×

Mean values of \(L\)-functions over function fields. (English) Zbl 0872.11038

The purpose of this paper is to study the average values of twists, by quadratic characters, of \(L\)-series associated with automorphic forms on \(GL_2\) over function fields. The method which the author develops here is the study of the Rankin-Selberg convolution of the form in question with an Eisenstein series on the metaplectic cover of \(GL_2\) of order 2. The author carries out this programme when the function field is rational and the field of constants is \(\mathbb{F}_q\) with \(q\) prime and \(q\equiv 1\pmod 4\). The infinite place is singled out and the theory developed in analogy to the classical case. The author develops a sieve to pick out the primitive characters and obtains an estimate for the averages of the \(L\)-functions at the centre of the critical strip with a good error term. This also leads to non-vanishing results for these values.

MSC:

11M41 Other Dirichlet series and zeta functions
11F70 Representation-theoretic methods; automorphic representations over local and global fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Atkin, A.; Lehner, J., Hecke operators on \(Γ_0m\), Math. Ann., 185, 134-160 (1970) · Zbl 0177.34901
[2] Bump, D.; Friedberg, S.; Hoffstein, J., Non vanishing theorems for \(L\), Invent. Math., 102, 543-618 (1990) · Zbl 0721.11023
[3] Friedberg, S.; Hoffstein, J., Nonvanishing theorems for automorphic \(L GL \), Ann. Math., 142, 385-423 (1995) · Zbl 0847.11026
[4] Gauss, C. F., Disquisitiones Arithmeticae (1966), Yale Univ. Press: Yale Univ. Press New Haven/London
[5] Gauss, C. F., Werke, I (1870)
[6] Goldfeld, D.; Hoffstein, J., Eisenstein series of 1/2 integral weight and the mean value of real Dirichlet \(L\), Invent. Math., 80, 185-208 (1985) · Zbl 0564.10043
[7] Goldfeld, D.; Hoffstein, J.; Patterson, S., On automorphic functions of half-integral weight with applications to elliptic curves, (Koblitz, N., Number Theory Related to Fermat’s Last Theorem (1983), Birkhäuser: Birkhäuser Boston/Basel/Stuttgart) · Zbl 0499.10029
[8] Goldfeld, D.; Viola, C., Mean values of \(L\), J. Number Theory, 11, 305-320 (1979) · Zbl 0409.10029
[9] Gekeler, E.; Reversat, M., Some results on the Jacobians of Drinfeld modular curves, (Goss, D.; Haynes, D.; Rosen, M., The Arithmetic of Function Fields (1992), de Gruyter: de Gruyter Berlin/New York) · Zbl 0827.11036
[10] Hoffstein, J., Theta functions on the \(n GL \), Invent. Math., 107, 61-86 (1992) · Zbl 0761.11024
[11] Hoffstein, J.; Rosen, M., Average value of \(L\), J. Reine Angew. Math., 426, 117-150 (1992) · Zbl 0754.11036
[13] Jutila, M., On the mean value of \(Lχ\), Analysis, 1, 149-161 (1981) · Zbl 0485.10029
[14] Kubota, T., Automorphic Forms and Reciprocity in a Number Field (1969), Kiyokuniya Book Store
[15] Kazhdan, D.; Patterson, S., Metaplectic forms, Inst. Hautes Études Sci. Publ. Math., 59, 35-142 (1984) · Zbl 0559.10026
[16] Landau, E., Elementary Number Theory (1958), Chelsea: Chelsea New York
[17] Lieman, D., Nonvanishing of \(L\), Ann. of Math., 140, 81-108 (1994) · Zbl 0817.11029
[18] Lipschitz, R., Sitzungsber. Akad. Berlin, 174-185 (1865)
[19] Mertens, F., Uber einige asymptotische Gesetze der Zahlentheorie, J. Mathematik, 77, 312-319 (1874)
[20] Murty, K.; Murty, R., Mean values of derivations of modular \(L\), Ann. of Math., 133, 447-475 (1991) · Zbl 0745.11032
[22] Shintani, T., Zeta functions associated with the vector space of quadratic forms, J. Fac. Sci. Univ. Tokyo, Sec. IA., 22, 25-65 (1975) · Zbl 0313.10041
[23] Siegel, C. L., The Average Measure of Quadratic Forms with Given Determinant and Signature. The Average Measure of Quadratic Forms with Given Determinant and Signature, Gesammelte Abhandlungen, Band II (1966), Springer-Verlag: Springer-Verlag New York/Berlin
[24] Suprunenko, D. A., Matrix Groups. Matrix Groups, Translations of Mathematical Monographs, Vol. 45 (1976), Amer. Math. Soc: Amer. Math. Soc Providence · Zbl 0317.20028
[25] Tahtadzjan, L. A.; Vinogradov, A. I., On the analogues of the Vinogradov-Gauss formula, Soviet Math. Dokl., 22, 555-559 (1980) · Zbl 0468.10022
[26] Vinogradov, I. M., Izv. Akad. Nauk SSSR, Ser. Mat., 13, 97 (1949)
[27] Selected Works. Selected Works, Izdat. Akad. Nauk SSSR, Vol. 366 (1952) · Zbl 0048.03104
[28] Waldspurger, J., Correspondences de Shimura et quaternions, Forum Math., 3, 219-307 (1991) · Zbl 0724.11026
[29] Weil, A., Sur certains groupes d’opérateurs unitaries, Acta Math., 111, 143-211 (1964) · Zbl 0203.03305
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.