Journal of the Mathematical Society of Japan

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

Martin DICKSON, Ameya PITALE, Abhishek SAHA, and Ralf SCHMIDT

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We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of $L$-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan–Gross–Prasad conjecture for Bessel periods as proposed by Liu. We note several consequences of our conjecture to arithmetic and analytic properties of $L$-functions and Fourier coefficients of Siegel modular forms.


The third author acknowledges the support of the EPSRC grant EP/L025515/1.

Article information

J. Math. Soc. Japan, Volume 72, Number 1 (2020), 251-301.

Received: 18 August 2017
Revised: 9 October 2018
First available in Project Euclid: 28 October 2019

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Mathematical Reviews number (MathSciNet)

Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11F30: Fourier coefficients of automorphic forms 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Siegel modular forms Bessel periods Fourier coefficients $L$-functions


DICKSON, Martin; PITALE, Ameya; SAHA, Abhishek; SCHMIDT, Ralf. Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level. J. Math. Soc. Japan 72 (2020), no. 1, 251--301. doi:10.2969/jmsj/78657865.

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