L-functions for holomorphic forms on GSp(4) x GL(2) and their special values

TL;DR

This paper develops an explicit integral representation for L-functions associated with pairs of holomorphic forms on GSp(4) and GL(2), extending previous results and proving algebraicity of certain critical values.

Contribution

It introduces a new explicit integral formula for L-functions of genus 2 Siegel and elliptic forms with squarefree levels, including level extension and algebraicity results.

Findings

Explicit integral representation for L-functions of (F,g) pairs.

Extension of known results to higher levels and more general cases.

Proof of algebraicity of critical special values of these L-functions.

Abstract

We provide an explicit integral representation for L-functions of pairs (F,g) where F is a holomorphic genus 2 Siegel newform and g a holomorphic elliptic newform, both of squarefree levels and of equal weights. When F,g have level one, this was earlier known by the work of Furusawa. The extension is not straightforward. Our methods involve precise double-coset and volume computations as well as an explicit formula for the Bessel model for GSp(4) in the Steinberg case; the latter is possibly of independent interest. We apply our integral representation to prove an algebraicity result for a critical special value of L(s, F \times g). This is in the spirit of known results on critical values of triple product L-functions, also of degree 8, though there are significant differences.