) \mid5 φ ( m ) ∣ 5
every prime divisor of m n mn mn
Then G \mathcal{G} G
Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n ∈ { 26 , 27 , 28 , 30 , 32 , 33 , 34 , 35 , 36 , 38 , 39 , 40 , 42 , 44 , 45 , 48 , 49 , 50 , 51 , 55 , 57 , 63 , 65 , 75 , 77 , 85 , 91 , 95 , 119 , 121 , 125 , 133 , 143 , 169 , 187 , 209 , 221 , 247 , 289 , 323 , 361 } n \in\{26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 48, 49, 50, 51, 55, 57, 63, 65, 75, 77, 85, 91, 95, 119, 121, 125, 133, 143, 169, 187, 209, 221, 247, 289, 323, 361\} n ∈ { 26 , 27 , 28 , 30 , 32 , 33 , 34 , 35 , 36 , 38 , 39 , 40 , 42 , 44 , 45 , 48 , 49 , 50 , 51 , 55 , 57 , 63 , 65 , 75 , 77 , 85 , 91 , 95 , 119 , 121 , 125 , 133 , 143 , 169 , 187 , 209 , 221 , 247 , 289 , 323 , 361 }
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 n ∈ { 9 , 10 , 11 , 12 } n \in\{9, 10, 11, 12\} n ∈ { 9 , 10 , 11 , 12 }
Moreover every group in Φ ( 5 ) ∖ Φ ∞ ( 5 ) \Phi(5) \setminus\Phi^\infty(5) Φ ( 5 ) ∖ Φ ∞ ( 5 ) G \mathcal{G} G
Proof. Conditions (2) and (3) are necessary for G G G Q ( ζ m ) ⊆ K \mathbb{Q}(\zeta_m) \subseteq K Q ( ζ m ) ⊆ K φ ( m ) ∣ 5 \varphi(m) \mid5 φ ( m ) ∣ 5 m ∈ { 1 , 2 } m \in\{1,2\} m ∈ { 1 , 2 } p ∣ m n p \mid mn p ∣ mn p p p p ∈ S ( 5 ) p \in S(5) p ∈ S ( 5 ) ≤ 19 \le19 ≤ 19 [8 Theorem 1.2] . The set Φ ∞ ( 5 ) \Phi^\infty(5) Φ ∞ ( 5 ) G ∉ Φ ∞ ( 5 ) G \notin\Phi^\infty(5) G ∈ / Φ ∞ ( 5 ) G \mathcal{G} G [9 §7] . □
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 1 ( m , n ) X_1(m,n) X 1 ( m , n ) ( E , K ) (E,K) ( E , K ) X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) K K K G ∉ Φ ( 1 ) G \notin\Phi(1) G ∈ / Φ ( 1 ) X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) G G G § 9 .
Three of the members of G \mathcal G G
Lemma 3.2. For 9 ≤ n ≤ 12 9 \le n \le12 9 ≤ n ≤ 12 Q \mathbb{Q} Q X 1 ( 4 n ) → X 1 ( 2 , 2 n ) X_1(4n) \to X_1(2,2n) X 1 ( 4 n ) → X 1 ( 2 , 2 n ) ( E , P ) ↦ ( E / ⟨ 2 n P ⟩ , Q , P m o d ⟨ 2 n P ⟩ ) (E,P) \mapsto(E/\langle2nP\rangle,Q,P\bmod\langle2nP\rangle) ( E , P ) ↦ ( E / ⟨ 2 n P ⟩ , Q , P mod ⟨ 2 n P ⟩) Q Q Q E [ 2 ] / ⟨ 2 n P ⟩ E[2]/\langle2nP\rangle E [ 2 ] / ⟨ 2 n P ⟩ X 1 ( 2 , 2 n ) X_1(2,2n) X 1 ( 2 , 2 n ) X 1 ( 4 n ) X_1(4n) X 1 ( 4 n ) Z / 4 n Z \mathbb{Z}/4n\mathbb{Z} Z /4 n Z
Proof. The morphism is the one described in [9 §2] (see also [12 §2] ). It maps non-cuspidal points to non-cuspidal points and does not increase the degree of a point. □
For n = 10 , 11 , 12 n=10,11,12 n = 10 , 11 , 12 § 6 and § 8 that X 1 ( 2 , 2 n ) X_1(2,2n) X 1 ( 2 , 2 n ) Z / 40 Z \mathbb{Z}/40\mathbb{Z} Z /40 Z Z / 44 Z \mathbb{Z}/44\mathbb{Z} Z /44 Z Z / 48 Z \mathbb{Z}/48\mathbb{Z} Z /48 Z n = 9 n=9 n = 9 Z / 36 Z \mathbb{Z}/36\mathbb{Z} Z /36 Z
The rest of the paper treats the 45 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 45 groups of Lemma 3.1.
The groups eliminated in [DN25]
Nineteen of the 45 groups of Lemma 3.1 were eliminated over quintic fields in [9 §7] , and three more follow from Lemma 3.2 once 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 n = 10 , 11 , 12 n=10,11,12 n = 10 , 11 , 12 d = 5 d=5 d = 5 d = 4 d=4 d = 4
The global method
4.1. The global method. The main tool is [9 Proposition 4.3] , due to Derickx: let n n n q q q q ∤ n q \nmid n q ∤ n H ⊆ ( Z / n Z ) × H \subseteq(\mathbb{Z}/n\mathbb{Z})^\times H ⊆ ( Z / n Z ) × − 1 -1 − 1 X H : = X 1 ( n ) / H X_H := X_1(n)/H X H := X 1 ( n ) / H J H : = Jac ( X H ) J_H := \operatorname{Jac}(X_H) J H := Jac ( X H ) a ∈ ( Z / n Z ) × / H a \in(\mathbb{Z}/n\mathbb{Z})^\times/H a ∈ ( Z / n Z ) × / H ( ⟨ a ⟩ − 1 ) J H ( Q ) \left(\langle a\rangle-1\right)J_H(\mathbb{Q}) ( ⟨ a ⟩ − 1 ) J H ( Q )
k q = { q + 1 if J H ( Q ) is finite , 2 q + 1 if a ∈ { q , q − 1 } , 2 ( q + 1 ) otherwise. k_q =
\begin{cases}
q+1 & \text{if } J_H(\mathbb{Q}) \text{ is finite},\\
2q+1 & \text{if } a \in\{q,q^{-1}\},\\
2(q+1) & \text{otherwise.}
\end{cases} k q = ⎩ ⎨ ⎧ q + 1 2 q + 1 2 ( q + 1 ) if J H ( Q ) is finite , if a ∈ { q , q − 1 } , otherwise. If k q d < gon Q X H k_qd < \operatorname{gon}_{\mathbb{Q}} X_H k q d < gon Q X H D ∈ X H ( d ) ( Q ) D \in X_H^{(d)}(\mathbb{Q}) D ∈ X H ( d ) ( Q ) ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ q = 2 q = 2 q = 2 ( ⟨ a ⟩ − 1 ) J H ( Q ) (\langle a\rangle- 1)J_H(\mathbb{Q}) (⟨ a ⟩ − 1 ) J H ( Q ) A 2 A_2 A 2 H = { ± 1 } H = \{\pm1\} H = { ± 1 } k q d < gon Q X 1 ( n ) k_qd < \operatorname{gon}_{\mathbb{Q}} X_1(n) k q d < gon Q X 1 ( n ) d < b ( n ) d < b(n) d < b ( n )
b ( n ) : = 325 2 15 ⋅ [ PSL 2 ( Z ) : Γ ‾ 1 ( n ) ] k q . (2) b(n) := \frac{325}{2^{15}} \cdot\frac{[\operatorname{PSL}_2(\mathbb{Z}) : \overline{\Gamma}_1(n)]}{k_q}. \tag*{(2)} b ( n ) := 2 15 325 ⋅ k q [ PSL 2 ( Z ) : Γ 1 ( n )] . (2) The diamond operators act freely on non-CM points, so if in addition ord ⟨ a ⟩ ∤ d \operatorname{ord}\langle a\rangle\nmid d ord ⟨ a ⟩ ∤ d ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ d d d d < d C M X 1 ( n ) d < d_{\mathrm{CM}}X_1(n) d < d CM X 1 ( n ) d = 5 d = 5 d = 5
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 ) ×
χ f ( a ) = 1 \chi_f(a) = 1 χ f ( a ) = 1 f f f n ′ n' n ′ n n n L ( f , 1 ) = 0 L(f,1) = 0 L ( f , 1 ) = 0 χ f \chi_f χ f f f f a a a ( Z / n ′ Z ) × (\mathbb{Z}/n'\mathbb{Z})^\times ( Z / n ′ Z ) ×
b ( n ) > 5 b(n) > 5 b ( n ) > 5 b ( n ) b(n) b ( n ) (2) with q = 3 q = 3 q = 3 k 3 k_3 k 3 a a a
ord ⟨ a ⟩ ∤ 5 \operatorname{ord}\langle a\rangle\nmid5 ord ⟨ a ⟩ ∤ 5 Δ ( n ) = ( Z / n Z ) × / { ± 1 } \Delta(n) = (\mathbb{Z}/n\mathbb{Z})^\times/\{\pm1\} Δ ( n ) = ( Z / n Z ) × / { ± 1 }
d C M X 1 ( n ) > 5 d_{\mathrm{CM}}X_1(n) > 5 d CM X 1 ( n ) > 5
Then X 1 ( n ) X_1(n) X 1 ( n ) Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z
Proof. We first show that condition (1) implies that ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle- 1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) 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 ) ∈ End ( A [ f ] ) ⊗ Q \chi_f(a) \in\operatorname{End}(A_{[f]}) \otimes\mathbb{Q} χ f ( a ) ∈ End ( A [ f ] ) ⊗ Q 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 ) [19 Corollary 14.3] , and for the remaining f f f χ f ( a ) = 1 \chi_f(a) = 1 χ f ( a ) = 1 ⟨ a ⟩ − 1 \langle a\rangle- 1 ⟨ a ⟩ − 1 A [ f ] A_{[f]} A [ f ] ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle- 1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) [9 Proposition 4.3] and the corollary following it, applied with d = 5 d = 5 d = 5 q = 3 q = 3 q = 3 H = { ± 1 } H = \{\pm1\} H = { ± 1 } □ \square □
Proposition 4.2. The groups Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z
n ∈ { 91 , 95 , 119 , 121 , 125 , 133 , 143 , 169 , 187 , 209 , 221 , 247 , 289 , 323 , 361 } n \in\{91, 95, 119, 121, 125, 133, 143, 169, 187, 209, 221, 247, 289, 323, 361\} n ∈ { 91 , 95 , 119 , 121 , 125 , 133 , 143 , 169 , 187 , 209 , 221 , 247 , 289 , 323 , 361 }
are not quintic torsion groups.
Proof. We apply Proposition 4.1 with q = 3 q = 3 q = 3 Table 2 . Note that 3 ∤ n 3 \nmid n 3 ∤ n d C M X 1 ( n ) d_{\mathrm{CM}}X_1(n) d CM X 1 ( n ) [6 ] , and all of them are > 5 > 5 > 5 b ( n ) b(n) b ( n ) (2) , and all of them are > 5 > 5 > 5 ord ⟨ a ⟩ \operatorname{ord}\langle a\rangle ord ⟨ a ⟩ Δ ( n ) \Delta(n) Δ ( n ) [23 ] . For 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 91. g 91.g 91. g 91. u 91.u 91. u 95. l 95.l 95. l 133. f 133.f 133. f 133. s 133.s 133. s 143. h 143.h 143. h 187. g 187.g 187. g 209. f 209.f 209. f 247. b e 247.be 247. b e 323. k 323.k 323. k a a a Table 2 lies in the kernel of all of them. All analytic ranks used are recorded in the LMFDB as proved. □ \square □
Table 2. Data for Proposition 4.1 : b ( n ) b(n) b ( n ) ord ⟨ a ⟩ \operatorname{ord}\langle a\rangle ord ⟨ a ⟩ Δ ( n ) \Delta(n) Δ ( n ) [9 Tables 2 and 3] , except for n = 289 n=289 n = 289 [9 Table 2] records the order 272 of 3 in ( Z / 289 Z ) × (\mathbb{Z}/289\mathbb{Z})^\times ( Z /289 Z ) × a a a [9 Tables 2 and 3] . All of them except a = 2 a=2 a = 2 n = 121 n=121 n = 121 a = 67 a=67 a = 67 n = 143 n=143 n = 143 n n n
Remark 4.3. The cases n ∈ { 91 , 121 , 143 , 169 , 187 , 221 , 289 } n\in\{91,121,143,169,187,221,289\} n ∈ { 91 , 121 , 143 , 169 , 187 , 221 , 289 } Proposition 4.2 are the ones covered by [9 Remark 6.2] , and the cases n ∈ { 95 , 119 , 125 , 133 , 209 , 247 , 323 , 361 } n\in\{95,119,125,133,209,247,323,361\} n ∈ { 95 , 119 , 125 , 133 , 209 , 247 , 323 , 361 } [9 Theorem 7.1] . Both are stated there for all degrees d ≤ ⌈ b ( n ) ⌉ d\le\lceil b(n)\rceil d ≤ ⌈ b ( n )⌉ d ≤ ⌊ b ( n ) ⌋ d\le\lfloor b(n)\rfloor d ≤ ⌊ b ( n )⌋
The criterion behind both is [9 Proposition 4.3] , whose hypothesis [9 (4.1)] reads d < b ( n ) d<b(n) d < b ( n ) b ( n ) b(n) b ( n ) [9 Tables 2 and 3] is an integer, this is the range d ≤ ⌊ b ( n ) ⌋ = ⌈ b ( n ) ⌉ − 1 d\le\lfloor b(n)\rfloor=\lceil b(n)\rceil-1 d ≤ ⌊ b ( n )⌋ = ⌈ b ( n )⌉ − 1
For d = 5 d=5 d = 5 ⌈ b ( n ) ⌉ = 5 \lceil b(n)\rceil=5 ⌈ b ( n )⌉ = 5 n = 77 n=77 n = 77 n = 85 n=85 n = 85 b ( 77 ) = 14625 3584 = 4.081 … b(77)=\frac{14625}{3584}=4.081\ldots b ( 77 ) = 3584 14625 = 4.081 … b ( 85 ) = 8775 2048 = 4.285 … b(85)=\frac{8775}{2048}=4.285\ldots b ( 85 ) = 2048 8775 = 4.285 … Z / 77 Z \mathbb{Z}/77\mathbb{Z} Z /77 Z Z / 85 Z \mathbb{Z}/85\mathbb{Z} Z /85 Z b ( n ) b(n) b ( n ) Table 2 exceeds 5, so Proposition 4.2 is unaffected.
Rank zero, cuspidal reduction at 2
The remaining four groups from [9 §7] are handled by [9 Proposition 7.3] .
Proposition 4.4 ([9 Proposition 7.3] ). The groups Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n ∈ { 49 , 51 , 55 , 75 } n\in\{49,51,55,75\} n ∈ { 49 , 51 , 55 , 75 }
Proof. We have 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)] , no elliptic curve over F 2 k \mathbb{F}_{2^k} F 2 k k ≤ 5 k\le5 k ≤ 5 n n n gon Q X 1 ( n ) ≥ 12 , 12 , 15 , 24 \operatorname{gon}_{\mathbb{Q}}X_1(n)\ge12,12,15,24 gon Q X 1 ( n ) ≥ 12 , 12 , 15 , 24 > 10 = 2 ⋅ 5 >10=2\cdot5 > 10 = 2 ⋅ 5 [9 Lemma 5.3 and Proposition 5.2 (b)(i)] with p = 2 p=2 p = 2 d = 5 d=5 d = 5
Propositions 4.2 and 4.4 together eliminate the nineteen groups Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n ∈ { 49 , 51 , 55 , 75 , 91 , 95 , 119 , 121 , 125 , 133 , 143 , 169 , 187 , 209 , 221 , 247 , 289 , 323 , 361 } n \in\{49, 51, 55, 75, 91, 95, 119, 121, 125, 133, 143, 169, 187, 209, 221, 247, 289, 323, 361\} n ∈ { 49 , 51 , 55 , 75 , 91 , 95 , 119 , 121 , 125 , 133 , 143 , 169 , 187 , 209 , 221 , 247 , 289 , 323 , 361 }
Once we have shown in § 6 and § 8 that X 1 ( 2 , 2 n ) X_1(2,2n) X 1 ( 2 , 2 n ) n = 10 , 11 , 12 n = 10, 11, 12 n = 10 , 11 , 12 Z / 40 Z \mathbb{Z}/40\mathbb{Z} Z /40 Z Z / 44 Z \mathbb{Z}/44\mathbb{Z} Z /44 Z Z / 48 Z \mathbb{Z}/48\mathbb{Z} Z /48 Z Z / 36 Z \mathbb{Z}/36\mathbb{Z} Z /36 Z [9 §7] . The remaining 22 groups are the subject of § 5 to § 8 .
The rank zero criterion
We recall [9 Proposition 5.2] in the form in which we use it.
Proposition 5.1 ([9 Proposition 5.2] ). Let m ∣ n m \mid n m ∣ n m n > 4 mn > 4 mn > 4 p ∤ m n p \nmid mn p ∤ mn d ≥ 1 d \ge1 d ≥ 1 rk J 1 ( m , n ) ( Q ( ζ m ) ) = 0 \operatorname{rk} J_1(m,n)(\mathbb{Q}(\zeta_m)) = 0 rk J 1 ( m , n ) ( Q ( ζ m )) = 0
(a) p > 2 p > 2 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
(b.i) 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
(b.ii) p = 2 p = 2 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 L L L Q ( ζ m ) ⊆ L ⊆ Q ( ζ n ) \mathbb{Q}(\zeta_m) \subseteq L \subseteq\mathbb{Q}(\zeta_n) Q ( ζ m ) ⊆ L ⊆ Q ( ζ n ) p L \mathfrak{p}_L p L L L L F p L \mathbb{F}_{\mathfrak{p}_L} F p L X 1 ( m , n ) X_1(m,n) X 1 ( m , n ) L L L p L \mathfrak{p}_L p L J 1 ( m , n ) ( L ) [ 2 ] J_1(m,n)(L)[2] J 1 ( m , n ) ( L ) [ 2 ]
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 ) K K K p p p
Combining Proposition 5.1 with Lemma 2.3 eliminates four groups outright.
Proposition 5.2. The groups Z / 27 Z \mathbb{Z}/27\mathbb{Z} Z /27 Z Z / 35 Z \mathbb{Z}/35\mathbb{Z} Z /35 Z Z / 39 Z \mathbb{Z}/39\mathbb{Z} Z /39 Z Z / 50 Z \mathbb{Z}/50\mathbb{Z} Z /50 Z
Proof. Let n ∈ { 27 , 35 , 39 , 50 } n \in\{27, 35, 39, 50\} n ∈ { 27 , 35 , 39 , 50 } E E E K K K n n n 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)] .
Take p = 2 p = 2 p = 2 n = 27 , 35 , 39 n = 27, 35, 39 n = 27 , 35 , 39 p = 3 p = 3 p = 3 n = 50 n = 50 n = 50 F p k \mathbb{F}_{p^k} F p k k ≤ 5 k \le5 k ≤ 5 n n n E E E K K K p p p d = 5 d = 5 d = 5 E E E K K K p p p
For n = 35 n = 35 n = 35 n = 39 n = 39 n = 39 gon Q X 1 ( 35 ) = 12 \operatorname{gon}_{\mathbb{Q}} X_1(35) = 12 gon Q X 1 ( 35 ) = 12 gon Q X 1 ( 39 ) = 14 \operatorname{gon}_{\mathbb{Q}} X_1(39) = 14 gon Q X 1 ( 39 ) = 14 [13 Theorem 4 and Table 1] , both > 10 = 2 ⋅ 5 > 10 = 2 \cdot5 > 10 = 2 ⋅ 5
For n = 50 n = 50 n = 50 [ PSL 2 ( Z ) : Γ ‾ 1 ( 50 ) ] = 900 [\operatorname{PSL}_2(\mathbb{Z}) : \overline{\Gamma}_1(50)] = 900 [ PSL 2 ( Z ) : Γ 1 ( 50 )] = 900 gon Q X 1 ( 50 ) ≥ gon C X 1 ( 50 ) > 8.9 \operatorname{gon}_{\mathbb{Q}} X_1(50) \ge\operatorname{gon}_{\mathbb{C}} X_1(50) > 8.9 gon Q X 1 ( 50 ) ≥ gon C X 1 ( 50 ) > 8.9 gon Q X 1 ( 50 ) > 5 \operatorname{gon}_{\mathbb{Q}} X_1(50) > 5 gon Q X 1 ( 50 ) > 5
For n = 27 n = 27 n = 27 L = Q L = \mathbb{Q} L = Q X 1 ( 27 ) X_1(27) X 1 ( 27 ) Q ( ζ 27 ) \mathbb{Q}(\zeta_{27}) Q ( ζ 27 ) F 2 \mathbb{F}_2 F 2 Q \mathbb{Q} Q J 1 ( 27 ) ( Q ) = Cl Q X 1 ( 27 ) ≃ Z / 3 Z × Z / 3 Z × Z / 24947 Z J_1(27)(\mathbb{Q}) = \operatorname{Cl}_{\mathbb{Q}} X_1(27) \simeq\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/3\mathbb{Z} \times\mathbb{Z}/24947\mathbb{Z} J 1 ( 27 ) ( Q ) = Cl Q X 1 ( 27 ) ≃ Z /3 Z × Z /3 Z × Z /24947 Z [7 Corollary 4.14 and Table 2] has no 2-torsion, so reduction is injective on J 1 ( 27 ) ( Q ) [ 2 ] J_1(27)(\mathbb{Q})[2] J 1 ( 27 ) ( Q ) [ 2 ] gon Q X 1 ( 27 ) = 6 > 5 \operatorname{gon}_{\mathbb{Q}} X_1(27) = 6 > 5 gon Q X 1 ( 27 ) = 6 > 5 [13 Theorem 4 and Table 1] , Proposition 5.1(b.ii) applies. □
Remark 5.3. Note that some of the enumerations in Proposition 5.2 can be avoided by using [Wat96, Theorem 4.1] instead.
The Hecke sieve in degree five
Let X = X 1 ( m , n ) X = X_1(m,n) X = X 1 ( m , n ) m ∈ { 1 , 2 } m \in\{1,2\} m ∈ { 1 , 2 } rk J ( Q ) = 0 \operatorname{rk} J(\mathbb{Q}) = 0 rk J ( Q ) = 0 p ∤ 2 n p \nmid2n p ∤ 2 n K K K x ∈ Y 1 ( m , n ) ( K ) x \in Y_1(m,n)(K) x ∈ Y 1 ( m , n ) ( 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} \hookrightarrow E(K) Z / m Z × Z / n Z ↪ E ( K ) x ( 5 ) : = ∑ σ x σ ∈ X ( 5 ) ( Q ) x^{(5)} := \sum_{\sigma} x^\sigma\in X^{(5)}(\mathbb{Q}) x ( 5 ) := ∑ σ x σ ∈ X ( 5 ) ( Q ) p p p
x F p ( 5 ) = D + C , D ∈ Y 1 ( m , n ) ( d ′ ) ( F p ) , C cuspidal of degree 5 − d ′ , x_{\mathbb{F}_p}^{(5)} = D + C,\qquad D \in Y_1(m,n)^{(d')}(\mathbb{F}_p),\qquad C\text{ cuspidal of degree }5-d', x F p ( 5 ) = D + C , D ∈ Y 1 ( m , n ) ( d ′ ) ( F p ) , C cuspidal of degree 5 − d ′ , where d ′ ≥ 1 d' \ge1 d ′ ≥ 1 E E E K K K p p p D D D d ′ d' d ′ type of the decomposition, and write it in brackets. Thus [ 5 ] [5] [ 5 ] D D D [ 3 , 2 ] [3,2] [ 3 , 2 ] D D D d ′ = 5 d'=5 d ′ = 5
The Hecke sieve [9 Proposition 5.5] says that if [ A q ( D ) ] ≠ 0 [A_q(D)] \ne0 [ A q ( D )] = 0 [ A q ( C ) ] = 0 [A_q(C)] = 0 [ A q ( C )] = 0 J ( F p ) J(\mathbb{F}_p) J ( F p ) q ∤ 2 p n q \nmid2pn q ∤ 2 p n A q A_q A q X X X D D D
The hypothesis [ A q ( C ) ] = 0 [A_q(C)] = 0 [ A q ( C )] = 0 C C C [9 Lemma 5.10] , which we restate below in the form in which our code checks it and in the form needed for the twisted operators of § 7 . We first fix notation for the cusps.
The aim is to lift a cuspidal divisor on X F p X_{\mathbb{F}_p} X F p p p p q q q Proposition 2.1 (2) applies.
By [7 Lemma 2.8] the cuspidal subscheme C C C X 1 ( n ) X_1(n) X 1 ( n ) Z [ 1 / 2 n ] \mathbb{Z}[1/2n] Z [ 1/2 n ] ∐ d ∣ n ( μ n / d ( Z / d Z ) ′ ) / [ − 1 ] \coprod_{d\mid n}(\mu_{n/d}(\mathbb{Z}/d\mathbb{Z})')/[-1] ∐ d ∣ n ( μ n / d ( Z / d Z ) ′ ) / [ − 1 ] d d d φ ( d ) φ ( n / d ) / 2 \varphi(d)\varphi(n/d)/2 φ ( d ) φ ( n / d ) /2 C C C Z [ 1 / 2 n ] \mathbb{Z}[1/2n] Z [ 1/2 n ] p p p C ( Q ‾ p ) → C ( F ‾ p ) C(\overline{\mathbb{Q}}_p) \to C(\overline{\mathbb{F}}_p) C ( Q p ) → C ( F p ) Gal ( Q ‾ p / Q p ) ↠ Gal ( F ‾ p / F p ) \operatorname{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p) \twoheadrightarrow\operatorname{Gal}(\overline{\mathbb{F}}_p/\mathbb{F}_p) Gal ( Q p / Q p ) ↠ Gal ( F p / F p ) Q ( ζ n ) \mathbb{Q}(\zeta_n) Q ( ζ n ) p p p C ( Q ‾ ) C(\overline{\mathbb{Q}}) C ( Q ) ( Z / n Z ) × (\mathbb{Z}/n\mathbb{Z})^\times ( Z / n Z ) × D p D_p D p p p p n n n C F p C_{\mathbb{F}_p} C F p D p D_p D p C ( Q ‾ ) C(\overline{\mathbb{Q}}) C ( Q )
Let C i C_i C i O i O_i O i D p D_p D p
H i : = { σ ∈ ( Z / n Z ) × : σ ( O i ) = O i } ⊇ D p , L i : = Q ( ζ n ) H i . H_i := \{\sigma\in(\mathbb{Z}/n\mathbb{Z})^\times: \sigma(O_i)=O_i\} \supseteq D_p,\qquad L_i := \mathbb{Q}(\zeta_n)^{H_i}. H i := { σ ∈ ( Z / n Z ) × : σ ( O i ) = O i } ⊇ D p , L i := Q ( ζ n ) H i . The divisor c i : = ∑ P ∈ O i P c_i := \sum_{P\in O_i} P c i := ∑ P ∈ O i P L i L_i L i C i C_i C i Q ( ζ n ) / Q \mathbb{Q}(\zeta_n)/\mathbb{Q} Q ( ζ n ) / Q L i / Q L_i/\mathbb{Q} L i / Q Gal ( L i / Q ) = ( Z / n Z ) × / H i \operatorname{Gal}(L_i/\mathbb{Q})=(\mathbb{Z}/n\mathbb{Z})^\times/H_i Gal ( L i / Q ) = ( Z / n Z ) × / H i Frob q = q \operatorname{Frob}_q=q Frob q = q H i H_i H i p p p L i L_i L i q q q L i L_i L i q q q n ∈ H i n \in H_i n ∈ H i
Lemma 6.1. Let X = X 1 ( n ) X=X_1(n) X = X 1 ( n ) n > 4 n>4 n > 4 p ∤ 2 n p\nmid2n p ∤ 2 n q ∤ 2 p n q\nmid2pn q ∤ 2 p n
(1) Let L ⊆ Q ( ζ n ) L\subseteq\mathbb{Q}(\zeta_n) L ⊆ Q ( ζ n ) p p p q q q C C C X F p X_{\mathbb{F}_p} X F p X L X_L X L [ A q ψ ( C ) ] = 0 [A_q\psi(C)]=0 [ A q ψ ( C )] = 0 J ( F p ) J(\mathbb{F}_p) J ( F p ) ψ ∈ Z [ Δ ( n ) ] \psi\in\mathbb{Z}[\Delta(n)] ψ ∈ Z [ Δ ( n )]
(2) Let C = ∑ i a i C i C=\sum_i a_iC_i C = ∑ i a i C i X F p X_{\mathbb{F}_p} X F p C i C_i C i q q q n n n H i H_i H i i i i [ A q ψ ( C ) ] = 0 [A_q\psi(C)]=0 [ A q ψ ( C )] = 0 ψ ∈ Z [ Δ ( n ) ] \psi\in\mathbb{Z}[\Delta(n)] ψ ∈ Z [ Δ ( n )] q = p j q=p^j q = p j n n n j ≥ 0 j\geq0 j ≥ 0
The case ψ = 1 \psi=1 ψ = 1 [9 Lemma 5.10] .
Proof. (1) Let c c c X L X_L X L C C C e : = deg ψ ( c ) e := \deg\psi(c) e := deg ψ ( c ) Q \mathbb{Q} Q ψ ( c ) \psi(c) ψ ( c ) L L L ψ ( c ) − e c 0 \psi(c)-ec_0 ψ ( c ) − e c 0 X L X_L X L J ( L ) t o r s J(L)_{\mathrm{tors}} J ( L ) tors q q q L L L Proposition 2.1 (2) gives [ A q ( ψ ( c ) − e c 0 ) ] = 0 [A_q(\psi(c)-ec_0)]=0 [ A q ( ψ ( c ) − e c 0 )] = 0 A q ( c 0 ) = 0 A_q(c_0)=0 A q ( c 0 ) = 0 Proposition 2.1 (3), so [ A q ψ ( c ) ] = 0 [A_q\psi(c)]=0 [ A q ψ ( c )] = 0 J ( L ) J(L) J ( L ) L L L p p p F p \mathbb{F}_p F p p p p L L L A q A_q A q ψ \psi ψ [ A q ψ ( C ) ] = 0 [A_q\psi(C)]=0 [ A q ψ ( C )] = 0
(2) Both ψ \psi ψ A q A_q A q [ A q ψ ( C ) ] = ∑ i a i [ A q ψ ( C i ) ] [A_q\psi(C)]=\sum_i a_i[A_q\psi(C_i)] [ A q ψ ( C )] = ∑ i a i [ A q ψ ( C i )] [ A q ψ ( C i ) ] = 0 [A_q\psi(C_i)]=0 [ A q ψ ( C i )] = 0 i i i i i i L = L i L=L_i L = L i C = C i C=C_i C = C i c i c_i c i L i L_i L i C i C_i C i p p p L i L_i L i D p ⊆ H i D_p\subseteq H_i D p ⊆ H i q q q L i L_i L i q m o d n ∈ H i q\bmod n\in H_i q mod n ∈ H i q ≡ p ( m o d n ) q\equiv p\pmod n q ≡ p ( mod n ) q m o d n ∈ D p ⊆ H i q\bmod n\in D_p\subseteq H_i q mod n ∈ D p ⊆ H i i i i
Since A q A_q A q Z [ Δ ( n ) ] \mathbb{Z}[\Delta(n)] Z [ Δ ( n )] A q ( C ) A_q(C) A q ( C ) [ A q ψ ( C ) ] = ψ ( [ A q ( C ) ] ) [A_q\psi(C)]=\psi([A_q(C)]) [ A q ψ ( C )] = ψ ([ A q ( C )]) ψ = 1 \psi=1 ψ = 1 □ \square □
Remark 6.2. Only the closed points that actually occur in a cuspidal complement matter, so in practice one only has to impose q m o d n ∈ H i q\bmod n\in H_i q mod n ∈ H i C i C_i C i 5 − d ′ 5-d' 5 − d ′ H i = ( Z / n Z ) × H_i=(\mathbb{Z}/n\mathbb{Z})^\times H i = ( Z / n Z ) × C i C_i C i q q q D p D_p D p O i O_i O i ( Z / n Z ) × (\mathbb{Z}/n\mathbb{Z})^\times ( Z / n Z ) × Q \mathbb{Q} Q p p p Q \mathbb{Q} Q p p p q q q q m o d n ∈ D p q\bmod n\in D_p q mod n ∈ D p D p ⊆ H i D_p\subseteq H_i D p ⊆ H i
We can now eliminate eight of the remaining groups.
Proposition 6.3. The groups Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n ∈ { 26 , 32 , 33 , 34 , 38 , 42 , 45 } n\in\{26,32,33,34,38,42,45\} n ∈ { 26 , 32 , 33 , 34 , 38 , 42 , 45 } Z / 2 Z × Z / 22 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/22\mathbb{Z} Z /2 Z × Z /22 Z Lemma 3.2 , Z / 44 Z \mathbb{Z}/44\mathbb{Z} Z /44 Z
Proof. Let X = X 1 ( m , n ) X=X_1(m,n) X = X 1 ( m , n ) p p p Table 3 . 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)] , since 26 , 32 , 33 , 34 , 38 , 42 , 45 26,32,33,34,38,42,45 26 , 32 , 33 , 34 , 38 , 42 , 45 S 0 S_0 S 0 X 1 ( 2 , 22 ) X_1(2,22) X 1 ( 2 , 22 ) [7 Theorem 3.1(3)] (or [DS17, Theorem 4.1).
Table 3. The Hecke sieve computations of Proposition 6.3 . The fourth column lists the number of non-cuspidal closed points of X F p X_{\mathbb{F}_p} X F p 1 , 2 , 3 , 4 , 5 1,2,3,4,5 1 , 2 , 3 , 4 , 5 [ 2 , 2 ] [2,2] [ 2 , 2 ] [ 3 , 2 ] [3,2] [ 3 , 2 ]
We first exclude the totally cuspidal reduction, i.e. d ′ = 0 d'=0 d ′ = 0 Proposition 5.1 (a) with d = 5 d=5 d = 5 gon Q X > 5 \operatorname{gon}_{\mathbb{Q}}X>5 gon Q X > 5 [13 Theorem 4 and Table 1] we have gon Q X 1 ( n ) = 6 , 8 , 10 , 10 , 12 \operatorname{gon}_{\mathbb{Q}}X_1(n)=6,8,10,10,12 gon Q X 1 ( n ) = 6 , 8 , 10 , 10 , 12 n = 26 , 32 , 33 , 34 , 38 n=26,32,33,34,38 n = 26 , 32 , 33 , 34 , 38 n = 42 n=42 n = 42 gon Q X 1 ( 42 ) > 8 \operatorname{gon}_{\mathbb{Q}}X_1(42)>8 gon Q X 1 ( 42 ) > 8 [13 proof of Proposition 6] , and for n = 45 n=45 n = 45 gon Q X 1 ( 45 ) ≥ 9 \operatorname{gon}_{\mathbb{Q}}X_1(45)\geq9 gon Q X 1 ( 45 ) ≥ 9 X 1 ( 2 , 22 ) X_1(2,22) X 1 ( 2 , 22 ) gon F 3 X 1 ( 2 , 22 ) > 5 \operatorname{gon}_{\mathbb{F}_3}X_1(2,22)>5 gon F 3 X 1 ( 2 , 22 ) > 5 [12 proof of Proposition 5.3] , so gon Q X 1 ( 2 , 22 ) ≥ gon F 3 X 1 ( 2 , 22 ) > 5 \operatorname{gon}_{\mathbb{Q}}X_1(2,22)\geq\operatorname{gon}_{\mathbb{F}_3}X_1(2,22)>5 gon Q X 1 ( 2 , 22 ) ≥ gon F 3 X 1 ( 2 , 22 ) > 5
It remains to exclude every decomposition x F p ( 5 ) = D + C x^{(5)}_{\mathbb{F}_p}=D+C x F p ( 5 ) = D + C deg D = d ′ ≥ 1 \deg D=d'\geq1 deg D = d ′ ≥ 1 Lemma 2.3 : a non-cuspidal closed point of degree e e e F p \mathbb{F}_p F p Table 3 . For one representative D D D [ A q ( D ) ] ≠ 0 [A_q(D)]\ne0 [ A q ( D )] = 0 q q q A q A_q A q D D D [ A q ( C ) ] = 0 [A_q(C)]=0 [ A q ( C )] = 0 C C C Lemma 6.1 (2) with ψ = 1 \psi=1 ψ = 1 X 1 ( n ) X_1(n) X 1 ( n ) X 1 ( 2 , 22 ) X_1(2,22) X 1 ( 2 , 22 ) [ 5 ] [5] [ 5 ] Table 3 ).
Remark 6.4. We spell out how Lemma 6.1 is applied for each curve.
For ( n , p ) = ( 26 , 7 ) , ( 34 , 3 ) , ( 38 , 3 ) (n,p)=(26,7),(34,3),(38,3) ( n , p ) = ( 26 , 7 ) , ( 34 , 3 ) , ( 38 , 3 ) p p p n n n D p = ( Z / n Z ) × D_p=(\mathbb{Z}/n\mathbb{Z})^\times D p = ( Z / n Z ) × q ∤ 2 p n q\nmid2pn q ∤ 2 p n
For ( n , p ) = ( 32 , 3 ) (n,p)=(32,3) ( n , p ) = ( 32 , 3 ) ( Z / 32 Z ) × (\mathbb{Z}/32\mathbb{Z})^\times ( Z /32 Z ) × 3 3 3 8 8 8 11 11 11 [ 3 ] [3] [ 3 ] [ 4 ] [4] [ 4 ] q = 11 q=11 q = 11 [ 5 ] [5] [ 5 ] q ∈ { 5 , 7 , 11 , 13 } q\in\{5,7,11,13\} q ∈ { 5 , 7 , 11 , 13 }
For X 1 ( 2 , 22 ) X_1(2,22) X 1 ( 2 , 22 ) p = 3 p=3 p = 3 [ 5 ] [5] [ 5 ] q q q
For X 1 ( 33 ) X_1(33) X 1 ( 33 ) p = 7 p=7 p = 7 Q \mathbb{Q} Q 1 10 , 2 5 , 10 2 1^{10},2^5,10^2 1 10 , 2 5 , 1 0 2 7 7 7 2 2 2 10 10 10 q m o d 33 ∈ D 7 = { 1 , 4 , 7 , 10 , 13 , 16 , 19 , 25 , 28 , 31 } q\bmod33\in D_7=\{1,4,7,10,13,16,19,25,28,31\} q mod 33 ∈ D 7 = { 1 , 4 , 7 , 10 , 13 , 16 , 19 , 25 , 28 , 31 } [ 3 ] [3] [ 3 ] [ 4 ] [4] [ 4 ] q ∈ { 13 , 19 , 37 , 43 , 67 } q\in\{13,19,37,43,67\} q ∈ { 13 , 19 , 37 , 43 , 67 } [ 5 ] [5] [ 5 ] q ∈ { 5 , 13 , 19 , 37 , 43 , 67 } q\in\{5,13,19,37,43,67\} q ∈ { 5 , 13 , 19 , 37 , 43 , 67 }
For X 1 ( 42 ) X_1(42) X 1 ( 42 ) p = 5 p=5 p = 5 1 12 , 2 6 , 6 4 1^{12},2^6,6^4 1 12 , 2 6 , 6 4 5 5 5 6 6 6 3 3 3 2 2 2 q q q 3 3 3 4 4 4 Y 1 ( 42 ) ( F 5 e ) = ∅ Y_1(42)(\mathbb{F}_{5^e})=\varnothing Y 1 ( 42 ) ( F 5 e ) = ∅ e ≤ 2 e\leq2 e ≤ 2
For X 1 ( 45 ) X_1(45) X 1 ( 45 ) p = 7 p=7 p = 7 1 12 , 2 4 , 3 6 , 6 2 , 8 1^{12},2^4,3^6,6^2,8 1 12 , 2 4 , 3 6 , 6 2 , 8 7 7 7 2 2 2 6 6 6 3 3 3 8 8 8 4 4 4 q m o d 45 ∈ D 7 = { 1 , 4 , 7 , 13 , 16 , 19 , 22 , 28 , 31 , 34 , 37 , 43 } q\bmod45\in D_7=\{1,4,7,13,16,19,22,28,31,34,37,43\} q mod 45 ∈ D 7 = { 1 , 4 , 7 , 13 , 16 , 19 , 22 , 28 , 31 , 34 , 37 , 43 } q ≡ 1 ( m o d 3 ) q\equiv1\pmod3 q ≡ 1 ( mod 3 ) [ 2 ] [2] [ 2 ] [ 3 ] [3] [ 3 ] [ 4 ] [4] [ 4 ] [ 2 , 2 ] [2,2] [ 2 , 2 ] q ∈ { 13 , 19 } q\in\{13,19\} q ∈ { 13 , 19 } [ 5 ] [5] [ 5 ] [ 3 , 2 ] [3,2] [ 3 , 2 ] q ∈ { 11 , 13 , 17 , 19 } q\in\{11,13,17,19\} q ∈ { 11 , 13 , 17 , 19 }
For X 1 ( 33 ) X_1(33) X 1 ( 33 ) X 1 ( 42 ) X_1(42) X 1 ( 42 ) X 1 ( 45 ) X_1(45) X 1 ( 45 ) [7 Lemma 2.8] . For the other four curves X 1 ( n ) X_1(n) X 1 ( n ) q m o d n ∈ D p q\bmod n\in D_p q mod n ∈ D p Lemma 6.1 is always sufficient.
The cases of positive analytic rank
We now treat the groups Z / n Z \mathbb{Z}/n\mathbb{Z} Z / n Z n ∈ { 57 , 63 , 65 , 77 , 85 } n\in\{57,63,65,77,85\} n ∈ { 57 , 63 , 65 , 77 , 85 } J 1 ( n ) J_1(n) J 1 ( n ) rk J 1 ( n ) ( Q ) = 0 \operatorname{rk}J_1(n)(\mathbb{Q})=0 rk J 1 ( n ) ( Q ) = 0 Proposition 5.1 nor the Hecke sieve of § 6 applies. What we use instead is the finiteness of ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle-1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) a a a n = 77 n=77 n = 77 n = 85 n=85 n = 85 n = 57 , 63 , 65 n=57,63,65 n = 57 , 63 , 65 rk J 1 ( n ) ( Q ) \operatorname{rk}J_1(n)(\mathbb{Q}) rk J 1 ( n ) ( Q )
Throughout this section we use the following consequence of Kato’s theorem, already used in the proof of Proposition 4.1 : if a ∈ ( Z / n Z ) × a\in(\mathbb{Z}/n\mathbb{Z})^\times a ∈ ( Z / n Z ) × χ f ( a ) = 1 \chi_f(a)=1 χ f ( a ) = 1 f f f 2 2 2 n n n L ( f , 1 ) = 0 L(f,1)=0 L ( f , 1 ) = 0 ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle-1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) a a a admissible for n n n n ∈ { 57 , 63 , 65 , 77 , 85 } n\in\{57,63,65,77,85\} n ∈ { 57 , 63 , 65 , 77 , 85 } [23 ] (all analytic ranks used are recorded there as proved).
Remark 7.1. The global method of Proposition 4.1 does not reach these five cases. For every odd prime q ∤ n q\nmid n q ∤ n b ( n ) < 5 b(n)<5 b ( n ) < 5 J 1 ( n ) ( Q ) J_1(n)(\mathbb{Q}) J 1 ( n ) ( Q ) q = 2 q=2 q = 2 n = 77 n=77 n = 77 b ( n ) > 5 b(n)>5 b ( n ) > 5 [9 Proposition 4.3] has the extra assumption on the 2-primary part of ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle-1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) § 4 , which we have not verified.
A diamond argument for the totally cuspidal reduction
Proposition 7.2. Let n > 4 n>4 n > 4 p ∤ n p\nmid n p ∤ n a ∈ ( Z / n Z ) × a\in(\mathbb{Z}/n\mathbb{Z})^\times a ∈ ( Z / n Z ) × admissible for n n n ε : = 1 \varepsilon:=1 ε := 1 p > 2 p>2 p > 2 ε : = 2 \varepsilon:=2 ε := 2 p = 2 p=2 p = 2 n n n
ord ⟨ a ⟩ ∤ 5 \operatorname{ord}\langle a\rangle\nmid5 ord ⟨ a ⟩ ∤ 5 Δ ( n ) \Delta(n) Δ ( n )
gon C X 1 ( n ) > 10 ε \operatorname{gon}_{\mathbb{C}}X_1(n)>10\varepsilon gon C X 1 ( n ) > 10 ε
d C M X 1 ( n ) > 5 d_{\mathrm{CM}}X_1(n)>5 d CM X 1 ( n ) > 5
Let K K K E / K E/K E / K n n n x ∈ Y 1 ( n ) ( K ) x \in Y_1(n)(K) x ∈ Y 1 ( n ) ( K ) [ Q ( x ) : Q ] = 5 [\mathbb{Q}(x) : \mathbb{Q}] = 5 [ Q ( x ) : Q ] = 5 E E E K K K p p p x F p ( 5 ) x_{\mathfrak{F}_p}^{(5)} x F p ( 5 )
Proof. Suppose the opposite. By Lemma 2.2 the reduction of E E E K K K p p p p \mathfrak{p} p Q ( ζ n ) \mathbb{Q}(\zeta_n) Q ( ζ n ) p p p x ( 5 ) x^{(5)} x ( 5 ) p \mathfrak{p} p [9 Proposition 5.2] , it lifts to an effective cuspidal divisor C C C X 1 ( n ) Q ( ζ n ) X_1(n)_{\mathbb{Q}(\zeta_n)} X 1 ( n ) Q ( ζ n ) Z [ 1 / n ] \mathbb{Z}[1/n] Z [ 1/ n ] X 1 ( n ) X_1(n) X 1 ( n ) Q ( ζ n ) \mathbb{Q}(\zeta_n) Q ( ζ n )
Let y : = ( ⟨ a ⟩ − 1 ) [ x ( 5 ) − C ] ∈ J 1 ( n ) ( Q ( ζ n ) ) y := (\langle a\rangle- 1)[x^{(5)} - C] \in J_1(n)(\mathbb{Q}(\zeta_n)) y := (⟨ a ⟩ − 1 ) [ x ( 5 ) − C ] ∈ J 1 ( n ) ( Q ( ζ n )) [ x ( 5 ) − C ] = [ x ( 5 ) − 5 c 0 ] + [ 5 c 0 − C ] [x^{(5)} - C] = [x^{(5)} - 5c_0] + [5c_0 - C] [ x ( 5 ) − C ] = [ x ( 5 ) − 5 c 0 ] + [ 5 c 0 − C ] [ x ( 5 ) − 5 c 0 ] ∈ J 1 ( n ) ( Q ) [x^{(5)} - 5c_0] \in J_1(n)(\mathbb{Q}) [ x ( 5 ) − 5 c 0 ] ∈ J 1 ( n ) ( Q ) ( ⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) (\langle a\rangle- 1)J_1(n)(\mathbb{Q}) (⟨ a ⟩ − 1 ) J 1 ( n ) ( Q ) [ 5 c 0 − C ] [5c_0 - C] [ 5 c 0 − C ] y y y y y y p \mathfrak{p} p p > 2 p > 2 p > 2 J 1 ( n ) ( Q ( ζ n ) ) t o r s J_1(n)(\mathbb{Q}(\zeta_n))_{\mathrm{tors}} J 1 ( n ) ( Q ( ζ n ) ) tors [18 Appendix] , so y = 0 y = 0 y = 0 p = 2 p = 2 p = 2 n n n [29 Proposition 2.4] , so 2 y = 0 2y = 0 2 y = 0 ε y = 0 \varepsilon y = 0 ε y = 0
ε ( ⟨ a ⟩ − 1 ) ( x ( 5 ) − C ) = div ( f ) \varepsilon(\langle a\rangle- 1)(x^{(5)} - C) = \operatorname{div}(f) ε (⟨ a ⟩ − 1 ) ( x ( 5 ) − C ) = div ( f ) for some f ∈ Q ( ζ n ) ( X 1 ( n ) ) f \in\mathbb{Q}(\zeta_n)(X_1(n)) f ∈ Q ( ζ n ) ( X 1 ( n ))
If f f f deg f ≤ ε deg ⟨ a ⟩ x ( 5 ) + C = 10 ε \deg f \le\varepsilon\deg\langle a\rangle x^{(5)} + C = 10\varepsilon deg f ≤ ε deg ⟨ a ⟩ x ( 5 ) + C = 10 ε
So f f f ε ( ⟨ a ⟩ − 1 ) ( x ( 5 ) − C ) = 0 \varepsilon(\langle a\rangle- 1)(x^{(5)} - C) = 0 ε (⟨ a ⟩ − 1 ) ( x ( 5 ) − C ) = 0 ⟨ a ⟩ x ( 5 ) = x ( 5 ) \langle a\rangle x^{(5)} = x^{(5)} ⟨ a ⟩ x ( 5 ) = x ( 5 ) ⟨ a ⟩ C = C \langle a\rangle C = C ⟨ a ⟩ C = C ⟨ a ⟩ \langle a\rangle ⟨ a ⟩ x ( 5 ) x^{(5)} x ( 5 ) ⟨ ⟨ a ⟩ ⟩ \langle\langle a\rangle\rangle ⟨⟨ a ⟩⟩ ord ⟨ a ⟩ \operatorname{ord}\langle a\rangle ord ⟨ a ⟩ ord ⟨ a ⟩ ∣ 5 \operatorname{ord}\langle a\rangle\mid5 ord ⟨ a ⟩ ∣ 5 x σ x^\sigma x σ ⟨ b ⟩ x σ = x σ \langle b\rangle x^\sigma= x^\sigma ⟨ b ⟩ x σ = x σ b ≢ ± 1 ( m o d n ) b \not\equiv\pm1 \pmod n b ≡ ± 1 ( mod n ) x σ = ( E ′ , P ′ ) x^\sigma= (E', P') x σ = ( E ′ , P ′ ) ( E ′ , P ′ ) ≃ ( E ′ , b P ′ ) (E', P') \simeq(E', bP') ( E ′ , P ′ ) ≃ ( E ′ , b P ′ ) E ′ E' E ′ ζ ≠ ± 1 \zeta\ne\pm1 ζ = ± 1 j ( E ′ ) ∈ { 0 , 1728 } j(E') \in\{0, 1728\} j ( E ′ ) ∈ { 0 , 1728 } x σ x^\sigma x σ □ \square □
Proposition 7.3. The groups Z / 77 Z \mathbb{Z}/77\mathbb{Z} Z /77 Z Z / 85 Z \mathbb{Z}/85\mathbb{Z} Z /85 Z
Proof. Let n ∈ { 77 , 85 } n \in\{77,85\} n ∈ { 77 , 85 } E E E K K K n n n Proposition 7.2 with p = 2 p = 2 p = 2 n n n ε = 2 \varepsilon= 2 ε = 2 F 2 k \mathbb{F}_{2^k} F 2 k k ≤ 5 k \le5 k ≤ 5 n n n n ≥ 45 n \ge45 n ≥ 45 ( 2 5 / 2 + 1 ) 2 < 45 (2^{5/2} + 1)^2 < 45 ( 2 5/2 + 1 ) 2 < 45 Lemma 2.3 the curve E E E K K K Proposition 7.2 once its assumptions are verified.
For n = 77 n = 77 n = 77 a = 3 a = 3 a = 3 ord ⟨ 3 ⟩ = 30 ∤ 5 \operatorname{ord}\langle3\rangle= 30 \nmid5 ord ⟨ 3 ⟩ = 30 ∤ 5 [ P S L 2 ( Z ) : Γ 1 ( 77 ) ] = 2880 [\mathrm{PSL}_2(\mathbb{Z}) : \Gamma_1(77)] = 2880 [ PSL 2 ( Z ) : Γ 1 ( 77 )] = 2880 gon C X 1 ( 77 ) > 28.56 > 20 \operatorname{gon}_{\mathbb{C}} X_1(77) > 28.56 > 20 gon C X 1 ( 77 ) > 28.56 > 20 d C M X 1 ( 77 ) = 60 > 5 d_{\mathrm{CM}}X_1(77) = 60 > 5 d CM X 1 ( 77 ) = 60 > 5
For n = 85 n = 85 n = 85 a = 2 a = 2 a = 2 [9 Table 2] ). It lies in the kernel of the character of 85.2. j . a 85.2.j.a 85.2. j . a a = 2 a = 2 a = 2 ord ⟨ 2 ⟩ = 8 ∤ 5 \operatorname{ord}\langle2\rangle= 8 \nmid5 ord ⟨ 2 ⟩ = 8 ∤ 5 gon C X 1 ( 85 ) > 34.27 > 20 \operatorname{gon}_{\mathbb{C}} X_1(85) > 34.27 > 20 gon C X 1 ( 85 ) > 34.27 > 20 d C M X 1 ( 85 ) = 32 > 5 d_{\mathrm{CM}}X_1(85) = 32 > 5 d CM X 1 ( 85 ) = 32 > 5 □ \square □
would read gon C X 1 ( 65 ) ≥ 2016 / 96 = 21 \operatorname{gon}_{\mathbb{C}} X_1(65) \ge2016/96 = 21 gon C X 1 ( 65 ) ≥ 2016/96 = 21 Z / 65 Z \mathbb{Z}/65\mathbb{Z} Z /65 Z ε = 1 \varepsilon= 1 ε = 1 gon C X 1 ( n ) > 10 \operatorname{gon}_{\mathbb{C}} X_1(n) > 10 gon C X 1 ( n ) > 10
The twisted Hecke sieve
For the remaining three groups we combine Proposition 7.2 at an odd prime (which excludes the totally cuspidal reduction) with a variant of the Hecke sieve in which A q A_q A q J 1 ( n ) ( Q ) J_1(n)(\mathbb{Q}) J 1 ( n ) ( Q )
Proposition 7.5 (Twisted Hecke sieve). Let n > 4 n > 4 n > 4 p ∤ 2 n p \nmid2n p ∤ 2 n q ∤ 2 p n q \nmid2pn q ∤ 2 p 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 )) t : = A q ψ t := A_q\psi t := A q ψ D + C ∈ X 1 ( n ) ( 5 ) ( F p ) D + C \in X_1(n)^{(5)}(\mathbb{F}_p) D + C ∈ X 1 ( n ) ( 5 ) ( F p ) D D D d ′ ≥ 1 d' \ge1 d ′ ≥ 1 C C C 5 − d ′ 5-d' 5 − d ′
(1) [ t ( D ) ] ≠ 0 [t(D)] \ne0 [ t ( D )] = 0 J 1 ( n ) ( F p ) J_1(n)(\mathbb{F}_p) J 1 ( n ) ( F p )
(2) q m o d n q \bmod n q mod n H i H_i H i C i C_i C i C C C
Then no z ∈ Y 1 ( n ) ( 5 ) ( Q ) z \in Y_1(n)^{(5)}(\mathbb{Q}) z ∈ Y 1 ( n ) ( 5 ) ( Q ) D + C D + C D + C
Proof. Suppose some z ∈ Y 1 ( n ) ( 5 ) ( Q ) z \in Y_1(n)^{(5)}(\mathbb{Q}) z ∈ Y 1 ( n ) ( 5 ) ( Q ) D + C D + C D + C A q A_q A q A q ( c 0 ) = 0 A_q(c_0) = 0 A q ( c 0 ) = 0 t ( c 0 ) = 0 t(c_0) = 0 t ( c 0 ) = 0 [ t ( z ) ] = A q ( ψ ( [ z − 5 c 0 ] ) ) [t(z)] = A_q(\psi([z - 5c_0])) [ t ( z )] = A q ( ψ ([ z − 5 c 0 ])) ψ ( [ z − 5 c 0 ] ) \psi([z - 5c_0]) ψ ([ z − 5 c 0 ]) ψ ( J 1 ( n ) ( Q ) ) \psi(J_1(n)(\mathbb{Q})) ψ ( J 1 ( n ) ( Q )) A q A_q A q L = Q L = \mathbb{Q} L = Q [ t ( z ) ] = 0 [t(z)] = 0 [ t ( z )] = 0 J 1 ( n ) ( Q ) J_1(n)(\mathbb{Q}) J 1 ( n ) ( Q )
As ψ ∈ Z [ Δ ( n ) ] \psi\in\mathbb{Z}[\Delta(n)] ψ ∈ Z [ Δ ( n )] [ t ( C ) ] = 0 [t(C)] = 0 [ t ( C )] = 0 J 1 ( n ) ( F p ) J_1(n)(\mathbb{F}_p) J 1 ( n ) ( F p ) p p p 0 = [ t ( z ) ] F p = [ t ( D ) ] + [ t ( C ) ] = [ t ( D ) ] 0 = [t(z)]_{\mathbb{F}_p} = [t(D)] + [t(C)] = [t(D)] 0 = [ t ( z ) ] F p = [ t ( D )] + [ t ( C )] = [ t ( D )]
We use two instances of t t t
t a : = A q ( ⟨ a ⟩ − 1 ) , t H : = A q π ∗ π ∗ = A q ( ∑ h ∈ H / { ± 1 } ⟨ h ⟩ ) . t_a := A_q(\langle a\rangle- 1), \qquad t_H := A_q\pi^*\pi_* = A_q\left(\sum_{h\in H/\{\pm1\}}\langle h\rangle\right). t a := A q (⟨ a ⟩ − 1 ) , t H := A q π ∗ π ∗ = A q h ∈ H / { ± 1 } ∑ ⟨ h ⟩ . Here a a a n n n ψ = ⟨ a ⟩ − 1 \psi= \langle a\rangle- 1 ψ = ⟨ a ⟩ − 1 H ⊆ ( Z / n Z ) × H \subseteq(\mathbb{Z}/n\mathbb{Z})^\times H ⊆ ( Z / n Z ) × − 1 -1 − 1 rk J H ( Q ) = 0 \operatorname{rk} J_H(\mathbb{Q}) = 0 rk J H ( Q ) = 0 π : X 1 ( n ) → X H \pi: X_1(n) \to X_H π : X 1 ( n ) → X H # H / 2 \#H/2 # H /2 ⟨ − 1 ⟩ \langle-1\rangle ⟨ − 1 ⟩ ψ = π ∗ π ∗ \psi= \pi^*\pi_* ψ = π ∗ π ∗ π ∗ J 1 ( n ) ( Q ) ⊆ J H ( Q ) \pi_*J_1(n)(\mathbb{Q}) \subseteq J_H(\mathbb{Q}) π ∗ J 1 ( n ) ( Q ) ⊆ J H ( Q )
Remark 7.6. A divisor D D D ⟨ a ⟩ D = D \langle a\rangle D = D ⟨ a ⟩ D = D t a ( D ) = 0 t_a(D) = 0 t a ( D ) = 0 D D D a a a t H t_H t H
Proposition 7.7. The groups Z / 57 Z \mathbb{Z}/57\mathbb{Z} Z /57 Z Z / 63 Z \mathbb{Z}/63\mathbb{Z} Z /63 Z Z / 65 Z \mathbb{Z}/65\mathbb{Z} Z /65 Z
Proof. Let n ∈ { 57 , 63 , 65 } n \in\{57, 63, 65\} n ∈ { 57 , 63 , 65 } p = 5 , 5 , 3 p = 5, 5, 3 p = 5 , 5 , 3 E E E K K K n n n x ∈ Y 1 ( n ) ( K ) x \in Y_1(n)(K) x ∈ Y 1 ( n ) ( K ) x ( 5 ) x^{(5)} x ( 5 ) p p p
The totally cuspidal reduction. We apply Proposition 7.2 with the odd prime p p p ε = 1 \varepsilon= 1 ε = 1 gon C X 1 ( n ) > 10 \operatorname{gon}_{\mathbb{C}} X_1(n) > 10 gon C X 1 ( n ) > 10 ord ⟨ a ⟩ ∤ 5 \operatorname{ord}\langle a\rangle\nmid5 ord ⟨ a ⟩ ∤ 5 d C M X 1 ( n ) > 5 d_{\mathrm{CM}}X_1(n) > 5 d CM X 1 ( n ) > 5 gon C X 1 ( n ) > 14.28 , 17.14 , 19.99 \operatorname{gon}_{\mathbb{C}} X_1(n) > 14.28, 17.14, 19.99 gon C X 1 ( n ) > 14.28 , 17.14 , 19.99 > 10 > 10 > 10 a = 2 , 5 , 2 a = 2, 5, 2 a = 2 , 5 , 2 Table 4 , and ord ⟨ a ⟩ = 9 , 3 , 6 \operatorname{ord}\langle a\rangle= 9, 3, 6 ord ⟨ a ⟩ = 9 , 3 , 6 d C M X 1 ( n ) = 12 , 36 , 24 d_{\mathrm{CM}}X_1(n) = 12, 36, 24 d CM X 1 ( n ) = 12 , 36 , 24 > 5 > 5 > 5 x F p ( 5 ) x^{(5)}_{\mathbb{F}_p} x F p ( 5 ) D D D d ′ ≥ 1 d' \ge1 d ′ ≥ 1
The remaining reductions. We rule out every possible D + C D + C D + C F p e \mathbb{F}_{p^e} F p e Y 1 ( n ) ( F p e ) Y_1(n)(\mathbb{F}_{p^e}) Y 1 ( n ) ( F p e ) e ≤ 5 e \le5 e ≤ 5 Table 5 , so D D D X 1 ( n ) F p X_1(n)_{\mathbb{F}_p} X 1 ( n ) F p
Table 4. The newforms of positive analytic rank. For n = 57 , 65 , 77 n=57,65,77 n = 57 , 65 , 77 a a a n = 63 n=63 n = 63 χ \chi χ { 1 , 5 , 25 , 38 , 58 , 62 } \{1,5,25,38,58,62\} { 1 , 5 , 25 , 38 , 58 , 62 } Δ ( 63 ) \Delta(63) Δ ( 63 ) ⟨ 5 ⟩ \langle5\rangle ⟨ 5 ⟩ n = 85 n=85 n = 85 χ \chi χ [9 Table 2] .
Table 5. The twisted Hecke sieve computations of Proposition 7.7 . The columns “points” and “orbits” list, for the degrees e e e X 1 ( n ) F p X_1(n)_{\mathbb{F}_p} X 1 ( n ) F p e e e
under the diamond action, are given in the table. For one representative D D D q q q t t t D D D
Condition (2) of Proposition 7.5 is vacuous in all three cases, as we now explain. By [7 Lemma 2.8] the Galois orbits of cusps over Q \mathbb{Q} Q 1 18 , 2 9 , 1 8 , 18 1^{18},2^9,1^8,18 1 18 , 2 9 , 1 8 , 18 n = 57 n=57 n = 57 Q \mathbb{Q} Q Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) Q ( ζ 19 ) \mathbb{Q}(\zeta_{19}) Q ( ζ 19 ) Q ( ζ 57 ) + \mathbb{Q}(\zeta_{57})^+ Q ( ζ 57 ) + 1 18 , 2 6 , 6 3 , 6 3 , 12 , 18 1^{18},2^6,6^3,6^3,12,18 1 18 , 2 6 , 6 3 , 6 3 , 12 , 18 n = 63 n=63 n = 63 Q \mathbb{Q} Q Q ( ζ 3 ) \mathbb{Q}(\zeta_3) Q ( ζ 3 ) Q ( ζ 7 ) \mathbb{Q}(\zeta_7) Q ( ζ 7 ) Q ( ζ 9 ) \mathbb{Q}(\zeta_9) Q ( ζ 9 ) Q ( ζ 21 ) \mathbb{Q}(\zeta_{21}) Q ( ζ 21 ) Q ( ζ 63 ) + \mathbb{Q}(\zeta_{63})^+ Q ( ζ 63 ) + 1 24 , 4 6 , 12 2 , 24 1^{24},4^6,12^{2},24 1 24 , 4 6 , 1 2 2 , 24 n = 65 n=65 n = 65 Q \mathbb{Q} Q Q ( ζ 5 ) \mathbb{Q}(\zeta_5) Q ( ζ 5 ) Q ( ζ 13 ) \mathbb{Q}(\zeta_{13}) Q ( ζ 13 ) Q ( ζ 65 ) + \mathbb{Q}(\zeta_{65})^+ Q ( ζ 65 ) + H i H_i H i Lemma 6.1 are all of ( Z / n Z ) × (\mathbb{Z}/n\mathbb{Z})^\times ( Z / n Z ) × p p p n = 57 , 63 n=57,63 n = 57 , 63 X 1 ( n ) F p X_1(n)_{\mathbb{F}_p} X 1 ( n ) F p ≤ 2 \le2 ≤ 2 Q \mathbb{Q} Q
It follows that the assumptions of Proposition 7.5 are satisfied for every possible D + C D+C D + C Y 1 ( n ) ( 5 ) ( Q ) Y_1(n)^{(5)}(\mathbb{Q}) Y 1 ( n ) ( 5 ) ( Q ) x ( 5 ) ∈ Y 1 ( n ) ( 5 ) ( Q ) x^{(5)}\in Y_1(n)^{(5)}(\mathbb{Q}) x ( 5 ) ∈ Y 1 ( n ) ( 5 ) ( Q ) □ \square □
Remark 7.8. For n = 63 n=63 n = 63 t H t_H t H H = ⟨ ± 8 ⟩ H=\langle\pm8\rangle H = ⟨ ± 8 ⟩ χ ( 8 ) = − 1 \chi(8)=-1 χ ( 8 ) = − 1 χ \chi χ J H J_H J H H = ⟨ ± 8 ⟩ H=\langle\pm8\rangle H = ⟨ ± 8 ⟩ J H J_H J H H H H [9 Proposition 6.4] ). We apply t 5 t_5 t 5 t ⟨ ± 8 ⟩ t_{\langle\pm8\rangle} t ⟨ ± 8 ⟩ t 5 t_5 t 5
Direct analysis via the cuspidal class group
The Hecke sieve shows that certain divisors on X 1 ( n ) F p X_1(n)_{\mathbb{F}_p} X 1 ( n ) F p X 1 ( 28 ) X_1(28) X 1 ( 28 ) X 1 ( 30 ) X_1(30) X 1 ( 30 ) X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) F p \mathbb{F}_p F p [7 §5.2] , which determines all rational effective divisors of a given degree.
Proposition 8.1. Let X = X 1 ( m , n ) X=X_1(m,n) X = X 1 ( m , n ) m ∈ { 1 , 2 } m\in\{1,2\} m ∈ { 1 , 2 } p ∤ 2 n p\nmid2n p ∤ 2 n rk J ( Q ) = 0 \operatorname{rk}J(\mathbb{Q})=0 rk J ( Q ) = 0 J ( Q ) = Cl Q X J(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}X J ( Q ) = Cl Q X c 1 , … , c r c_1,\ldots,c_r c 1 , … , c r Q \mathbb{Q} Q c 1 c_1 c 1 Q \mathbb{Q} Q φ : Z r → Cl ( X F p ) \varphi:\mathbb{Z}^r\to\operatorname{Cl}(X_{\mathbb{F}_p}) φ : Z r → Cl ( X F p ) i i i c i ‾ − deg ( c i ) c 1 ‾ \overline{c_i}-\deg(c_i)\overline{c_1} c i − deg ( c i ) c 1
(a) Z r / ker φ ≃ J ( Q ) \mathbb{Z}^r/\ker\varphi\simeq J(\mathbb{Q}) Z r / ker φ ≃ J ( Q ) p p p J ( Q ) J(\mathbb{Q}) J ( Q ) im φ \operatorname{im}\varphi im φ
(b) if D ∈ X ( 5 ) ( Q ) D\in X^{(5)}(\mathbb{Q}) D ∈ X ( 5 ) ( Q ) red p ( [ D − 5 c 1 ] + [ 5 c 1 ‾ ] ) \operatorname{red}_p([D-5c_1]+[5\overline{c_1}]) red p ([ D − 5 c 1 ] + [ 5 c 1 ]) S p S_p S p X F p X_{\mathbb{F}_p} X F p
(c) for every γ ∈ J ( Q ) \gamma\in J(\mathbb{Q}) γ ∈ J ( Q ) Q \mathbb{Q} Q 5 c 1 + γ 5c_1+\gamma 5 c 1 + γ ∣ 5 c 1 + γ ∣ \lvert5c_1+\gamma\rvert ∣ 5 c 1 + γ ∣
Proof. Reduction modulo p p p J ( Q ) = J ( Q ) t o r s J(\mathbb{Q})=J(\mathbb{Q})_{\mathrm{tors}} J ( Q ) = J ( Q ) tors [18 Appendix] , since p > 2 p>2 p > 2 X X X Z r → J ( Q ) \mathbb{Z}^{r}\to J(\mathbb{Q}) Z r → J ( Q ) i i i [ c i − deg ( c i ) c 1 ] [c_i-\deg(c_i)c_1] [ c i − deg ( c i ) c 1 ] φ \varphi φ red p \operatorname{red}_p red p ker φ \ker\varphi ker φ Z r → J ( Q ) \mathbb{Z}^{r}\to J(\mathbb{Q}) Z r → J ( Q ) red p \operatorname{red}_p red p D D D X F p X_{\mathbb{F}_p} X F p
Remark 8.2. Proposition 8.1 turns the determination of X ( 5 ) ( Q ) X^{(5)}(\mathbb{Q}) X ( 5 ) ( Q ) Cl ( X F p ) \operatorname{Cl}(X_{\mathbb{F}_p}) Cl ( X F p ) X F p X_{\mathbb{F}_p} X F p S p S_p S p J ( Q ) J(\mathbb{Q}) J ( Q ) Z r / ker φ \mathbb{Z}^{r}/\ker\varphi Z r / ker φ γ \gamma γ p p p γ \gamma γ γ \gamma γ L ( 5 c 1 + γ ) L(5c_1+\gamma) L ( 5 c 1 + γ ) Q \mathbb{Q} Q ∣ 5 c 1 + γ ∣ |5c_1+\gamma| ∣5 c 1 + γ ∣ X ( 5 ) ( Q ) X^{(5)}(\mathbb{Q}) X ( 5 ) ( Q )
Remark 8.3. The choice of base point in Proposition 8.1 matters: c 1 c_1 c 1 Q \mathbb{Q} Q F p \mathbb{F}_p F p p p p > 1 >1 > 1 F p \mathbb{F}_p F p F p \mathbb{F}_p F p Q \mathbb{Q} Q Cl ( X F p ) \operatorname{Cl}(X_{\mathbb{F}_p}) Cl ( X F p ) [ c ˉ i − deg ( c i ) c ˉ ] [\bar{c}_i-\deg(c_i)\bar{c}] [ c ˉ i − deg ( c i ) c ˉ ] Cl Q c X \operatorname{Cl}_{\mathbb{Q}}^{\mathrm{c}}X Cl Q c X X 1 ( 30 ) X_1(30) X 1 ( 30 ) Q \mathbb{Q} Q F 7 \mathbb{F}_7 F 7 F 7 \mathbb{F}_7 F 7 Q \mathbb{Q} Q c ˉ \bar{c} c ˉ [ 136 , 8160 ] [136,8160] [ 136 , 8160 ] [ 4 , 8160 ] [4,8160] [ 4 , 8160 ]
The hypothesis J ( Q ) = Cl Q c X J(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^{\mathrm{c}}X J ( Q ) = Cl Q c X X 1 ( n ) X_1(n) X 1 ( n ) n ≤ 55 n\le55 n ≤ 55 n ≠ 37 , 43 , 53 , 54 n\ne37,43,53,54 n = 37 , 43 , 53 , 54 [7 Corollary 4.14] , and for X 1 ( 2 , 2 N ) X_1(2,2N) X 1 ( 2 , 2 N ) N ≤ 16 N\le16 N ≤ 16 [7 Theorem 3.1(3) and Proposition 4.16] .
We also explain, once and for all, how to count the effective Q \mathbb{Q} Q X X X a e a_e a e Q \mathbb{Q} Q e e e d d d t d t^d t d
∏ e ≥ 1 ( 1 − t e ) − a e . (3) \prod_{e\ge1}(1-t^e)^{-a_e}. \tag*{(3)} e ≥ 1 ∏ ( 1 − t e ) − a e . (3) In each proof below the degrees of the Galois orbits of cusps are listed, and the number of effective Q \mathbb{Q} Q t 5 t^5 t 5 gon Q X > 5 \operatorname{gon}_{\mathbb{Q}}X>5 gon Q X > 5 X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 )
Proposition 8.4. The groups Z / 2 Z × Z / 20 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/20\mathbb{Z} Z /2 Z × Z /20 Z Z / 2 Z × Z / 24 Z \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z} Z /2 Z × Z /24 Z Lemma 3.2 , neither are Z / 40 Z \mathbb{Z}/40\mathbb{Z} Z /40 Z Z / 48 Z \mathbb{Z}/48\mathbb{Z} Z /48 Z
Proof. Let X = X 1 ( 2 , 20 ) X=X_1(2,20) X = X 1 ( 2 , 20 ) J ( Q ) = Cl Q c X ≃ Z / 4 Z × Z / 60 Z × Z / 120 Z J(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^{\mathrm{c}}X\simeq\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/60\mathbb{Z}\times\mathbb{Z}/120\mathbb{Z} J ( Q ) = Cl Q c X ≃ Z /4 Z × Z /60 Z × Z /120 Z gon Q X > 5 \operatorname{gon}_{\mathbb{Q}}X>5 gon Q X > 5 gon F 3 X > 5 \operatorname{gon}_{\mathbb{F}_3}X>5 gon F 3 X > 5 [12 proof of Proposition 5.3] . There are 16 Galois orbits of cusps over Q \mathbb{Q} Q 1 8 1^8 1 8 2 4 2^4 2 4 4 4 4^4 4 4 p = 3 p=3 p = 3 p = 7 p=7 p = 7 [ 4 , 60 , 120 ] [4,60,120] [ 4 , 60 , 120 ] F 3 \mathbb{F}_3 F 3 F 7 \mathbb{F}_7 F 7 J ( Q ) J(\mathbb{Q}) J ( Q ) p = 3 p=3 p = 3 p = 7 p=7 p = 7 Q \mathbb{Q} Q Q \mathbb{Q} Q gon Q X > 5 \operatorname{gon}_{\mathbb{Q}} X > 5 gon Q X > 5 X X X
The argument for X = X 1 ( 2 , 24 ) X=X_1(2,24) X = X 1 ( 2 , 24 ) p = 5 p=5 p = 5 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}_{\mathbb{Q}}^{\mathrm{c}}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 1 8 1^8 1 8 2 6 2^6 2 6 4 5 4^5 4 5 gon F 5 X > 5 \operatorname{gon}_{\mathbb{F}_5}X>5 gon F 5 X > 5 [12 proof of Proposition 5.3] , so gon Q X > 5 \operatorname{gon}_{\mathbb{Q}}X>5 gon Q X > 5 F 5 \mathbb{F}_5 F 5 J ( Q ) J(\mathbb{Q}) J ( Q ) Q \mathbb{Q} Q
Remark 8.5. The same computation carried out on X 1 ( 26 ) X_1(26) X 1 ( 26 ) p = 7 p=7 p = 7 Proposition 6.3 for n = 26 n=26 n = 26 J ( Q ) = Cl Q c X 1 ( 26 ) ≃ Z / 133 Z × Z / 1995 Z J(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^{\mathrm{c}}X_1(26)\simeq\mathbb{Z}/133\mathbb{Z}\times\mathbb{Z}/1995\mathbb{Z} J ( Q ) = Cl Q c X 1 ( 26 ) ≃ Z /133 Z × Z /1995 Z ( 16 5 ) = 4368 \binom{16}{5}=4368 ( 5 16 ) = 4368
We now turn to the three curves that do have quintic points.
Proposition 8.6. The non-cuspidal closed points of degree 5 on X 1 ( 28 ) X_1(28) X 1 ( 28 ) ⟨ a ⟩ x 0 \langle a\rangle x_0 ⟨ a ⟩ x 0 a ∈ Δ ( 28 ) a\in\Delta(28) a ∈ Δ ( 28 )
x 0 = ( α , α 3 − 1 ) ∈ X 1 ( 28 ) , α 5 − α 4 − 2 α 3 − α 2 + 2 α + 2 = 0 , x_0=(\alpha,\alpha^3-1)\in X_1(28),\qquad\alpha^5-\alpha^4-2\alpha^3-\alpha^2+2\alpha+2=0, x 0 = ( α , α 3 − 1 ) ∈ X 1 ( 28 ) , α 5 − α 4 − 2 α 3 − α 2 + 2 α + 2 = 0 , in Sutherland’s coordinates [31 ] .
Proof. We have rk J 1 ( 28 ) ( Q ) = 0 \operatorname{rk}J_1(28)(\mathbb{Q})=0 rk J 1 ( 28 ) ( Q ) = 0 J 1 ( 28 ) ( Q ) = Cl Q c X 1 ( 28 ) ≃ Z / 2 Z × Z / 4 Z × Z / 12 Z × Z / 936 Z J_1(28)(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^{\mathrm{c}}X_1(28)\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/12\mathbb{Z}\times\mathbb{Z}/936\mathbb{Z} J 1 ( 28 ) ( Q ) = Cl Q c X 1 ( 28 ) ≃ Z /2 Z × Z /4 Z × Z /12 Z × Z /936 Z [7 Theorem 3.1(2), Corollary 4.14 and Table 2] , and gon Q X 1 ( 28 ) = 6 \operatorname{gon}_{\mathbb{Q}}X_1(28)=6 gon Q X 1 ( 28 ) = 6 [13 Theorem 4 and Table 1] , so every class of degree 5 contains at most one effective divisor. We use Proposition 8.1 with p = 3 p=3 p = 3 Q \mathbb{Q} Q 1 9 1^9 1 9 2 3 2^3 2 3 3 3 3 6 6 6 6 6 6 Cl Q c X 1 ( 28 ) \operatorname{Cl}_{\mathbb{Q}}^{\mathrm{c}}X_1(28) Cl Q c X 1 ( 28 ) Cl ( X F 3 ) \operatorname{Cl}(X_{\mathbb{F}_3}) Cl ( X F 3 ) [ 2 , 4 , 12 , 936 ] [2,4,12,936] [ 2 , 4 , 12 , 936 ] F 3 \mathbb{F}_3 F 3 1 , … , 5 1,\ldots,5 1 , … , 5 J ( Q ) J(\mathbb{Q}) J ( Q ) Proposition 8.1 . Of these, 1884 are the classes of the 1884 effective Q \mathbb{Q} Q ⟨ a ⟩ x 0 ( 5 ) \langle a\rangle x_0^{(5)} ⟨ a ⟩ x 0 ( 5 ) x 0 x_0 x 0 Q ( α ) \mathbb{Q}(\alpha) Q ( α ) Q ( α ) \mathbb{Q}(\alpha) Q ( α ) x 0 ( 5 ) x_0^{(5)} x 0 ( 5 ) Q \mathbb{Q} Q X 1 ( 28 ) X_1(28) X 1 ( 28 ) ⟨ a ⟩ x 0 ( 5 ) \langle a\rangle x_0^{(5)} ⟨ a ⟩ x 0 ( 5 )
Remark 8.7. The Hecke sieve at p = 3 p=3 p = 3 1 + 2 + 9 1+2+9 1 + 2 + 9 [ D + C − 5 C ‾ ] [D+C-5\overline{C}] [ D + C − 5 C ] J 1 ( 28 ) ( Q ) J_1(28)(\mathbb{Q}) J 1 ( 28 ) ( Q ) C C C [9 Proposition 5.13] .
Proposition 8.8. The non-cuspidal closed points of degree 5 on X 1 ( 30 ) X_1(30) X 1 ( 30 ) ⟨ a ⟩ x i \langle a\rangle x_i ⟨ a ⟩ x i a ∈ Δ ( 30 ) a\in\Delta(30) a ∈ Δ ( 30 ) i = 1 , 2 i=1,2 i = 1 , 2
x 1 = ( α , 2 α 4 + α 3 − 6 α 2 + 4 α + 4 ) , x 2 = ( β , 1 53 ( 3 β 4 + 7 β 3 + 6 β 2 + 11 β − 73 ) ) , x_1=(\alpha,2\alpha^4+\alpha^3-6\alpha^2+4\alpha+4),\qquad x_2=\left(\beta,\frac{1}{53}(3\beta^4+7\beta^3+6\beta^2+11\beta-73)\right), x 1 = ( α , 2 α 4 + α 3 − 6 α 2 + 4 α + 4 ) , x 2 = ( β , 53 1 ( 3 β 4 + 7 β 3 + 6 β 2 + 11 β − 73 ) ) , where α 5 + α 4 − 3 α 3 + 3 α + 1 = 0 \alpha^5+\alpha^4-3\alpha^3+3\alpha+1=0 α 5 + α 4 − 3 α 3 + 3 α + 1 = 0 β 5 + β 4 − 7 β 3 + β 2 + 12 β + 3 = 0 \beta^5+\beta^4-7\beta^3+\beta^2+12\beta+3=0 β 5 + β 4 − 7 β 3 + β 2 + 12 β + 3 = 0 Q ( α ) \mathbb{Q}(\alpha) Q ( α ) Q ( β ) \mathbb{Q}(\beta) Q ( β )
Proof. We have rk J 1 ( 30 ) ( Q ) = 0 \operatorname{rk} J_1(30)(\mathbb{Q})=0 rk J 1 ( 30 ) ( Q ) = 0 J 1 ( 30 ) ( Q ) = Cl Q c X 1 ( 30 ) ≃ Z / 4 Z × Z / 8160 Z J_1(30)(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^c X_1(30)\simeq\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/8160\mathbb{Z} J 1 ( 30 ) ( Q ) = Cl Q c X 1 ( 30 ) ≃ Z /4 Z × Z /8160 Z 32640 32640 32640 [7 Theorem 3.1(2), Corollary 4.14 and Table 2] , and gon Q X 1 ( 30 ) = 6 \operatorname{gon}_{\mathbb{Q}}X_1(30)=6 gon Q X 1 ( 30 ) = 6 [13 Theorem 4 and Table 1] . We use Proposition 8.1 with p = 7 p=7 p = 7 Q \mathbb{Q} Q 1 8 1^8 1 8 2 4 2^4 2 4 4 4 4^4 4 4 Cl Q c \operatorname{Cl}_{\mathbb{Q}}^c Cl Q c Cl ( X F 7 ) \operatorname{Cl}(X_{\mathbb{F}_7}) Cl ( X F 7 ) [ 4 , 8160 ] [4,8160] [ 4 , 8160 ] F 7 \mathbb{F}_7 F 7 1 , … , 5 1,\ldots,5 1 , … , 5 J ( Q ) J(\mathbb{Q}) J ( Q ) Q \mathbb{Q} Q ⟨ a ⟩ x i ( 5 ) \langle a\rangle x_i^{(5)} ⟨ a ⟩ x i ( 5 ) i = 1 , 2 i=1,2 i = 1 , 2 a ∈ Δ ( 30 ) a\in\Delta(30) a ∈ Δ ( 30 ) x 1 , x 2 x_1,x_2 x 1 , x 2 gon Q X 1 ( 30 ) = 6 > 5 \operatorname{gon}_{\mathbb{Q}}X_1(30)=6>5 gon Q X 1 ( 30 ) = 6 > 5
Proposition 8.9. The curve X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) j j j Q \mathbb{Q} Q
Proof. We have rk J 1 ( 2 , 18 ) ( Q ) = 0 \operatorname{rk} J_1(2,18)(\mathbb{Q})=0 rk J 1 ( 2 , 18 ) ( Q ) = 0 J 1 ( 2 , 18 ) ( Q ) = Cl Q c X 1 ( 2 , 18 ) ≃ Z / 2 Z × Z / 42 Z × Z / 126 Z J_1(2,18)(\mathbb{Q})=\operatorname{Cl}_{\mathbb{Q}}^c X_1(2,18)\simeq\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/42\mathbb{Z}\times\mathbb{Z}/126\mathbb{Z} J 1 ( 2 , 18 ) ( Q ) = Cl Q c X 1 ( 2 , 18 ) ≃ Z /2 Z × Z /42 Z × Z /126 Z [7 Theorem 3.1(3), Proposition 4.16 and Table 3] . The gonality of X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) Propositions 8.6 and 8.8 . Instead we use the fact, proved by Derickx and Sutherland [12 Proposition 5.4] , that Q ( X 1 ( 2 , 18 ) ) \mathbb{Q}(X_1(2,18)) Q ( X 1 ( 2 , 18 ))
We run Proposition 8.1 at p = 5 p=5 p = 5 Q \mathbb{Q} Q 1 9 1^9 1 9 2 3 2^3 2 3 3 3 3^3 3 3 Cl Q c \operatorname{Cl}_{\mathbb{Q}}^c Cl Q c Cl ( X F 5 ) \operatorname{Cl}(X_{\mathbb{F}_5}) Cl ( X F 5 ) [ 2 , 42 , 126 ] [2,42,126] [ 2 , 42 , 126 ] F 5 \mathbb{F}_5 F 5 1 , … , 5 1,\ldots,5 1 , … , 5 J ( Q ) J(\mathbb{Q}) J ( Q ) Proposition 8.1 .
By (3) there are 1980 effective Q \mathbb{Q} Q
We claim that a class containing an effective cuspidal divisor C C C D ∼ C D\sim C D ∼ C Q \mathbb{Q} Q D D D f ∈ Q ( X 1 ( 2 , 18 ) ) × f\in\mathbb{Q}(X_1(2,18))^\times f ∈ Q ( X 1 ( 2 , 18 ) ) × div ( f ) = D − C \operatorname{div}(f)=D-C div ( f ) = D − C D D D C C C D D D C C C f f f C C C f f f deg C = 5 \deg C=5 deg C = 5 [12 Proposition 5.4] .
There remain 1944 − 1926 = 18 1944-1926=18 1944 − 1926 = 18 L ( 5 c 1 + γ ) L(5c_1+\gamma) L ( 5 c 1 + γ ) Q \mathbb{Q} Q j j j
Remark 8.10. A supplementary computation shows that 27 of the 1944 candidate classes contain more than one effective rational divisor of degree 5, forming a pencil whose fixed part is a Q \mathbb{Q} Q X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) [12 Proposition 5.4] for this curve.
Table 6. The computations of this section. “Candidates” are the classes of J ( Q ) J(\mathbb{Q}) J ( Q ) Proposition 8.1 at every prime p p p Q \mathbb{Q} Q
The three sporadic groups and the proof of the main theorem
We now make the three exceptional cases explicit. Let x , y x,y x , y X 1 ( n ) X_1(n) X 1 ( n ) [31 ] . Following [32 ] , from a non-cuspidal point ( x 0 , y 0 ) (x_0,y_0) ( x 0 , y 0 )
r = x 0 2 y 0 − x 0 y 0 + y 0 − 1 x 0 2 y 0 − x 0 , s = x 0 y 0 − y 0 + 1 x 0 y 0 , b = r s ( r − 1 ) , c = s ( r − 1 ) , r=\frac{x_0^2y_0-x_0y_0+y_0-1}{x_0^2y_0-x_0},\qquad s=\frac{x_0y_0-y_0+1}{x_0y_0},\qquad b=rs(r-1),\qquad c=s(r-1), r = x 0 2 y 0 − x 0 x 0 2 y 0 − x 0 y 0 + y 0 − 1 , s = x 0 y 0 x 0 y 0 − y 0 + 1 , b = r s ( r − 1 ) , c = s ( r − 1 ) , and then
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) has P = ( 0 , 0 ) P=(0,0) P = ( 0 , 0 ) n n n X 1 ( 2 , 2 n ) X_1(2,2n) X 1 ( 2 , 2 n ) [11 ] a point yields b , c b,c b , c
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 P = ( 0 , 0 ) P=(0,0) P = ( 0 , 0 ) Q = ( b , b ) Q=(b,b) Q = ( b , b ) 2 n 2n 2 n
Proposition 9.1. (1) Let K 28 : = Q ( α ) K_{28}:=\mathbb{Q}(\alpha) K 28 := Q ( α ) α 5 − α 4 − 2 α 3 − α 2 + 2 α + 2 = 0 \alpha^5-\alpha^4-2\alpha^3-\alpha^2+2\alpha+2=0 α 5 − α 4 − 2 α 3 − α 2 + 2 α + 2 = 0 E 28 = E ( b , c ) E_{28}=E(b,c) E 28 = E ( b , c ) (4) with
b = 419 α 4 − 759 α 3 − 222 α 2 − 239 α + 1032 , c = 17 α 4 − 31 α 3 − 8 α 2 − 11 α + 42. (6) b=419\alpha^4-759\alpha^3-222\alpha^2-239\alpha+1032,\qquad c=17\alpha^4-31\alpha^3-8\alpha^2-11\alpha+42. \tag*{(6)} b = 419 α 4 − 759 α 3 − 222 α 2 − 239 α + 1032 , c = 17 α 4 − 31 α 3 − 8 α 2 − 11 α + 42. (6) Then P = ( 0 , 0 ) P=(0,0) P = ( 0 , 0 ) E 28 ( K 28 ) t o r s ≃ Z / 28 Z E_{28}(K_{28})_{\mathrm{tors}}\simeq\mathbb{Z}/28\mathbb{Z} E 28 ( K 28 ) tors ≃ Z /28 Z
(2) Let K 30 : = Q ( α ) K_{30}:=\mathbb{Q}(\alpha) K 30 := Q ( α ) α 5 + α 4 − 3 α 3 + 3 α + 1 = 0 \alpha^5+\alpha^4-3\alpha^3+3\alpha+1=0 α 5 + α 4 − 3 α 3 + 3 α + 1 = 0 E 30 , E 30 ′ E_{30},E'_{30} E 30 , E 30 ′ E ( b , c ) E(b,c) E ( b , c ) (4) with, respectively,
b = − 6378 α 4 − 1917 α 3 + 20475 α 2 − 14321 α − 9118 , b=-6378\alpha^4-1917\alpha^3+20475\alpha^2-14321\alpha-9118, b = − 6378 α 4 − 1917 α 3 + 20475 α 2 − 14321 α − 9118 , c = − 114 α 4 − 34 α 3 + 366 α 2 − 257 α − 162 , (7) c=-114\alpha^4-34\alpha^3+366\alpha^2-257\alpha-162, \tag*{(7)} c = − 114 α 4 − 34 α 3 + 366 α 2 − 257 α − 162 , (7) b = 190 α 4 − 38 α 3 − 382 α 2 + 238 α + 295 , b=190\alpha^4-38\alpha^3-382\alpha^2+238\alpha+295, b = 190 α 4 − 38 α 3 − 382 α 2 + 238 α + 295 , c = 7 α 4 + 13 α 3 − 36 α 2 + 9 α + 28. (8) c=7\alpha^4+13\alpha^3-36\alpha^2+9\alpha+28. \tag*{(8)} c = 7 α 4 + 13 α 3 − 36 α 2 + 9 α + 28. (8) Then in both cases P = ( 0 , 0 ) P=(0,0) P = ( 0 , 0 ) Z / 30 Z \mathbb{Z}/30\mathbb{Z} Z /30 Z K 30 K_{30} K 30 K 30 K_{30} K 30
(3) Let K 2 , 18 : = Q ( w ) K_{2,18} := \mathbb{Q}(w) K 2 , 18 := Q ( w ) w 5 − 2 w 4 + 2 w 3 − 3 w 2 − w + 4 = 0 w^5 - 2w^4 + 2w^3 - 3w^2 - w + 4 = 0 w 5 − 2 w 4 + 2 w 3 − 3 w 2 − w + 4 = 0 E 2 , 18 = E ′ ( b , c ) E_{2,18} = E'(b,c) E 2 , 18 = E ′ ( b , c ) (5) with
b = 1 3 ( 480 w 4 − 1364 w 3 + 2108 w 2 − 3216 w + 2248 ) , c = 1 3 ( − 940 w 4 + 2668 w 3 − 4128 w 2 + 6296 w − 4389 ) . (9) \begin{aligned}
b &= \frac{1}{3}(480w^4 - 1364w^3 + 2108w^2 - 3216w + 2248),\\
c &= \frac{1}{3}(-940w^4 + 2668w^3 - 4128w^2 + 6296w - 4389). \tag*{(9)}
\end{aligned} b c = 3 1 ( 480 w 4 − 1364 w 3 + 2108 w 2 − 3216 w + 2248 ) , = 3 1 ( − 940 w 4 + 2668 w 3 − 4128 w 2 + 6296 w − 4389 ) . (9) Then P = ( 0 , 0 ) P=(0,0) P = ( 0 , 0 ) Q = ( b , b ) Q=(b,b) Q = ( b , b ) E 2 , 18 ( K 2 , 18 ) t o r s ≃ Z / 2 Z × Z / 18 Z E_{2,18}(K_{2,18})_{\mathrm{tors}} \simeq\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} E 2 , 18 ( K 2 , 18 ) tors ≃ Z /2 Z × Z /18 Z
In all four cases the curve has no complex multiplication and [ Q ( j ) : Q ] = 5 [\mathbb{Q}(j):\mathbb{Q}]=5 [ Q ( j ) : Q ] = 5 Table 7 .
Proof. We verify this directly in Magma [2 ] . The curves of (1) and (2) are obtained from van Hoeij’s points of Propositions 8.6 and 8.8 through the formulas above, and we check that the values of b b b c c c Q ( α ) \mathbb{Q}(\alpha) Q ( α ) Q ( α ) \mathbb{Q}(\alpha) Q ( α ) Q ( w ) \mathbb{Q}(w) Q ( w )
Proposition 9.2. We use the notation of Proposition 9.1. Let K K K E / K E/K E / K
If E ( K ) E(K) E ( K ) ( K , E ) ≃ ( K 28 , E 28 ) (K,E) \simeq(K_{28},E_{28}) ( K , E ) ≃ ( K 28 , E 28 ) E ( K ) t o r s ≃ Z / 28 Z E(K)_{\mathrm{tors}} \simeq\mathbb{Z}/28\mathbb{Z} E ( K ) tors ≃ Z /28 Z
If E ( K ) E(K) E ( K ) ( K , E ) (K,E) ( K , E ) ( K 30 , E 30 ) (K_{30},E_{30}) ( K 30 , E 30 ) ( K 30 , E 30 ′ ) (K_{30},E'_{30}) ( K 30 , E 30 ′ ) E ( K ) t o r s ≃ Z / 30 Z E(K)_{\mathrm{tors}} \simeq\mathbb{Z}/30\mathbb{Z} E ( K ) tors ≃ Z /30 Z
If E ( K ) E(K) E ( K ) Z / 2 Z × Z / 18 Z \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} Z /2 Z × Z /18 Z ( K , E ) ≃ ( K 2 , 18 , E 2 , 18 ) (K,E) \simeq(K_{2,18},E_{2,18}) ( K , E ) ≃ ( K 2 , 18 , E 2 , 18 ) E ( K ) t o r s ≃ Z / 2 Z × Z / 18 Z E(K)_{\mathrm{tors}} \simeq\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} E ( K ) tors ≃ Z /2 Z × Z /18 Z
In particular no group properly containing 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
Proof. We prove (1) and (2) together, and then (3), where an extra step is needed. As in § 3 , a point of order 28 on E ( K ) E(K) E ( K ) x ˉ \bar{x} x ˉ Y 1 ( 28 ) Y_1(28) Y 1 ( 28 ) Q ( x ˉ ) ≃ K \mathbb{Q}(\bar{x}) \simeq K Q ( x ˉ ) ≃ K Proposition 8.6 x ˉ \bar{x} x ˉ ⟨ a ⟩ x 0 \langle a\rangle x_0 ⟨ a ⟩ x 0 K 28 K_{28} K 28 X 1 ( 28 ) X_1(28) X 1 ( 28 ) Q ( x ˉ ) \mathbb{Q}(\bar{x}) Q ( x ˉ ) ( E 28 , a P ) (E_{28},aP) ( E 28 , a P ) a a a K 28 K_{28} K 28 E ≃ E 28 E \simeq E_{28} E ≃ E 28 E 28 ( K ) t o r s ≃ Z / 28 Z E_{28}(K)_{\mathrm{tors}} \simeq\mathbb{Z}/28\mathbb{Z} E 28 ( K ) tors ≃ Z /28 Z Proposition 8.8 in place of Proposition Proposition 8.6 .
For (3), put E : = E 2 , 18 E := E_{2,18} E := E 2 , 18 K : = K 2 , 18 K := K_{2,18} K := K 2 , 18 E E E j j j K K K E E E Q Q Q P P P ⟨ Q ⟩ \langle Q\rangle ⟨ Q ⟩ 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 ) Z / 2 Z × Z / 18 Z ↪ E ( K ) \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} \hookrightarrow E(K) Z /2 Z × Z /18 Z ↪ E ( K ) K K K Y 1 ( 2 , 18 ) Y_1(2,18) Y 1 ( 2 , 18 ) σ ∈ Gal ( Q / Q ) \sigma\in\operatorname{Gal}(\mathbb{Q}/\mathbb{Q}) σ ∈ Gal ( Q / Q ) j ( E ) j(E) j ( E ) K = Q ( j ( E ) ) K = \mathbb{Q}(j(E)) K = Q ( j ( E )) E E E K K K X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 )
Now let E ′ / K ′ E'/K' E ′ / K ′ Z / 2 Z × Z / 18 Z ⊆ E ′ ( K ′ ) \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} \subseteq E'(K') Z /2 Z × Z /18 Z ⊆ E ′ ( K ′ ) X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) ( K ′ , E ′ ) ≃ ( K , E ) (K',E') \simeq(K,E) ( K ′ , E ′ ) ≃ ( K , E ) E ( K ) t o r s ≃ Z / 2 Z × Z / 18 Z E(K)_{\mathrm{tors}} \simeq\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} E ( K ) tors ≃ Z /2 Z × Z /18 Z
In all three cases the last sentence follows, since a group G G G E ( K ) t o r s E(K)_{\mathrm{tors}} E ( K ) tors E E E G G G
Proposition 9.3. Z / 36 Z \mathbb{Z}/36\mathbb{Z} Z /36 Z not a quintic torsion group.
Proof. Suppose E 1 E_1 E 1 K K K P 1 ∈ E 1 ( K ) P_1 \in E_1(K) P 1 ∈ E 1 ( K ) Lemma 3.2 the triple ( E ′ , Q ′ , P 1 m o d ⟨ 18 P 1 ⟩ ) (E',Q',P_1 \bmod\langle18P_1\rangle) ( E ′ , Q ′ , P 1 mod ⟨ 18 P 1 ⟩) E ′ = E 1 / ⟨ 18 P 1 ⟩ E' = E_1/\langle18P_1\rangle E ′ = E 1 / ⟨ 18 P 1 ⟩ Q ′ Q' Q ′ E 1 [ 2 ] / ⟨ 18 P 1 ⟩ E_1[2]/\langle18P_1\rangle E 1 [ 2 ] / ⟨ 18 P 1 ⟩ K K K Y 1 ( 2 , 18 ) Y_1(2,18) Y 1 ( 2 , 18 ) Q \mathbb{Q} Q Z / 2 Z × Z / 18 Z ∉ Φ ( 1 ) \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} \notin\Phi(1) Z /2 Z × Z /18 Z ∈ / Φ ( 1 ) Proposition 9.2 (3) we may assume K = Q ( w ) K = \mathbb{Q}(w) K = Q ( w ) E ′ = E 2 , 18 E' = E_{2,18} E ′ = E 2 , 18 Q ′ ∈ E 2 , 18 [ 2 ] Q' \in E_{2,18}[2] Q ′ ∈ E 2 , 18 [ 2 ]
Let ϕ : E 1 → E ′ \phi: E_1 \to E' ϕ : E 1 → E ′ ⟨ 18 P 1 ⟩ \langle18P_1\rangle ⟨ 18 P 1 ⟩ ker ϕ ^ = ϕ ( E 1 [ 2 ] ) = ⟨ Q ′ ⟩ \ker\hat{\phi} = \phi(E_1[2]) = \langle Q'\rangle ker ϕ ^ = ϕ ( E 1 [ 2 ]) = ⟨ Q ′ ⟩ E 1 ≃ E ′ / ⟨ Q ′ ⟩ E_1 \simeq E'/\langle Q'\rangle E 1 ≃ E ′ / ⟨ Q ′ ⟩ E 2 , 18 E_{2,18} E 2 , 18 K K K K K K Z / 18 Z \mathbb{Z}/18\mathbb{Z} Z /18 Z
Proof of Theorems 1.1 to 1.3. By Lemma Lemma 3.1 , if G ∈ Φ ( 5 ) ∖ Φ ∞ ( 5 ) G \in\Phi(5) \setminus\Phi^{\infty}(5) G ∈ Φ ( 5 ) ∖ Φ ∞ ( 5 ) G G G G G G Table 1 lists, for each of the 45 members of G G G Proposition 9.2 shows that any elliptic curve over a quintic field containing one of them is one of the four curves of Proposition Proposition 9.1, whose torsion groups are Z / 28 Z \mathbb{Z}/28\mathbb{Z} Z /28 Z Z / 30 Z \mathbb{Z}/30\mathbb{Z} Z /30 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 G ∈ { Z / 28 Z , Z / 30 Z , Z / 2 Z × Z / 18 Z } G \in\{\mathbb{Z}/28\mathbb{Z}, \mathbb{Z}/30\mathbb{Z}, \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z}\} G ∈ { Z /28 Z , Z /30 Z , Z /2 Z × Z /18 Z } Theorem 1.1 , and Propositions Proposition 9.1 and Proposition 9.2 prove Theorem Theorem 1.2 .
We now prove Theorem Theorem 1.3 . The counts of closed points are Propositions Proposition 8.6 , Proposition 8.8 and Proposition 8.9. On X 1 ( 28 ) X_1(28) X 1 ( 28 ) x 0 x_0 x 0 # Δ ( 28 ) = φ ( 28 ) / 2 = 6 \#\Delta(28) = \varphi(28)/2 = 6 #Δ ( 28 ) = φ ( 28 ) /2 = 6 E 28 ( K ) t o r s ≃ Z / 28 Z E_{28}(K)_{\mathrm{tors}} \simeq\mathbb{Z}/28\mathbb{Z} E 28 ( K ) tors ≃ Z /28 Z # Δ ( 30 ) = φ ( 30 ) / 2 = 4 \#\Delta(30) = \varphi(30)/2 = 4 #Δ ( 30 ) = φ ( 30 ) /2 = 4 X 1 ( 30 ) X_1(30) X 1 ( 30 ) E 30 E_{30} E 30 E 30 ′ E'_{30} E 30 ′ X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) Z / 2 Z × Z / 18 Z ↪ E 2 , 18 ( K ) \mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/18\mathbb{Z} \hookrightarrow E_{2,18}(K) Z /2 Z × Z /18 Z ↪ E 2 , 18 ( K ) Proposition 9.2 . □
Points of degree five on X 1 ( n ) X_1(n) X 1 ( n ) X 1 ( 2 , 2 n ) X_1(2, 2n) X 1 ( 2 , 2 n )
We can now prove Corollary 1.4 and describe the sporadic and isolated points involved.
Proof of Corollary 1.4 . For n ≤ 4 n \le4 n ≤ 4 X 1 ( n ) X_1(n) X 1 ( n ) n ≥ 5 n \ge5 n ≥ 5 n n n n n n Φ ( 5 ) \Phi(5) Φ ( 5 ) 1 , … , 22 , 24 , 25 , 28 , 30 1,\ldots,22,24,25,28,30 1 , … , 22 , 24 , 25 , 28 , 30 X 1 ( n ) X_1(n) X 1 ( n ) n ≤ 25 n \le25 n ≤ 25 n ≠ 23 n \ne23 n = 23 [13 Theorem 3] , and the counts for n = 28 , 30 n = 28,30 n = 28 , 30 Theorem 1.3 . The same argument gives (2), using [12 Proposition 5.3] for n ≤ 8 n \le8 n ≤ 8 n ≥ 10 n \ge10 n ≥ 10 E ( K ) t o r s ∈ Φ ( 5 ) E(K)_{\mathrm{tors}} \in\Phi(5) E ( K ) tors ∈ Φ ( 5 ) 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 Φ ( 5 ) \Phi(5) Φ ( 5 ) 18 < 2 n 18 < 2n 18 < 2 n X 1 ( 28 ) X_1(28) X 1 ( 28 ) X 1 ( 30 ) X_1(30) X 1 ( 30 ) gon Q = 6 \operatorname{gon}_{\mathbb{Q}} = 6 gon Q = 6 ≤ 5 \le5 ≤ 5 [12 Propositions 2.2 and 2.3] , and X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 )
Remark 9.4. The word non-cuspidal cannot be dropped from Corollary 1.4 (3). On X 1 ( 44 ) X_1(44) X 1 ( 44 ) d = 2 d = 2 d = 2 φ ( 22 ) φ ( 2 ) / 2 = 5 \varphi(22)\varphi(2)/2 = 5 φ ( 22 ) φ ( 2 ) /2 = 5 [7 Lemma 2.8] , on which Gal ( Q ‾ / Q ) \operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) Gal ( Q / Q ) ( Z / 22 Z ) × (\mathbb{Z}/22\mathbb{Z})^\times ( Z /22 Z ) × { ± 1 } \{\pm1\} { ± 1 } Q ( ζ 11 ) \mathbb{Q}(\zeta_{11}) Q ( ζ 11 ) rk J 1 ( 44 ) ( Q ) = 0 \operatorname{rk} J_1(44)(\mathbb{Q}) = 0 rk J 1 ( 44 ) ( Q ) = 0 [7 Theorem 3.1(2)] and gon Q X 1 ( 44 ) > 8 \operatorname{gon}_{\mathbb{Q}} X_1(44) > 8 gon Q X 1 ( 44 ) > 8 [13 proof of Proposition 6] , the curve has only finitely many points of degree ≤ 5 \le5 ≤ 5 [12 Proposition 2.3] , so this cuspidal point is sporadic.
The quintic points of Theorem 1.3 (3) are not sporadic, since gon Q X 1 ( 2 , 18 ) = 4 \operatorname{gon}_{\mathbb{Q}} X_1(2,18) = 4 gon Q X 1 ( 2 , 18 ) = 4 X 1 ( 2 , 18 ) X_1(2,18) X 1 ( 2 , 18 ) isolated in the sense of Bourdon, Ejder, Liu, Odumodu and Viray [3 Definition 4.1] . The curve has only finitely many points of degree 5, so by [3 Theorem 4.2 (1)] all of them are isolated.
Remark 9.5. The curve E 2 , 18 E_{2,18} E 2 , 18 K K K
The computations
All of our computations were performed in Magma V2.29-4 [2 ] on the Mordell server at the Department of Mathematics, University of Zagreb, with an AMD EPYC 9175F CPU and 384 GB of RAM, using the package mdmagma [26 ] for the moduli-theoretic operations (the routines ModuliPoint, LevelStructure, HeckeOperator, DiamondOperator, CuspOrbitsQ), Sutherland’s optimized models of X 1 ( n ) X_1(n) X 1 ( n ) [30 , 31 ] and the Derickx–Sutherland models of X 1 ( 2 , 2 n ) X_1(2, 2n) X 1 ( 2 , 2 n ) [11 ] . The total cost of the computations reported here was about 203 hours of CPU time, of which 197 hours were spent on the four curves X 1 ( 45 ) X_1(45) X 1 ( 45 ) X 1 ( 57 ) X_1(57) X 1 ( 57 ) X 1 ( 63 ) X_1(63) X 1 ( 63 ) X 1 ( 65 ) X_1(65) X 1 ( 65 ) X 1 ( 42 ) X_1(42) X 1 ( 42 )
Fast Hecke operators and diamond orbits
On X 1 ( 57 ) X_1(57) X 1 ( 57 ) X 1 ( 63 ) X_1(63) X 1 ( 63 ) X 1 ( 65 ) X_1(65) X 1 ( 65 ) mdm: the Hecke operator, which builds the splitting field of the whole q q q
Lemma 10.1. Let q ∤ p n q \nmid pn q ∤ p n z z z X 1 ( n ) F p X_1(n)_{\mathbb{F}_p} X 1 ( n ) F p k k k ( E , P ) (E,P) ( E , P ) κ : = F p k \kappa:= \mathbb{F}_{p^k} κ := F p k O \mathcal{O} O Gal ( κ ‾ / κ ) \operatorname{Gal}(\overline{\kappa}/\kappa) Gal ( κ / κ ) q + 1 q+1 q + 1 q q q E [ q ] E[q] E [ q ] O ∈ O O \in\mathcal{O} O ∈ O C ∈ O C \in O C ∈ O L L L C C C y O y_O y O X 1 ( n ) F p X_1(n)_{\mathbb{F}_p} X 1 ( n ) F p ( E / C , P m o d C ) (E/C,P\bmod C) ( E / C , P mod C )
T q ( z ) = ∑ O ∈ O [ L : κ ] ⋅ k deg y O y O , T_q(z)=\sum_{O\in\mathcal{O}}\frac{[L:\kappa]\cdot k}{\deg y_O}\,y_O, T q ( z ) = O ∈ O ∑ deg y O [ L : κ ] ⋅ k y O , and [ L : κ ] = # O [L:\kappa]=\#O [ L : κ ] = # O L L L κ \kappa κ g C = ∏ Q ∈ ( C ∖ { 0 } ) / ± ( T − x ( Q ) ) g_C=\prod_{Q\in(C\setminus\{0\})/\pm}(T-x(Q)) g C = ∏ Q ∈ ( C ∖ { 0 }) / ± ( T − x ( Q )) x ( Q ) x(Q) x ( Q ) x x x Q Q Q E E E
Proof. By definition T q ( z ) = ∑ σ ∈ Gal ( κ ‾ / F p ) ∑ C σ ( E / C , P m o d C ) T_q(z)=\sum_{\sigma\in\operatorname{Gal}(\overline{\kappa}/\mathbb{F}_p)}\sum_C \sigma(E/C,P\bmod C) T q ( z ) = ∑ σ ∈ Gal ( κ / F p ) ∑ C σ ( E / C , P mod C ) C C C q + 1 q+1 q + 1 q q q E [ q ] E[q] E [ q ] k ( q + 1 ) k(q+1) k ( q + 1 ) T q ( z ) = ∑ y ( m y k / deg y ) y T_q(z)=\sum_y(m_yk/\deg y)y T q ( z ) = ∑ y ( m y k / deg y ) y m y = # { C : ( E / C , P m o d C ) ∈ y } m_y=\#\{C:(E/C,P\bmod C)\in y\} m y = # { C : ( E / C , P mod C ) ∈ y } [26 ] . Now E E E P P P κ \kappa κ C ′ = F i ( C ) C'=F^i(C) C ′ = F i ( C ) F F F κ \kappa κ ( E / C ′ , P m o d C ′ ) = F i ( E / C , P m o d C ) (E/C',P\bmod C')=F^i(E/C,P\bmod C) ( E / C ′ , P mod C ′ ) = F i ( E / C , P mod C ) X X X m y = ∑ O : y O = y # O m_y=\sum_{O:y_O=y}\#O m y = ∑ O : y O = y # O C C C # O \#O # O # O = [ L : κ ] \#O=[L:\kappa] # O = [ L : κ ] L L L C C C q q q x x x C C C g C g_C g C L = κ ( coefficients of g C ) L=\kappa(\text{coefficients of }g_C) L = κ ( coefficients of g C )
In the implementation, which assumes q q q C C C x 1 x_1 x 1 q q q E E E κ \kappa κ x x x x ( j Q ) x(jQ) x ( j Q ) 1 ≤ j ≤ ( q − 1 ) / 2 1\le j\le(q-1)/2 1 ≤ j ≤ ( q − 1 ) /2 x 1 x_1 x 1 κ \kappa κ κ ( x 1 ) \kappa(x_1) κ ( x 1 ) E → E / C E\to E/C E → E / C L L L y O y_O y O O O O N L / κ ( g C ) = ∏ C ′ ∈ O g C ′ N_{L/\kappa}(g_C)=\prod_{C'\in O}g_{C'} N L / κ ( g C ) = ∏ C ′ ∈ O g C ′ ∑ O # O = q + 1 \sum_O\#O=q+1 ∑ O # O = q + 1 q = 2 q=2 q = 2 q = 2 q=2 q = 2 g C g_C g C
Lemma 10.2. Let x , x ′ x,x' x , x ′ X 1 ( n ) F p X_1(n)_{\mathbb{F}_p} X 1 ( n ) F p k k k ( E 1 , P 1 ) (E_1,P_1) ( E 1 , P 1 ) κ 1 \kappa_1 κ 1 ( E 2 , P 2 ) (E_2,P_2) ( E 2 , P 2 ) κ 2 \kappa_2 κ 2 x ′ = ⟨ a ⟩ x x'=\langle a\rangle x x ′ = ⟨ a ⟩ x a ∈ ( Z / n Z ) × a\in(\mathbb{Z}/n\mathbb{Z})^\times a ∈ ( Z / n Z ) × κ 2 → κ 1 \kappa_2\to\kappa_1 κ 2 → κ 1 σ \sigma σ κ 1 \kappa_1 κ 1 ι : E 2 σ → E 1 \iota:E_2^\sigma\to E_1 ι : E 2 σ → E 1 κ 1 \kappa_1 κ 1 ι ( P 2 σ ) \iota(P_2^\sigma) ι ( P 2 σ ) ⟨ P 1 ⟩ \langle P_1\rangle ⟨ P 1 ⟩
Proof. The diamond operators commute with the Galois action, so x ′ = ⟨ a ⟩ x x'=\langle a\rangle x x ′ = ⟨ a ⟩ x ( E 1 , a P 1 ) (E_1,aP_1) ( E 1 , a P 1 ) Gal ( F ‾ p / F p ) \operatorname{Gal}(\overline{\mathbb{F}}_p/\mathbb{F}_p) Gal ( F p / F p ) ( E 2 , P 2 ) (E_2,P_2) ( E 2 , P 2 ) σ \sigma σ ι : ( E 2 , P 2 ) σ → ( E 1 , a P 1 ) \iota:(E_2,P_2)^\sigma\to(E_1,aP_1) ι : ( E 2 , P 2 ) σ → ( E 1 , a P 1 ) F ‾ p \overline{\mathbb{F}}_p F p ι \iota ι κ 1 \kappa_1 κ 1 ι F ι − 1 \iota F\iota^{-1} ι F ι − 1 F F F κ 1 \kappa_1 κ 1 x ↦ x p k x\mapsto x^{p^k} x ↦ x p k E 1 E_1 E 1 a P 1 aP_1 a P 1 κ 1 \kappa_1 κ 1 n ≥ 5 n\ge5 n ≥ 5 ζ \zeta ζ ker ( ζ − 1 ) \ker(\zeta-1) ker ( ζ − 1 ) ζ = [ − 1 ] \zeta=[-1] ζ = [ − 1 ] ker ( ζ − 1 ) = E [ 2 ] \ker(\zeta-1)=E[2] ker ( ζ − 1 ) = E [ 2 ]
The implementation loops over the k k k σ \sigma σ E 2 σ → E 1 E_2^\sigma\to E_1 E 2 σ → E 1 κ 1 \kappa_1 κ 1 E 1 E_1 E 1 a ∈ ( Z / n Z ) × a \in(\mathbb{Z}/n\mathbb{Z})^{\times} a ∈ ( Z / n Z ) × j j j F p \mathbb{F}_p F p X 1 ( 45 ) F 7 X_1(45)_{\mathbb{F}_7} X 1 ( 45 ) F 7 T 11 T_{11} T 11 X 1 ( 57 ) F 5 X_1(57)_{\mathbb{F}_5} X 1 ( 57 ) F 5 T q ( x ) T_q(x) T q ( x ) ( n , p , deg x , q ) (n,p,\deg x,q) ( n , p , deg x , q ) X 1 ( 26 ) F 7 X_1(26)_{\mathbb{F}_7} X 1 ( 26 ) F 7 X 1 ( 28 ) F 5 X_1(28)_{\mathbb{F}_5} X 1 ( 28 ) F 5 X 1 ( 30 ) F 7 X_1(30)_{\mathbb{F}_7} X 1 ( 30 ) F 7
University of Zagreb Faculty of Science, Department of Mathematics, Bijenička cesta 30, 10000 Zagreb, Croatia
Email address: fnajman@math.hr
