Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

Petrow, Ian

doi:10.2140/ant.2018.12.2471

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Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

Ian Petrow

Vol. 12 (2018), No. 10, 2471–2498

DOI: 10.2140/ant.2018.12.2471

Abstract

We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.

Keywords

traces of Hecke operators, modular curves over a finite field, elliptic curves over a finite field, Petersson formula for newforms, Tsfasman–Vlăduţ–Zink theorem

Mathematical Subject Classification 2010

Primary: 11F25

Secondary: 11F11, 11F72, 11G20, 14G15

Milestones

Received: 6 May 2018

Revised: 22 July 2018

Accepted: 23 August 2018

Published: 1 February 2019

Authors

Ian Petrow
Departement Mathematik ETH Zürich Zürich Switzerland

Vol. 12, No. 10, 2018