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Double Dirichlet series over function fields

Published online by Cambridge University Press:  04 December 2007

Benji Fisher  and
Solomon Friedberg
Benji Fisher
Affiliation:
Mathematics Department, Boston College, Chestnut Hill, MA 02467-3806, [email protected], [email protected]
Solomon Friedberg
Affiliation:
Mathematics Department, Boston College, Chestnut Hill, MA 02467-3806, [email protected], [email protected]
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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.

Keywords

algebraic curveL-functionquadratic twistrational functionsum of L-functions11M38 (primary)11G2011L0511L0711R4711R5814H05 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 3 , May 2004 , pp. 613 - 630
Copyright
Foundation Compositio Mathematica 2004