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# Title: 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.
 Comments: 47 pages. This paper supersedes arxiv:1702.05004 Subjects: Number Theory (math.NT) Cite as: arXiv:1803.06227 [math.NT] (or arXiv:1803.06227v1 [math.NT] for this version)

## Submission history

From: Abhishek Saha [view email]
[v1] Thu, 15 Mar 2018 16:18:28 GMT (45kb)