## Duke Mathematical Journal

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

#### 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 in Project Euclid: 26 May 2004

https://projecteuclid.org/euclid.dmj/1085598404

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

Mathematical Reviews number (MathSciNet)
MR1979053

Zentralblatt MATH identifier
1048.11039

#### Citation

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

#### References

• D. Bump, S. Friedberg, and J. Hoffstein, Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic $L$-functions and their derivatives, Ann. of Math. (2) 131 (1990), 53--127.
• --. --. --. --., Nonvanishing theorems for $L$-functions of modular forms and their derivatives, Invent. Math. 102 (1990), 543--618.
• --. --. --. --., On some applications of automorphic forms to number theory, Bull. Amer. Math. Soc. (N.S.) 33 (1996), 157--175.
• --------, Sums of twisted $\GL(3)$ automorphic $L$-functions'' to appear in Contributions to Automorphic Forms, Geometry and Arithmetic, ed. H. Hida, D. Ramakrishnan, and F. Shahidi, Johns Hopkins Univ. Press., Baltimore, 2003.
• C. J. Bushnell, G. M. Henniart, and P. C. Kutzko, Local Rankin-Selberg convolutions for ${\GL}\sb n$: Explicit conductor formula, J. Amer. Math. Soc. 11 (1998), 703--730.
• J. W. S. Cassels and A. Fröhlich, eds., Algebraic Number Theory, Academic Press, London, 1967.
• A. Diaconu, D. Goldfeld, and J. Hoffstein, Multiple Dirichlet series and moments of zeta and $L$-functions, preprint.
• V. G. Drinfel'd, Proof of the Petersson conjecture for $\GL(2)$ over a global field of characteristic $p$, Funktsional. Anal. i Prilozhen. 22, no. 1 (1988), 34--54.; English translation in Funct. Anal. Appl. 22 (1988), 28--43.
• S. Dutta Gupta, Mean values of $L$-functions over function fields, J. Number Theory 63 (1997), 101--131.
• B. Fisher and S. Friedberg, Double Dirichlet series over function fields, to appear in Compositio Math.
• S. Friedberg and J. Hoffstein, Nonvanishing theorems for automorphic $L$-functions on $\GL(2)$, Ann. of Math. (2) 142 (1995), 385--423.
• J. Hoffstein and M. Rosen, Average values of L-series in function fields, J. Reine Angew. Math. 426 (1992), 117--150.
• L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3d ed., North-Holland Math. Library 7, North-Holland, Amsterdam, 1990.
• J. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math. 9, Springer, New York, 1972.