## Journal of the Mathematical Society of Japan

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

#### Abstract

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.

#### Note

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

#### Article information

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

Dates
Revised: 9 October 2018
First available in Project Euclid: 28 October 2019

https://projecteuclid.org/euclid.jmsj/1572249776

Digital Object Identifier
doi:10.2969/jmsj/78657865

Mathematical Reviews number (MathSciNet)
MR4055095

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1572249776

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