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1 results

  • Double Dirichlet series over function fields

  • Benji Fisher, Solomon Friedberg
  • Journal:
    Compositio Mathematica / Volume 140 / Issue 3 / May 2004
    Published online by Cambridge University Press:
    04 December 2007, pp. 613-630
    Print publication:
    May 2004
    • Article
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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.