On the standard L-function for \(\mathrm{GSp}_{2n} \times \mathrm{GL}_1\) and algebraicity of symmetric fourth L-values for \(\mathrm{GL}_2\)

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 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 values of the symmetric fourth L-function of a classical newform.