The second moment of GL(3) × GL(2) L- functions at special points from GL(3) forms

Abstract

For a fixed even SL(\mathrm{2},\mathbb{Z}) Hecke{Maass form f, we get an estimate for the second moment of L(s,{{\displaystyle \phi }}_j\times f) at special points, where {{\displaystyle \phi }}_j runs over an orthogonal basis of Hecke{Maass cusp forms for {{\displaystyle \mathrm{S}\mathrm{L}}}_{\mathrm{3}}(\mathbb{Z}).

Key words： Rankin{Selberg L-functions; Hecke{Maass forms; Kuznetsov trace formula

References

1 Blomer V, Buttcane J. Global decomposition of GL(3) Kloosterman sums and the spectral large sieve. J Reine Angew Math, 2019, 757: 51{88

2 Blomer V, Buttcane J. On the subconvexity problem for L-functions on GL(3): Ann Sci Éc Norm Super (4) (to appear)

3 Brumley F. Second order average estimates on local data of cusp forms. Arch Math, 2006, 87: 19–32

4 Buttcane J. The spectral Kuznetsov formula on SL(3). Trans Amer Math Soc, 2016, 368: 6683–6714

5 Erdélyi A, Magnus W, Oberhettinger F, Tricomi F. Higher Transcendental Functions, Vol II. New York: McGraw-Hill, 1953

6 Goldfeld D. Automorphic Forms and L-Functions for the Group GL(n,ℝ). Cambridge Stud Adv Math, Vol 99. Cambridge: Cambridge Univ Press, 2006

7 Gradshteyn I S, Ryzhik I M. Table of Integrals, Series, and Products. 7th ed. New York: Academic Press, 2007

8 Kim H H. Functoriality for the exterior square of GL₄ and the symmetric fourth of GL₂. Kim H H, Sarnak P. Appendix 2: Refined estimates towards the Ramanujan and Selberg conjectures. J Amer Math Soc, 2003, 16: 175–181

9 Li X. Upper bounds on L-functions at the edge of the critical strip. Int Math Res Not IMRN, 2010, 2010: 727–755

10 Li X, Young M. The L² restriction norm of a GL(3) Maass form. Compos Math, 2012, 148: 675–717

11 Miller S D. On the existence and temperedness of cusp forms for SL(3,ℤ). J Reine Angew Math, 2001, 533: 127C-169

12 Young M. The second moment of GL(3) × GL(2) L-functions at special points. Math Ann, 2013, 356: 1005–1028