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Double Dirichlet series over function fields

Published online by Cambridge University Press:  04 December 2007

Benji Fisher
Affiliation:
Mathematics Department, Boston College, Chestnut Hill, MA 02467-3806, USAbenji@member.ams.org, friedber@bc.edu
Solomon Friedberg
Affiliation:
Mathematics Department, Boston College, Chestnut Hill, MA 02467-3806, USAbenji@member.ams.org, friedber@bc.edu
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Abstract

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We construct a finite-dimensional vector space of functions of two complex variables attached to a smooth algebraic curve C over a finite field $\mathbb{F}_q$, q odd, and a level. These functions collect the analytic information about the cohomology of the curve and its quadratic twists that is encoded in the corresponding L-functions; they are double Dirichlet series in two independent complex variables s and w. We prove that these series satisfy a finite, non-abelian group of functional equations in the two complex variables (s, w) and are rational functions in q-s and q-w with a specified denominator. The group is D6, the dihedral group of order 12.

Type
Research Article
Copyright
Foundation Compositio Mathematica 2004