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arXiv · 1810.02006

A note on the dimension of the largest simple Hecke submodule

Sandro Bettin·Corentin Perret-Gentil·Maksym Radziwiłł
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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.

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Sandro Bettin, Corentin Perret-Gentil, Maksym Radziwiłł. 2018-10-03. A note on the dimension of the largest simple Hecke submodule. https://arxiv.org/abs/1810.02006

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