math.NT — Number Theory

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

Contributed by David Niedbala Giraudin

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.