Fourier coefficients of sextic theta series
HTML articles powered by AMS MathViewer
- by Reinier Bröker and Jeff Hoffstein PDF
- Math. Comp. 85 (2016), 1901-1927 Request permission
Abstract:
This article focuses on the theta series on the 6-fold cover of $\mathrm {GL}_2$. We investigate the Fourier coefficients $\tau (r)$ of the theta series, and give partially proven, partially conjectured values for $\tau (\pi )^2$, $\tau (\pi ^2)$ and $\tau (\pi ^4)$ for prime values $\pi$.References
- Ben Brubaker, Alina Bucur, Gautam Chinta, Sharon Frechette, and Jeffrey Hoffstein, Nonvanishing twists of GL(2) automorphic $L$-functions, Int. Math. Res. Not. 78 (2004), 4211–4239. MR 2111362, DOI 10.1155/S1073792804133473
- J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
- Gautam Chinta, Solomon Friedberg, and Jeffrey Hoffstein, Double Dirichlet series and theta functions, Contributions in analytic and algebraic number theory, Springer Proc. Math., vol. 9, Springer, New York, 2012, pp. 149–170. MR 3060459, DOI 10.1007/978-1-4614-1219-9_{6}
- Henri Cohen, Number theory. Vol. II. Analytic and modern tools, Graduate Texts in Mathematics, vol. 240, Springer, New York, 2007. MR 2312338
- C. Eckhardt, Eine Vermuting über biquadratische Thetareihen und ihre numerische Untersuchung, PhD thesis, University of Göttingen, 1989.
- C. Eckhardt and S. J. Patterson, On the Fourier coefficients of biquadratic theta series, Proc. London Math. Soc. (3) 64 (1992), no. 2, 225–264. MR 1143226, DOI 10.1112/plms/s3-64.2.225
- S. Friedberg, D. Ginzburg, Descent and theta functions for metaplectic groups, preprint, 2014.
- Jeff Hoffstein, Eisenstein series and theta functions on the metaplectic group, Theta functions: from the classical to the modern, CRM Proc. Lecture Notes, vol. 1, Amer. Math. Soc., Providence, RI, 1993, pp. 65–104. MR 1224051, DOI 10.1090/crmp/001/02
- D. A. Kazhdan and S. J. Patterson, Metaplectic forms, Inst. Hautes Études Sci. Publ. Math. 59 (1984), 35–142. MR 743816
- T. Kubota, On Automorphic Forms and the Reciprocity Law in a Number Field, Springer, Lecture Notes Math., (1973), p. 348.
- Jürgen Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Springer-Verlag, Berlin, 1999.
- S. J. Patterson, A cubic analogue of the theta series, J. Reine Angew. Math. 296 (1977), 125–161. MR 563068, DOI 10.1515/crll.1977.296.125
- S. J. Patterson, A heuristic principle and applications to Gauss sums, J. Indian Math. Soc. (N.S.) 52 (1987), 1–22 (1988). MR 989227
- S. J. Patterson, The distribution of general Gauss sums and similar arithmetic functions at prime arguments, Proc. London Math. Soc. (3) 54 (1987), no. 2, 193–215. MR 872805, DOI 10.1112/plms/s3-54.2.193
- D. R. Heath-Brown and S. J. Patterson, The distribution of Kummer sums at prime arguments, J. Reine Angew. Math. 310 (1979), 111–130. MR 546667
- Toshiaki Suzuki, Some results on the coefficients of the biquadratic theta series, J. Reine Angew. Math. 340 (1983), 70–117. MR 691962, DOI 10.1515/crll.1983.340.70
- J. T. Tate, Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305–347. MR 0217026
- G. Wellhausen, Fourierkoeffizienten von Thetafunktionen sechster Ordnung, PhD thesis, University of Göttingen, 1996.
Additional Information
- Reinier Bröker
- Affiliation: Brown University, Department of Mathematics, Box 1917, Providence, Rhode Island
- MR Author ID: 759393
- Email: reinier@math.brown.edu
- Jeff Hoffstein
- Affiliation: Brown University, Department of Mathematics, Box 1917, Providence, Rhode Island
- MR Author ID: 87085
- Email: jhoff@math.brown.edu
- Received by editor(s): February 21, 2014
- Received by editor(s) in revised form: October 18, 2014, and January 21, 2015
- Published electronically: October 21, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 1901-1927
- MSC (2010): Primary 11Y35
- DOI: https://doi.org/10.1090/mcom3044
- MathSciNet review: 3471113