A note on the dimension of the largest simple Hecke submodule

TL;DR

This paper provides new lower bounds on the size of the largest simple Hecke submodule in spaces of modular forms, using an analytic approach to improve understanding of their structure for large levels.

Contribution

It introduces a simple analytic method to establish lower bounds on the dimension of the largest simple Hecke submodule, extending results to more general settings.

Findings

Lower bounds grow roughly as log log N / log(2p).

Results apply to various modular form spaces, including those with nebentypus.

Bounds are stronger or comparable to previous results for specific N subsets.

Abstract

For $k\ge 2$ even, let $d_{k,N}$ denote the dimension of the largest simple Hecke submodule of $S_{k}(\Gamma_0(N); \mathbb{Q})^\text{new}$. We show, using a simple analytic method, that $d_{k,N} \gg_k \log\log N / \log(2p)$ with $p$ the smallest prime co-prime to $N$. Previously, bounds of this quality were only known for $N$ in certain subsets of the primes. We also establish similar (and sometimes stronger) results concerning $S_{k}(\Gamma_0(N), \chi)$, with $k \geq 2$ an integer and $\chi$ an arbitrary nebentypus.