Hecke eigenvalues in level 1 modulo powers of 2 and 3: Hatada's question (1981) and the Coleman-Stein conjecture
Source: hexagonmath.org/abs/2611.00035v1
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 $a_p\equiv 1+p^{k-1}\pmod{2^{10}}$…
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