The spectral decomposition of shifted convolution sums over number fields

References

[1] V. Blomer and F. Brumley, On the Ramanujan conjecture over number fields, Ann. of Math. (2) 174 (2011), no. 1, 581–605. 10.4007/annals.2011.174.1.18Search in Google Scholar

[2] V. Blomer and G. Harcos, The spectral decomposition of shifted convolution sums, Duke Math. J. 144 (2008), no. 2, 321–339. 10.1215/00127094-2008-038Search in Google Scholar

[3] V. Blomer and G. Harcos, Twisted L-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), no. 1, 1–52; erratum, www.renyi.hu/~gharcos/hilbert_erratum.pdf. 10.1007/s00039-010-0063-xSearch in Google Scholar

[4] R. W. Bruggeman and Y. Motohashi, A note on the mean value of the zeta and L-functions. XIII, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 6, 87–91. 10.3792/pjaa.78.87Search in Google Scholar

[5] R. W. Bruggeman and Y. Motohashi, Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field, Funct. Approx. Comment. Math. 31 (2003), 23–92. 10.7169/facm/1538186640Search in Google Scholar

[6] R. W. Bruggeman and Y. Motohashi, A new approach to the spectral theory of the fourth moment of the Riemann zeta-function, J. reine angew. Math. 579 (2005), 75–114. 10.1515/crll.2005.2005.579.75Search in Google Scholar

[7] D. Bump, Automorphic forms and representations, Cambridge Stud. Adv. Math. 55, Cambridge University Press, Cambridge 1997. 10.1017/CBO9780511609572Search in Google Scholar

[8] D. Bump, Lie groups, Grad. Texts in Math. 225, Springer, New York 2004. 10.1007/978-1-4757-4094-3Search in Google Scholar

[9] W. Casselman, On some results of Atkin and Lehner, Math. Ann. 201 (1973), 301–314. 10.1007/BF01428197Search in Google Scholar

[10] J. W. Cogdell and I. Piatetski-Shapiro, The arithmetic and spectral analysis of Poincaré series, Perspect. Math. 13, Academic Press, Boston 1990. Search in Google Scholar

[11] J. W. Cogdell, I. Piatetski-Shapiro and P. Sarnak, Estimates on the critical line for Hilbert modular L-functions and applications, preprint (2001). Search in Google Scholar

[12] J. Dixmier and P. Malliavin, Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2) 102 (1978), no. 4, 307–330. Search in Google Scholar

[13] W. Duke, J. Friedlander and H. Iwaniec, Bounds for automorphic L-functions, Invent. Math. 112 (1993), no. 1, 1–8. 10.1007/BF01232422Search in Google Scholar

[14] W. Duke, J. B. Friedlander and H. Iwaniec, The subconvexity problem for Artin L-functions, Invent. Math. 149 (2002), no. 3, 489–577. 10.1007/s002220200223Search in Google Scholar

[15] T. Estermann, Über die Darstellungen einer Zahl als Differenz von zwei Produkten, J. reine angew. Math. 164 (1931), 173–182. 10.1515/crll.1931.164.173Search in Google Scholar

[16] D. Flath, Decomposition of representations into tensor products, Automorphic forms, representations and L-functions (Oregon 1977), Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence (1979), 179–183. 10.1090/pspum/033.1/546596Search in Google Scholar

[17] S. Gelbart and H. Jacquet, Forms of GL(2) from the analytic point of view, Automorphic forms, representations and L-functions (Oregon 1977), Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence (1979), 213–251. 10.1090/pspum/033.1/546600Search in Google Scholar

[18] A. Good, Beiträge zur Theorie der Dirichletreihen, die Spitzenformen zugeordnet sind, J. Number Theory 13 (1981), no. 1, 18–65. 10.1016/0022-314X(81)90028-7Search in Google Scholar

[19] A. Good, Cusp forms and eigenfunctions of the Laplacian, Math. Ann. 255 (1981), no. 4, 523–548. 10.1007/BF01451932Search in Google Scholar

[20] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier, Amsterdam 2007. Search in Google Scholar

[21] D. R. Heath-Brown, The fourth power moment of the Riemann zeta function, Proc. Lond. Math. Soc. (3) 38 (1979), no. 3, 385–422. 10.1112/plms/s3-38.3.385Search in Google Scholar

[22] A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. Lond. Math. Soc. (2) 27 (1926), no. 1, 273–300. 10.1112/plms/s2-27.1.273Search in Google Scholar

[23] A. E. Ingham, Some asymptotic formulae in the theory of numbers, J. Lond. Math. Soc. (2) (1927), no. 3, 202–208. 10.1112/jlms/s1-2.3.202Search in Google Scholar

[24] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math. 114, Springer, Berlin 1970. 10.1007/BFb0058988Search in Google Scholar

[25] K. Khuri-Makdisi, On the Fourier coefficients of nonholomorphic Hilbert modular forms of half-integral weight, Duke Math. J. 84 (1996), no. 2, 399–452. 10.1215/S0012-7094-96-08414-8Search in Google Scholar

[26] H. Lokvenec-Guleska, Sum formula for SL(2) over imaginary quadratic fields, Ph.D. thesis, Utrecht University, Utrecht, 2004. Search in Google Scholar

[27] P. Maga, A semi-adelic Kuznetsov formula over number fields, Int. J. Number Theory 9 (2013), no. 7, 1649–1681. 10.1142/S1793042113500498Search in Google Scholar

[28] P. Maga, Subconvexity and shifted convolution sums over number fields, Ph.D. thesis, Central European University, Budapest, 2013. Search in Google Scholar

[29] P. Michel and A. Venkatesh, The subconvexity problem for GL2, Publ. Math. Inst. Hautes Études Sci. 111 (2010), no. 1, 171–271. 10.1007/s10240-010-0025-8Search in Google Scholar

[30] T. Miyake, On automorphic forms on GL⁢(2) and Hecke operators, Ann. of Math. (2) 94 (1971), 174–189. 10.2307/1970741Search in Google Scholar

[31] Y. Motohashi, The binary additive divisor problem, Ann. Sci. Éc. Norm. Supér. (4) 27 (1994), no. 5, 529–572. 10.24033/asens.1700Search in Google Scholar

[32] Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge Tracts in Math. 127, Cambridge University Press, Cambridge 1997. 10.1017/CBO9780511983399Search in Google Scholar

[33] P. Sarnak, Estimates for Rankin–Selberg L-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), no. 2, 419–453. 10.1006/jfan.2001.3783Search in Google Scholar

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

[35] M. Sugiura, Unitary representations and harmonic analysis. An introduction, 2nd ed., North-Holland Math. Libr. 44, North-Holland Publishing, Amsterdam 1990. Search in Google Scholar

[36] A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2) 172 (2010), no. 2, 989–1094. 10.4007/annals.2010.172.989Search in Google Scholar

[37] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Math. Lib., Cambridge University Press, Cambridge 1995. Search in Google Scholar

[38] A. Weil, Basic number theory, 3rd ed., Grundlehren Math. Wiss. 144, Springer, Berlin 1974. 10.1007/978-3-642-61945-8Search in Google Scholar

[39] E. T. Whittaker and G. N. Watson, A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Cambridge Math. Libr., Cambridge University Press, Cambridge 1996. Search in Google Scholar

[40] H. Wu, Burgess-like subconvex bounds for GL2×GL1, Geom. Funct. Anal. 24 (2014), no. 3, 968–1036. 10.1007/s00039-014-0277-4Search in Google Scholar