Compositio Mathematica

Double Dirichlet series over function fields

Benji Fisher a1 and Solomon Friedberg a1
a1 Mathematics Department, Boston College, Chestnut Hill, MA 02467-3806, USA,

Article author query
fisher b   [Google Scholar] 
friedberg s   [Google Scholar] 


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.

(Received November 9 2001)
(Accepted September 17 2002)

Key Words: algebraic curve; L-function; quadratic twist; rational function; sum of L-functions.

Maths Classification

11M38 (primary); 11G20; 11L05; 11L07; 11R47; 11R58; 14H05 (secondary).