On the standard $L$-function for $GSp_{2n} \times GL_1$ and algebraicity of symmetric fourth $L$-values for $GL_2$

Abstract

We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case $n=2$ we are able to prove a reciprocity law -- predicted by Deligne's conjecture -- for the critical special values of the twisted standard $L$-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan--Shahidi lift, we obtain a reciprocity law for the critical special values of the symmetric fourth $L$-function of a classical newform.