Skip to main content
×
Home
    • Aa
    • Aa

Simultaneous non-vanishing of GL(3) × GL(2) and GL(2) L-functions

  • RIZWANUR KHAN (a1)
Abstract
Abstract

Fix g a Hecke–Maass form for SL3(). In the family of holomorphic newforms f of fixed weight and large prime level q, we find the average value of the product . From this we derive a result on the simultaneous non-vanishing of these L-functions at the central point.

Copyright
References
Hide All
[1]Atkin A. O. L. and Lehner J.. Hecke operators on Γ0(m). Math. Ann. 185 (1970), 134160.
[2]Blomer V.. Subconvexity for twisted L-functions on GL(3). Amer. J. Math. (to appear).
[3]Blomer V. and Brumley F.. On the Ramanujan conjecture over number fields, Ann. of Math. (2) (to appear).
[4]Duke W., Friedlander J. and Iwaniec H.. Erratum: “Bounds for automorphic L-functions. II” [Invent. Math. 115 (1994), no. 2, 219239; MR1258904 (95a:11044)], Invent. Math. 140 (2000), no. 1, 227–242.
[5]Duke W., Friedlander J. B. and Iwaniec H.. Bounds for automorphic L-functions. II. Invent. Math. 115 (1994), no. 2, 219239.
[6]Goldfeld D.. Automorphic forms and L-functions for the group GL(n, R). Cambridge Studies in Advanced Mathematics, vol. 99 (Cambridge University Press, 2006), with an appendix by Broughan Kevin A..
[7]Goldfeld D. and Li X.. Voronoi formulas on GL(n). Int. Math. Res. Not. (2006), Art. ID 86295, 25.
[8]Gradshteyn I. S. and Ryzhik I. M.. Table of Integrals, Series, and Products, sixth ed. (Academic Press Inc., 2000), Translated from the Russian, Translation edited and with a preface by Jeffrey Alan and Zwillinger Daniel.
[9]Ivić A.. On the ternary additive divisor problem and the sixth moment of the zeta-function. Sieve methods, exponential sums and their applications in number theory (Cardiff, 1995). London Math. Soc. Lecture Note Ser., vol. 237 (Cambridge University Press, 1997) pp. 205243.
[10]Iwaniec H.. Topics in classical automorphic forms. Graduate Studies in Mathematics, vol. 17 (American Mathematical Society, 1997).
[11]Iwaniec H. and Kowalski E.. Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, 2004).
[12]Iwaniec H., Luo W. and Sarnak P.. Low lying zeros of families of L-functions. Inst. Hautes Études Sci. Publ. Math. (2000), no. 91, 55131 (2001).
[13]Iwaniec H. and Sarnak P.. The non-vanishing of central values of automorphic L-functions and Landau–Siegel zeros. Israel J. Math. 120 (2000), no. part A, 155177.
[14]Jacquet H. and Shalika J. A.. A non-vanishing theorem for zeta functions of GLn. Invent. Math. 38 (1976/77), no. 1, 116.
[15]Kim H. H.. Functoriality for the exterior square of GL4 and the symmetric fourth of GL2. J. Amer. Math. Soc. 16 (2003), no. 1, 139–183 (electronic). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak.
[16]Kowalski E., Michel P. and Vanderkam J.. Mollification of the fourth moment of automorphic L-functions and arithmetic applications. Invent. Math. 142 (2000), no. 1, 95151.
[17]Kowalski E., Michel P. and Vanderkam J.. Rankin–Selberg L-functions in the level aspect. Duke Math. J. 114 (2002), no. 1, 123191.
[18]Li X.. The central value of the Rankin–Selberg L-functions. Geom. Funct. Anal. 18 (2009), no. 5, 16601695.
[19]Li X.. Bounds for GL(3) × GL(2) L-functions and GL(3) L-functions. Ann. of Math. (2) 173 (2011), no. 1, 301336.
[20]Liu S.-C.. Determination of GL(3) cusp forms by central values of GL(3) × GL(2) L-functions, level aspect. J. Number Theory 131 (2011), no. 8, 13971408.
[21]Michel P.. The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. Ann. of Math. (2) 160 (2004), no. 1, 185236.
[22]Miller S. D. and Schmid W.. Automorphic distributions, L-functions and Voronoi summation for GL(3). Ann. of Math. (2) 164 (2006), no. 2, 423488.
[23]Soudry D.. on Langlands functoriality from classical groups to GLn. Astérisque (2005), no. 298, 335–390, Automorphic forms. I.
[24]Watson G. N.. A treatise on the theory of Bessel functions. Cambridge Mathematical Library. (Cambridge University Press, 1995). Reprint of the second (1944) edition.
[25]Young M.. The second moment of GL(3) × GL(2) L-functions, integrated. Adv. Math. 226 (2011), no. 4, 35503578.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 18 *
Loading metrics...

Abstract views

Total abstract views: 105 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th October 2017. This data will be updated every 24 hours.