A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)

References

[1] A. Adolphson and S. Sperber, Exponential sums and Newton polyhedra: Cohomology and estimates, Ann. of Math. (2) 130 (1989), 367–406. 10.2307/1971424Search in Google Scholar

[2] V. Blomer, Shifted convolution sums and subconvexity bounds for automorphic L-functions, Int. Math. Res. Not. IMRN 2004 (2004), no. 73, 3905–3926. 10.1155/S1073792804142505Search in Google Scholar

[3] V. Blomer, Rankin–Selberg L-functions on the critical line, Manuscripta Math. 117 (2005), 111–133. 10.1007/s00229-005-0557-2Search in Google Scholar

[4] V. Blomer and G. Harcos, The spectral decomposition of shifted convolution sums, Duke Math. J. 144 (2008), 321–339. 10.1215/00127094-2008-038Search in Google Scholar

[5] V. Blomer, G. Harcos and P. Michel, A Burgess-like subconvex bound for twisted L-functions. Appendix 2 by Z. Mao, Forum Math. 19 (2007), 61–105. 10.1515/FORUM.2007.003Search in Google Scholar

[6] E. Bombieri, J. B. Friedlander and H. Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986), 203–251. 10.1007/BF02399204Search in Google Scholar

[7] E. Bombieri and S. Sperber, On the estimation of certain exponential sums, Acta Arith. 69 (1995), 329–358. 10.4064/aa-69-4-329-358Search in Google Scholar

[8] A. R. Booker, Numerical tests of modularity, J. Ramanujan Math. Soc. 20 (2005), 283–339. Search in Google Scholar

[9] J.-L. Brylinski, Transformations canoniques, dualité projective, théeorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Astérisque 140–141 (1986), 3–134. Search in Google Scholar

[10] P. Deligne, La conjecture de Weil. II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252. 10.1007/BF02684780Search in Google Scholar

[11] W. Duke, J. B. Friedlander and H. Iwaniec, A quadratic divisor problem, Invent. Math. 115 (1994), 209–217. 10.1007/BF01231758Search in Google Scholar

[12] É. Fouvry, Sur le problème des diviseurs de Titchmarsh, J. Reine Angew. Math. 357 (1985), 51–76. 10.1515/crll.1985.357.51Search in Google Scholar

[13] É. Fouvry, E. Kowalski and P. Michel, Algebraic twists of modular forms and Hecke orbits, Geom. Funct. Anal. 25 (2015), 580–657. 10.1007/s00039-015-0310-2Search in Google Scholar

[14] D. Goldfeld, Automorphic Forms and L-Functions for the Group GL⁢(n,𝐑). With an Appendix by Kevin A. Broughan, Cambridge Stud. Adv. Math. 99, Cambridge University Press, Cambridge, 2006. Search in Google Scholar

[15] D. Goldfeld and X. Li, Voronoi formulas on GLn), Int. Math. Res. Not. IMRN 2006 (2006), Article ID 86295. 10.1155/IMRN/2006/86295Search in Google Scholar

[16] G. Harcos, An additive problem in the Fourier coefficients of cusp forms, Math. Ann. 326 (2003), 347–365. 10.1007/s00208-003-0421-1Search in Google Scholar

[17] G. Harcos and P. Michel, The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II, Invent. Math. 163 (2006), 581–655. 10.1007/s00222-005-0468-6Search in Google Scholar

[18] D. R. Heath-Brown, The square sieve and consecutive square-free numbers, Math. Ann. 266 (1984), 251–259. 10.1007/BF01475576Search in Google Scholar

[19] D. R. Heath-Brown, A new form of the circle method, and its application to quadratic forms, J. Reine Angew. Math. 481 (1996), 149–206. 10.1515/crll.1996.481.149Search in Google Scholar

[20] R. Holowinsky, A sieve method for shifted convolution sums, Duke Math. J. 146 (2009), 401–448. 10.1215/00127094-2009-002Search in Google Scholar

[21] M. Jutila, Transformations of exponential sums, Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori 1989), Universitá di Salerno, Salerno (1992), 263–270. Search in Google Scholar

[22] M. Jutila, A variant of the circle method, Sieve Methods, Exponential Sums, and Their Applications in Number Theory (Cardiff 1995), London Math. Soc. Lecture Note Ser. 237, Cambridge University Press, Cambridge (1997), 245–254. 10.1017/CBO9780511526091.016Search in Google Scholar

[23] N. M. Katz, Sommes Exponentielles, Astérisque 79, Société Mathématique de France, Paris, 1980. Search in Google Scholar

[24] N. M. Katz, Gauss Sums, Kloosterman Sums, And Monodromy Groups, Ann. of Math. Stud. 116, Princeton University Press, Princeton, 1988. 10.1515/9781400882120Search in Google Scholar

[25] E. Kowalski, P. Michel and J. VanderKam, Rankin–Selberg L-functions in the level aspect, Duke Math. J. 114 (2002), 123–191. 10.1215/S0012-7094-02-11416-1Search in Google Scholar

[26] G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil, Publ. Math. Inst. Hautes Études Sci. 65 (1987), 131–210. 10.1007/BF02698937Search in Google Scholar

[27] W. Luo and P. Sarnak, Mass equidistribution for Hecke eigenforms, Comm. Pure Appl. Math. 56 (2003), 874–891. 10.1002/cpa.10078Search in Google Scholar

[28] K. Matomäki, M. Radziwiłł and T. Tao, An averaged form of Chowla’s conjecture, Algebra Number Theory 9 (2015), 2167–2196. 10.2140/ant.2015.9.2167Search in Google Scholar

[29] T. Meurman, On exponential sums involving the Fourier coefficients of Maass wave forms, J. Reine Angew. Math. 384 (1988), 192–207. 10.1515/crll.1988.384.192Search in Google Scholar

[30] P. Michel, The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points, Ann. of Math. (2) 160 (2004), 185–236. 10.4007/annals.2004.160.185Search in Google Scholar

[31] S. D. Miller, Cancellation in additively twisted sums on GLn), Amer. J. Math. 128 (2006), 699–729. 10.1353/ajm.2006.0027Search in Google Scholar

[32] S. D. Miller and W. Schmid, Automorphic distributions, L-functions, and Voronoi summation for GL⁢(3), Ann. of Math. (2) 164 (2006), 423–488. 10.4007/annals.2006.164.423Search in Google Scholar

[33] G. Molteni, Upper and lower bounds at s=1 for certain Dirichlet series with Euler product, Duke Math. J. 111 (2002), 133–158. 10.1215/S0012-7094-02-11114-4Search in Google Scholar

[34] R. Munshi, Shifted convolution sums for GL⁢(3)×GL⁢(2), Duke Math. J. 162 (2013), 2345–2362. 10.1215/00127094-2371416Search in Google Scholar

[35] R. Munshi, Shifted convolution of divisor function d3 and Ramanujan τ function, The Legacy of Srinivasa Ramanujan, Ramanujan Math. Soc. Lect. Notes Ser. 20, Ramanujan Mathematical Society, Mysore (2013), 251–260. Search in Google Scholar

[36] N. J. E. Pitt, On shifted convolution sums of ζ3⁢(s) with automorphic L-functions, Duke Math J. 77 (1995), 383–406. 10.1215/S0012-7094-95-07711-4Search in Google Scholar

[37] P. Sarnak, Estimates for Rankin–Selberg L-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), 419–453. 10.1006/jfan.2001.3783Search in Google Scholar

[38] A. Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math. 8 (1965), 1–15. 10.1090/pspum/008/0182610Search in Google Scholar

[39] Q. Sun, Averages of shifted convolution sums for GL⁢(3)×GL⁢(2), preprint (2017), https://arxiv.org/abs/1701.02018. 10.1016/j.jnt.2017.07.005Search in Google Scholar

[40] H. Tang, Shifted convolution sum of d3 and the Fourier coefficient of Hecke–Maass forms, Bull. Aust. Math. Soc. 92 (2015), 195–204. 10.1017/S000497271500043XSearch in Google Scholar

[41] J. R. Wilton, A note on Ramanujan’s function τ⁢(n), Math. Proc. Cambridge Philos. Soc. 25 (1929), 121–129. 10.1017/S0305004100018636Search in Google Scholar

[42] J. Wu and P. Xi, Arithmetic exponent pairs for algebraic trace functions and applications, preprint (2016), https://arxiv.org/abs/1603.07060. 10.2140/ant.2021.15.2123Search in Google Scholar