Sums of twisted GL(2) \(L\)-functions over function fields. (English) Zbl 1048.11039

Fisher, Benji; Friedberg, Solomon

Duke Math. J. 117, No. 3, 543-570 (2003).

This paper concerns a weighted sum of twisted \(L\)-functions, \[ \sum_D L(s, \pi \otimes \chi_D) a_0(s, \pi, D) \eta(D) | D| ^{-w}. \] Here \(\pi\) (respectively, \(\eta\)) is a cuspidal automorphic representation of \(\text{GL}_2(\mathbb{A}_K)\) (respectively, \(\text{GL}_1(\mathbb{A}_K)\)) and \(\chi_D\) are quadratic characters. The field \(K\) is a function field of a smooth projective curve \(C\) over a finite field of odd characteristic. The sum runs over a certain set of effective divisors of \(C\). The weight functions \(a_0(s, \pi, D)\) have been defined by D. Bump, S. Friedberg and J. Hoffstein [Sums of twisted GL(3) automorphic \(L\)-functions, in Contributions to automorphic forms, geometry and arithmetic, Johns Hopkins Univ. Press, Baltimore, MD, 131–162 (2004)], who proved the uniqueness. That is, there is a unique family of weight functions such that the sum satisfies a certain nonabelian group of functional equations.
This paper presents a different approach. The authors rewrite the sum as a multiple Dirichlet series and give a new description of the weight functions. They prove the sum has a finite, nonabelian group of functional equations. They also establish a similar result in the noncuspidal case, where the natural object is a sum in three complex variables.
The main result is the proof that the sums are rational functions. The authors specify the possible denominators and the degrees of the numerators of these rational functions.

Reviewer: Dubravka Ban (Münster)

MSC:

11F66	Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F70	Representation-theoretic methods; automorphic representations over local and global fields
11G20	Curves over finite and local fields
11M38	Zeta and \(L\)-functions in characteristic \(p\)

Keywords:

twisted \(L\)-function; double Dirichlet series; automorphic forms

References:

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