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

David Niedbala Giraudin

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

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Contributed by David Niedbala Giraudin ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

Let

f

be a normalized Hecke eigenform of level

1

and weight

k

, with eigenvalues

a_n(f)

. In 1981 Hatada asked whether

a_p(f)\equiv 1+p \pmod{2^a}

whenever

p\equiv\pm1\pmod{2^{a-1}}

and

6\le a\le 13

. Rustom's congruence modulo

128

answers this for

a\le 7

. We show that the answer is affirmative for

a\le 10

and negative for

a=11

,

12

and

13

, the counterexamples coming from weight

16

and

p=12289

. The proof combines Rustom's theorem with a certificate for

T_5

modulo

512

on period polynomials, a

2

-adic Galois law (for instance

a_p\equiv 1+p^{k-1}\pmod{2^{10}}

for every

p\equiv 1\pmod 8

), and a reduction to a computation in a finite group. As a consequence,

a_p(f)=0

implies

p\equiv -1\pmod{2^{11}}

. At the prime

3

we show that, on Hatada's lattice of period polynomials,

T_2^2\equiv 9(k-1)^2\pmod{27}

, with refinements modulo

81

. This gives an independent proof of Conjecture 9.1 of Rustom without any ramification hypothesis and of the level

1

case of the Coleman--Stein conjecture, both of which also follow from Rustom's recent classification modulo

81

. We also prove that

a_n(f)\equiv\sigma_{k-1}(n)\pmod{81}

for every

n\equiv 1\pmod 3

, and a

3

-adic Galois law modulo

243

on the Galois group of

\mathbb{Q}(\sqrt{-3})

, which gives congruences modulo

243

and

729

for suitable split primes and a bilinear law for primes

\ell\equiv 2\pmod 3

. Beyond

729

, for every prime

\ell\equiv 2\pmod 3

we prove

a_\ell^2\equiv\sigma_{k-1}(\ell)^2\pmod{3^{\min(2j+2,10)}}

, where

j=v_3(\ell+1)

; the exponent

2j+2

is attained for

j\le 4

and fails for

j=5

. Each statement is marked as proved, known, verified or open. The finite computations are certified by seven Python scripts, provided as ancillary files.

Provenance statement

The research and the writing were carried out by the author with Claude (Anthropic). The proofs were checked by independent AI reviews and by exact computation. No professional mathematician has reviewed the note yet.

Tools used

Anthropic: Claude Opus

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Version history

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