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Published April 2021 | Submitted
Journal Article Open

A Note on the Dimension of the Largest Simple Hecke Submodule


For k ≥ 2 even, let d_(k,N) denote the dimension of the largest simple Hecke submodule of S_k(Γ₀ (N);Q)^(new)⁠. We show, using a simple analytic method, that d_(k,N) ≫ 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(Γ₀ (N),χ)⁠, with k ≥ 2 an integer and χ an arbitrary nebentypus.

Additional Information

© The Author(s) 2018. Published by Oxford University Press. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model). Received October 3, 2018; Revised December 5, 2018; Accepted December 7, 2018. Published: 26 December 2018. We would like to thank Nicolas Billerey, Armand Brumer, and Ricardo Menares for comments on the manuscript. We would like to thank the referees for the careful reading of the paper and useful suggestions. This work was partially supported by PRIN (Progetti di Ricerca di Interesse Nazionale) 2015 "Number Theory and Arithmetic Geometry" [to S.B.]; Sloan fellowship [to M.R.].

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