Duke Mathematical Journal

Sums of twisted GL(2) L-functions over function fields

Benji Fisher and Solomon Friedberg

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Abstract

Let $K$ be a function field of odd characteristic, and let $\pi$ (resp., $\eta$) be a cuspidal automorphic representation of ${\rm GL}\sb 2(\mathbb {A}\sb K)$ (resp., ${\rm GL}\sb 1(\mathbb {A}\sb K)$). Then we show that a weighted sum of the twists of $L(s,\pi)$ by quadratic characters $\chi\sb D,\sum \sb DL(s,\pi\otimes \sp \chi\sb D)a\sb 0(s,\pi,D)\eta(D)|D|\sp {-w}$, is a rational function and has a finite, nonabelian group of functional equations. A similar construction in the noncuspidal cases gives a rational function of three variables. We specify the possible denominators and the degrees of the numerators of these rational functions. By rewriting this object as a multiple Dirichlet series, we also give a new description of the weight functions $a\sb 0(s,\pi,D)$ originally considered by D. Bump, S. Friedberg and J. Hoffstein.

Article information

Source
Duke Math. J. Volume 117, Number 3 (2003), 543-570.

Dates
First available: 26 May 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1085598404

Mathematical Reviews number (MathSciNet)
MR1979053

Digital Object Identifier
doi:10.1215/S0012-7094-03-11735-4

Zentralblatt MATH identifier
1048.11039

Subjects
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11M38: Zeta and $L$-functions in characteristic $p$

Citation

Fisher, Benji; Friedberg, Solomon. Sums of twisted GL(2) L -functions over function fields. Duke Mathematical Journal 117 (2003), no. 3, 543--570. doi:10.1215/S0012-7094-03-11735-4. http://projecteuclid.org/euclid.dmj/1085598404.


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References

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