The Spectral Kuznetsov Formula on 𝑆𝐿(3)

Buttcane, Jack

doi:10.1090/tran/6833

The Spectral Kuznetsov Formula on $SL(3)$
HTML articles powered by AMS MathViewer

by Jack Buttcane PDF

Trans. Amer. Math. Soc. 368 (2016), 6683-6714 Request permission

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

W. N. Bailey, Generalized hypergeometric series, Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Stechert-Hafner, Inc., New York, 1964. MR 0185155
Valentin Blomer, Subconvexity for twisted $L$-functions on $\textrm {GL}(3)$, Amer. J. Math. 134 (2012), no. 5, 1385–1421. MR 2975240, DOI 10.1353/ajm.2012.0032
Valentin Blomer, Applications of the Kuznetsov formula on $GL(3)$, Invent. Math. 194 (2013), no. 3, 673–729. MR 3127065, DOI 10.1007/s00222-013-0454-3
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, DOI 10.4171/CMH/337
Daniel Bump, Automorphic forms on $\textrm {GL}(3,\textbf {R})$, Lecture Notes in Mathematics, vol. 1083, Springer-Verlag, Berlin, 1984. MR 765698, DOI 10.1007/BFb0100147
Daniel Bump, Solomon Friedberg, and Dorian Goldfeld, Poincaré series and Kloosterman sums for $\textrm {SL}(3,\textbf {Z})$, Acta Arith. 50 (1988), no. 1, 31–89. MR 945275, DOI 10.4064/aa-50-1-31-89
Jack Buttcane, On sums of $SL(3,\Bbb {Z})$ Kloosterman sums, Ramanujan J. 32 (2013), no. 3, 371–419. MR 3130656, DOI 10.1007/s11139-013-9488-9
Solomon Friedberg, Poincaré series for $\textrm {GL}(n)$: Fourier expansion, Kloosterman sums, and algebreo-geometric estimates, Math. Z. 196 (1987), no. 2, 165–188. MR 910824, DOI 10.1007/BF01163653
Dorian Goldfeld, Automorphic forms and $L$-functions for the group $\textrm {GL}(n,\mathbf R)$, Cambridge Studies in Advanced Mathematics, vol. 99, Cambridge University Press, Cambridge, 2006. With an appendix by Kevin A. Broughan. MR 2254662, DOI 10.1017/CBO9780511542923
Dorian Goldfeld and Alex Kontorovich, On the $\textrm {GL}(3)$ 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
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
Xiaoqing Li, A spectral mean value theorem for $\textrm {GL}(3)$, J. Number Theory 130 (2010), no. 11, 2395–2403. MR 2678854, DOI 10.1016/j.jnt.2010.04.008
Freydoon Shahidi, Eisenstein series and automorphic $L$-functions, American Mathematical Society Colloquium Publications, vol. 58, American Mathematical Society, Providence, RI, 2010. MR 2683009, DOI 10.1090/coll/058
Audrey Terras, Harmonic analysis on symmetric spaces and applications. II, Springer-Verlag, Berlin, 1988. MR 955271, DOI 10.1007/978-1-4612-3820-1