Bessel periods and anticyclotomic $p$-adic spinor $L$-functions

Abstract

We construct the anticyclotomic p-adic L-function that interpolates a square root of central values of twisted spinor L-functions of a quadratic base change of a Siegel cusp form of genus 2 with respect to a paramodular group of square-free level, assuming the Böcherer conjecture for the central L-values with anticyclotomic twists.

On the Gross-Prasad conjecture with its refinement for $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the generalized Böcherer conjecture

“…Assuming that the endoscopic classification holds for G = / , we may apply Theorem 1.2 to holomorphic modular forms on G (A). In this case, Hsieh-Yamana [51] compute local Bessel periods and show an explicit formula for Bessel periods such as (8.3.1) for scalar valued holomorphic modular forms, including the case when = and is an even squarefree integer. Meanwhile we shall maintain to be odd in Theorem 8.1, since our computation of the local Bessel period at the real place in the vector valued case in Appendix B is performed under the assumption that is odd.…”

“…= 0, the explicit computation of ∞ is done in Dickson et al [18, 3.5] using the explicit formula for matrix coefficients when ≥ 3. Meanwhile Hsieh and Yamana [51,Proposition 5.7] compute ∞ in a different way when ≥ 2, based on Shimura's work on confluent hypergeometric functions.…”

“…The formula (1.8.2) and its generalization (8.3.1) are expected to have a broad spectrum of interesting applications both arithmetic and analytic. Some of the examples are [10], [18, Section 3], [19], [51], [86] and [100].…”