Algebra & Number Theory

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

Ian Petrow

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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.

Article information

Algebra Number Theory, Volume 12, Number 10 (2018), 2471-2498.

Received: 6 May 2018
Revised: 22 July 2018
Accepted: 23 August 2018
First available in Project Euclid: 14 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F25: Hecke-Petersson operators, differential operators (one variable)
Secondary: 11F11: Holomorphic modular forms of integral weight 11F72: Spectral theory; Selberg trace formula 11G20: Curves over finite and local fields [See also 14H25] 14G15: Finite ground fields

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


Petrow, Ian. Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field. Algebra Number Theory 12 (2018), no. 10, 2471--2498. doi:10.2140/ant.2018.12.2471.

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