Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Spectral Kuznetsov Formula on $ SL(3)$


Author: Jack Buttcane
Journal: Trans. Amer. Math. Soc. 368 (2016), 6683-6714
MSC (2010): Primary 11F72; Secondary 44A20, 33E20
DOI: https://doi.org/10.1090/tran/6833
Published electronically: January 13, 2016
MathSciNet review: 3461048
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The $ SL(3)$ Kuznetsov formula exists in several versions and has been employed with some success to study automorphic forms on $ SL(3)$. In each version, the weight functions on the geometric side are given by multiple integrals with complicated oscillating factors; this is the primary obstruction to its use. By describing them as solutions to systems of differential equations, we give power series and Mellin-Barnes integral representations of minimal dimension for these weight functions. This completes the role of harmonic analysis on symmetric spaces on the geometric side of the Kuznetsov formula, so that further study may be done through classical analytic techniques and should immediately open the door for results in the study of $ GL(3)$ $ L$-functions.


References [Enhancements On Off] (What's this?)

  • [1] W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. MR 0185155 (32 #2625)
  • [2] Valentin Blomer, Subconvexity for twisted $ L$-functions on $ {\rm GL}(3)$, Amer. J. Math. 134 (2012), no. 5, 1385-1421. MR 2975240, https://doi.org/10.1353/ajm.2012.0032
  • [3] Valentin Blomer, Applications of the Kuznetsov formula on $ GL(3)$, Invent. Math. 194 (2013), no. 3, 673-729. MR 3127065, https://doi.org/10.1007/s00222-013-0454-3
  • [4] Valentin Blomer, Jack Buttcane, and Nicole Raulf, A Sato-Tate law for $ \rm GL (3)$, Comment. Math. Helv. 89 (2014), no. 4, 895-919. MR 3284298, https://doi.org/10.4171/CMH/337
  • [5] Daniel Bump, Automorphic forms on $ {\rm GL}(3,{\bf R})$, Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. MR 765698 (86g:11028)
  • [6] Daniel Bump, Solomon Friedberg, and Dorian Goldfeld, Poincaré series and Kloosterman sums for $ {\rm SL}(3,{\bf Z})$, Acta Arith. 50 (1988), no. 1, 31-89. MR 945275 (89j:11047)
  • [7] Jack Buttcane, On sums of $ SL(3,\mathbb{Z})$ Kloosterman sums, Ramanujan J. 32 (2013), no. 3, 371-419. MR 3130656, https://doi.org/10.1007/s11139-013-9488-9
  • [8] Solomon Friedberg, Poincaré series for $ {\rm GL}(n)$: Fourier expansion, Kloosterman sums, and algebreo-geometric estimates, Math. Z. 196 (1987), no. 2, 165-188. MR 910824 (88m:11032), https://doi.org/10.1007/BF01163653
  • [9] Dorian Goldfeld, Automorphic forms and $ L$-functions for the group $ {\rm GL}(n,\mathbf {R})$, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662 (2008d:11046)
  • [10] Dorian Goldfeld and Alex Kontorovich, On the $ {\rm GL}(3)$ Kuznetsov formula with applications to symmetry types of families of $ L$-functions, Automorphic representations and $ L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22, Tata Inst. Fund. Res., Mumbai, 2013, pp. 263-310. MR 3156855
  • [11] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010 (2008g:00005)
  • [12] Xiaoqing Li, A spectral mean value theorem for $ {\rm GL}(3)$, J. Number Theory 130 (2010), no. 11, 2395-2403. MR 2678854 (2012a:11062), https://doi.org/10.1016/j.jnt.2010.04.008
  • [13] Freydoon Shahidi, Eisenstein series and automorphic $ L$-functions, American Mathematical Society Colloquium Publications, vol. 58, American Mathematical Society, Providence, RI, 2010. MR 2683009 (2012d:11119)
  • [14] Audrey Terras, Harmonic analysis on symmetric spaces and applications. II, Springer-Verlag, Berlin, 1988. MR 955271 (89k:22017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11F72, 44A20, 33E20

Retrieve articles in all journals with MSC (2010): 11F72, 44A20, 33E20


Additional Information

Jack Buttcane
Affiliation: Mathematisches Institut, Georg-August Universität Göttingen, Bunsenstrasse 3-5, D-37073 Göttingen, Germany
Address at time of publication: Department of Mathematics, SUNY Buffalo, Buffalo, New York 14260
Email: buttcane@buffalo.edu

DOI: https://doi.org/10.1090/tran/6833
Received by editor(s): January 6, 2015
Published electronically: January 13, 2016
Additional Notes: During the time of this research, the author was supported by European Research Council Starting Grant agreement No. 258713.
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society