Hecke eigenvalues in level 1 modulo powers of 2 and 3: Hatada's question (1981) and the Coleman-Stein conjecture

The question

For even $k$, let $S_k$ be the space of cusp forms of weight $k$ for $\mathrm{SL}_2(\mathbb{Z})$, and let $\lambda_p$ be an eigenvalue of the Hecke operator $T_k(p)$, $p$ an odd prime. Hatada proved that $\lambda_p \equiv1+p \pmod8$ for every odd $p$ [9], and then that

[10 p. 305]. He then asks the following question [10 p. 306]:

Is it true that $\lambda_p \equiv1+p \pmod{2^a}$ if $p\equiv\pm1\pmod{2^{a-1}}$ and $6\leq a\leq13$?

together with a stronger version concerning the lattice $L_{w+2}$ of forms whose odd periods are integers (stability of $L_{w+2}$ under $(T_{w+2}(p)-1-p)/2^a$), which implies the former. In his Remark 1, Hatada points out that for $a\geq14$ the congruence may fail when $p\equiv-1\pmod{2^a}$, after Swinnerton-Dyer [22]. According to Rustom [17], the question was asked at Serre’s suggestion. Rustom proved $\lambda_p\equiv1+p^{k-1}\pmod{128}$ for every odd $p$ and every $k$ [17 Thm 8.1], which gives the affirmative answer for $a\leq7$.¹

In residue characteristic 3, Hatada proved $\lambda_p\equiv1+p\pmod3$ for $p\ne3$, $\lambda_3\equiv0\pmod9$, and $\lambda_3\equiv0\pmod{27}$ if $3\nmid k$ [9], congruences (5) to (7). Coleman and Stein [6] conjectured that there are only five residue classes modulo 9 of eigenforms (levels 1, 3, 9), and checked it up to weight 74 in level 1. Rustom [17, Thm 9.2] states that finiteness modulo 9 in level 1 follows from the following statement.

Chiriac [4] proved unconditionally that $a_2 \equiv0$ (mod $9\overline{\mathbb{Z}}_3$) implies $k \equiv4$ (mod 6) [4 Thm 1.2], and the whole of Conjecture 9.1 when 3 is unramified in the coefficient field [4 Cor. 1.3]; since Rustom checked this non-ramification for $k \le1000$ [17], Conjecture 9.1 is proved for $k \le1000$.

Related work. The recent preprint [18] of Rustom classifies, in every weight, the prime-to-$p$ Hecke eigensystems of level 1 modulo $2^8$, $3^4$, $5^3$ and $7^2$, by Hecke-equivariant transfer maps between modular-symbol modules. At $p=3$, [18 Thm 1.5] gives in particular the literal congruence modulo 27 of every prime-to-3 eigensystem with an Eisenstein twist; this implies Conjecture 9.1 of [17] without any ramification hypothesis, and the level 1 case of the Coleman–Stein conjecture. Our Corollary L and Theorem M were obtained independently, by a different route (explicit certificates on Hatada’s lattice), and [18] has priority for these two statements. At $p=2$, [18 Thm 1.4] is a classification modulo $2^8$ in the valuative sense ($v_2(a_\ell-\alpha)>7$); it does not address Hatada’s question, and for $p\equiv\pm1$ (mod $2^7$) it gives the valuative form of the case $a=8$. The congruences modulo $2^9$ and $2^{10}$ of § 8, the negative answer for $a=11,12,13$, and the 3-adic results beyond 81 of §§§ 7–§ 9 are not covered by [18]. Earlier, Calegari and Emerton [3, Lemma 3.3] had proved $a_\ell\equiv1+\ell$ (mod 9) for $\ell\equiv\pm1$ (mod 9).

The main results of this paper are the following. We first give again (Theorem F, § 4), on Hatada’s lattice, his congruences $a_3(f)\equiv0$ (mod 9), and (mod 27) if $3\nmid k$ [9]. Next (§ 5), on Hatada’s lattice, $T_2^2\equiv9(k-1)^2$ (mod 27), and, more precisely, modulo 81, $T_2^2\equiv\sigma_{k-1}(2)^2$ if $k\equiv0,2$ (mod 6) and $T_2^2+9T_2\equiv0$ if $k\equiv4$ (mod 6) (Theorems I and K). It follows that Conjecture 9.1 of [17] holds for every eigenform of level 1, without any ramification hypothesis (this also follows from [18 Thm 1.5], see above), and even that $a_2(f)\equiv\pm\sigma_{k-1}(2)$ (mod 27) when $k\equiv0,2$ (mod 6) (Corollary L). Together with [17], we obtain the level 1 case of the Coleman–Stein conjecture (Theorem M). Next, § 6 extends these congruences to the operators $T_\ell$, $\ell\le19$, and deduces, with the structure theorem for the 3-adic Hecke algebra of [17], a law valid in every weight: for every eigenform $f$ of level 1 and every $n\equiv1$ (mod 3), $a_n(f)\equiv\sigma_{k-1}(n)$ (mod 81), and, if $k\not\equiv4$ (mod 6), $a_n(f)\equiv\psi(n)\sigma_{k-1}(n)$ (mod 27) for every $n$ prime to 3, with $\psi$ trivial or equal to the character of conductor 3 (Theorem O). Section 7 derives a Galois law: on the Galois group of $\mathbb{Q}(\sqrt{-3})$, $\operatorname{tr}\rho_f=1+\chi^{k-1}+(a_2^2-\sigma_{k-1}(2)^2)u^2$ (mod 243), where $u$ measures the action on $\zeta_9$ (Theorem P); hence $a_\ell\equiv\sigma_{k-1}(\ell)$ (mod 243) if $\ell\equiv1$ (mod 9), a congruence modulo 729 for the other split completely in $\mathbb{Q}(\zeta_9,\sqrt[3]{3})$ (Theorem R), a bilinear law for $\ell,\ell'=2$ (mod 3) (Theorem S), and, if $k\not\equiv4$ (mod 6), the determination of all $a_n$, $3\nmid n$, modulo 81 by $a_2$ modulo 81 (Corollary T). Then § 8 establishes a 2-adic Galois law (Theorem U), from the modulo 128 theorem of [17] and a certificate for $T_5$ modulo 512: $a_p\equiv1+p^{k-1}$ (mod $2^{10}$) for every $p\equiv1$ (mod 8), and explicit laws on the other classes; with a reduction to a computation in a finite group (Theorem W), Hatada’s question is completely answered: affirmatively for $a\le10$, negatively for $a=11,12,13$ (Corollary X), and $a_p(f)=0$ implies $p\equiv-1$ (mod $2^{11}$) for every eigenform of level 1 (Corollary Y). Finally, § 9 extends the 3-adic Galois law beyond 729: for every prime $\ell\equiv2$ (mod 3), $a_\ell^2\equiv\sigma_{k-1}(\ell)^2$ (mod $3^{\min(2j+2,10)}$), where $j=v_3(\ell+1)$ (Theorem R$'$); the exponent $2j+2$ is extended for $j\le4$ and no longer holds for $j=5$. It follows that $v_3(a_\ell)\ge v_3(\ell+1)$ if $v_3(\ell+1)\le4$ and $k\not\equiv4$ (mod 6), and that, for such $k$, $a_\ell(f)=0$ implies $\ell\equiv-1$ (mod $2^{11}\cdot3^5$) (Corollary Y$'$). For $\ell\equiv1$ (mod 27) with 3 a cube modulo $\ell$, $a_\ell=\sigma_{k-1}(\ell)$ (mod $3^7$), and (mod $3^8$) if $k\equiv4$ (mod 6) and $\ell=X^2+XY+547Y^2$ (Theorem R$''$).