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Mathematics > Number Theory

Title: Integral representation and critical $L$-values for holomorphic forms on $GSp_{2n} \times GL_1$

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. The proof of this application uses our recent structure theorem [arXiv:1501.00524] for the space of nearly holomorphic Siegel modular forms of degree 2 and arbitrary level.
Comments: 43 pages
Subjects: Number Theory (math.NT)
Cite as: arXiv:1702.05004 [math.NT]
  (or arXiv:1702.05004v1 [math.NT] for this version)

Submission history

From: Abhishek Saha [view email]
[v1] Thu, 16 Feb 2017 15:04:17 GMT (40kb)