Detecting large simple rational Hecke modules for $\Gamma_0(N)$ via congruences

TL;DR

The paper introduces a new method to bound the size of simple Hecke modules in modular forms spaces without relying on eigenvalue distribution, providing improved bounds for certain prime levels and proposing conjectures on Hecke module structure.

Contribution

A novel congruence-based approach to bound the dimension of simple Hecke submodules, improving existing bounds and suggesting new conjectures on Hecke module simplicity.

Findings

Unconditional bound: d ≥ log₂(log₂(N/8)) for N ≡ 7 mod 8

Improved bounds over previous √log log N estimates

Proposed conjectures on Hecke module structure and simplicity

Abstract

We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_2(\Gamma_0(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_0(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. For prime levels $N\equiv 7\mod 8$ our method yields an unconditional bound of $d\ge\log_2\log_2(N/8)$, improving the known bound of $d\gg\sqrt{\log\log N}$ due to Murty--Sinha and Royer. We also discuss conditional bounds, the strongest of which is $d\gg_\epsilon N^{1/2-\epsilon}$ over a large set of primes $N$, contingent on Soundararajan's heuristics for the class number problem and Artin's conjecture on primitive roots. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_k(\mathrm{SL}_2(\mathbb{Z});\mathbb{Q})$.