An explicit formula for the second moment of Maass form symmetric square L-functions
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Olga Balkanova
Steklov Mathematical Institute (Moscou, Rússia)
Dmitry Frolenkov
Steklov Mathematical Institute (Moscou, Rússia)
We prove an explicit formula for the second moment of symmetric square L-functions associated to Maass forms for the full modular group. In particular, we show how to express the considered second moment in terms of dual second moments of symmetric square L-functions associated to Maass cusp forms of levels 4, 16, and 64.
Paraules clau
L-functions, Kloosterman sums, Kuznetsov trace formula, regularization
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Balkanova, Olga; and Frolenkov, Dmitry. “An explicit formula for the second moment of Maass form symmetric square L-functions”. Publicacions Matemàtiques, vol.VOL 67, no. 2, pp. 611-60, https://raco.cat/index.php/PublicacionsMatematiques/article/view/418413.
Referències
N. Andersen and E. M. Kıral, Level reciprocity in the twisted second moment of Rankin–Selberg L-functions, Mathematika 64(3) (2018), 770–784. DOI: 10.1112/s0025579318000256
O. Balkanova, Spectral decomposition formula and moments of symmetric square L-functions, Preprint (2020). arXiv:2011.05952
O. Balkanova, The first moment of Maaß form symmetric square L-functions, Ramanujan J. 55(2) (2021), 761–781. DOI: 10.1007/s11139-020-00272-z
O. Balkanova and D. Frolenkov, The mean value of symmetric square L-functions, Algebra Number Theory 12(1) (2018), 35–59. DOI: 10.2140/ant.2018.12.35
O. Balkanova and D. Frolenkov, Non-vanishing of Maass form symmetric square L-functions, J. Math. Anal. Appl. 500(2) (2021), Paper No. 125148,
23 pp. DOI: 10.1016/j.jmaa.2021.125148
O. Balkanova, D. Frolenkov, and H. Wu, On Weyl’s subconvex bound for cube-free Hecke characters: Totally real case, Preprint (2021). arXiv:2108.
12283
S. Bettin, On the reciprocity law for the twisted second moment of Dirichlet L-functions, Trans. Amer. Math. Soc. 368(10) (2016), 6887–6914.
DOI: 10.1090/tran/6771
A. Biro´, A duality relation for certain triple products of automorphic forms, Israel J. Math. 192(2) (2012), 587–636. DOI: 10.1007/s11856-012-0041-0
V. Blomer, P. Humphries, R. Khan, and M. B. Milinovich, Motohashi’s fourth moment identity for non-archimedean test functions and applications,
Compos. Math. 156(5) (2020), 1004–1038. DOI: 10.1112/s0010437x20007101
V. Blomer and R. Khan, Twisted moments of L-functions and spectral reciprocity, Duke Math. J. 168(6) (2019), 1109–1177.
DOI: 10.1215/00127094-2018-0060
V. Blomer and R. Khan, Uniform subconvexity and symmetry breaking reciprocity, J. Funct. Anal. 276(7) (2019), 2315–2358. DOI: 10.1016/j.jfa.2018.11.009
V. Blomer, X. Li, and S. D. Miller, A spectral reciprocity formula and nonvanishing for L-functions on GL(4) × GL(2), J. Number Theory 205 (2019), 1–43. DOI: 10.1016/j.jnt.2019.05.011
R. W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45(1) (1978), 1–18. DOI: 10.1007/BF01406220
V. A. Bykovskiï, Density theorems and the mean value of arithmetic functions on short intervals (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 212(12) (1994), 56–70, 196; translation in: J. Math. Sci. (New York) 83(6) (1997), 720–730. DOI: 10.1007/BF02439199
J. B. Conrey, The mean-square of Dirichlet L-functions, Preprint (2007). arXiv:0708.2699
S. Drappeau, Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method, Proc. Lond. Math. Soc. (3) 114(4) (2017), 684–732. DOI: 10.1112/plms.12022
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi ´ , “Tables of Integral Transforms”, Vol. I, Based, in part, on notes left by Harry Bateman,
McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954.
D. Frolenkov, The cubic moment of automorphic L-functions in the weight aspect, J. Number Theory 207 (2020), 247–281.
DOI: 10.1016/j.jnt.2019.07.009
I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products”, Translated from the Russian, Translation edited and with a preface by Alan
Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX), Seventh edition, Elsevier/Academic Press, Amsterdam, 2007.
P. Humphries, Density theorems for exceptional eigenvalues for congruence subgroups, Algebra Number Theory 12(7) (2018), 1581–1610. DOI: 10.2140/ant.2018.12.1581
P. Humphries and R. Khan, On the random wave conjecture for dihedral Maaß forms, Geom. Funct. Anal. 30(1) (2020), 34–125.
DOI: 10.1007/s00039-020-00526-4
H. Iwaniec, “Topics in Classical Automorphic Forms”, Graduate Studies in Mathematics 17, American Mathematical Society, Providence, RI, 1997.
DOI: 10.1090/gsm/017
S. Jana and R. Nunes, Spectral reciprocity for GL(n) and simultaneous nonvanishing of central L-values, Preprint (2021). arXiv:2111.02297
I. Kaneko, Motohashi’s formula for the fourth moment of individual Dirichlet L-functions and applications, Forum Math. Sigma 10 (2022), Paper No. e41, 43 pp. DOI: 10.1017/fms.2022.33
R. Khan and M. P. Young, Moments and hybrid subconvexity for symmetricsquare L-functions, J. Inst. Math. Jussieu, First view, 1–45. DOI: 10.1017/S1474748021000566
E. M. Kıral and M. P. Young, Kloosterman sums and Fourier coefficients of Eisenstein series, Ramanujan J. 49(2) (2019), 391–409.
DOI: 10.1007/s11139-018-0031-x
E. M. Kıral and M. P. Young, The fifth moment of modular L-functions, J. Eur. Math. Soc. (JEMS) 23(1) (2021), 237–314. DOI: 10.4171/jems/1011
N. V. Kuznecov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums (Russian), Mat. Sb. (N.S.) 111(153) (1980), no. 3, 334–383, 479; English translation: Math. USSR-Sb 39(3) (1981), 299–342. DOI: 10.1070/SM1981v039n03ABEH001518
N. V. Kuznetsov, Sums of Kloosterman sums and the eighth power moment of the Riemann zeta-function, in: “Number Theory and Related Topics” (Bombay, 1988), Tata Inst. Fund. Res. Stud. Math. 12, Tata Inst. Fund. Res., Bombay, 1989, pp. 57–117.
N. V. Kuznetsov, Hecke series at the center of the critical strip (Russian), Preprint FEB RAS Khabarovsk Division of the Institute for Applied Mathematics, No. 21, Vladivostok: Dalnauka (1999), 1–27
N. V. Kuznetsov, Trace formulas and average value of arithmetic functions in short intervals (Russian), Vladivostok: Dalnauka (2003), 1–160.
C.-H. Kwan, Spectral moment formulae for GL(3)×GL(2) L-functions, Preprint (2021). arXiv:2112.08568
X. Miao, Spectral reciprocity for the product of Rankin–Selberg L-functions, Preprint (2021). arXiv:2110.11529
Y. Motohashi, Kuznetsov’s paper on the eighth power moment of the Riemann zeta-function (revised), Part I, Manuscript, June 22, 1991.
Y. Motohashi, Spectral mean values of Maass waveform L-functions, J. Number Theory 42(3) (1992), 258–284. DOI: 10.1016/0022-314X(92)90092-4
Y. Motohashi, An explicit formula for the fourth power mean of the Riemann zeta-function, Acta Math. 170(2) (1993), 181–220. DOI: 10.1007/BF0239278
Y. Motohashi, “Spectral Theory of the Riemann Zeta-Function”, Cambridge Tracts in Mathematics 127, Cambridge University Press, Cambridge, 1997.
DOI: 10.1017/CBO9780511983399
Y. Motohashi, A functional equation for the spectral fourth moment of modular Hecke L-functions, in: “Proceedings of the Session in Analytic Number Theory and Diophantine Equations”, Bonner Math. Schriften 360, Univ. Bonn, Bonn, 2003, 19 pp.
M. R. Murty, Oscillations of Fourier coefficients of modular forms, Math. Ann. 262(4) (1983), 431–446. DOI: 10.1007/BF01456059
P. D. Nelson, Eisenstein series and the cubic moment for PGL2, Preprint (2019). arXiv:1911.06310
R. M. Nunes, Spectral reciprocity via integral representations, Preprint (2020). arXiv:2002.01993
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clarke, ed., “NIST Handbook of Mathematical Functions”, With 1 CD-ROM (Windows, Macintosh and UNIX), U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010
I. N. Petrow, A twisted Motohashi formula and Weyl-subconvexity for L-functions of weight two cusp forms, Math. Ann. 363(1-2) (2015), 175–216. DOI: 10.1007/s00208-014-1166-8
I. Petrow, Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field, Algebra Number Theory 12(10) (2018), 2471–2498. DOI: 10.2140/ant.2018.12.2471
I. Petrow and M. P. Young, The fourth moment of Dirichlet L-functions along a coset and the Weyl bound, Preprint (2019). arXiv:1908.10346
R. Schulze-Pillot and A. Yenirce, Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients, Int. J. Number Theory 14(8) (2018), 2277–2290. DOI: 10.1142/S1793042118501385
G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31(1) (1975), 79–98. DOI: 10.1112/plms/s3-31.1.79
K. Soundararajan and M. P. Young, The prime geodesic theorem, J. Reine Angew. Math. 2013(676) (2013), 105–120. DOI: 10.1515/crelle.2012.002
H. Wu, On Motohashi’s formula, Preprint (2020). arXiv:2001.09733.
M. P. Young, The reciprocity law for the twisted second moment of Dirichlet L-functions, Forum Math. 23(6) (2011), 1323–1337. DOI: 10.1515/FORM.2011.053
R. Zacharias, Periods and reciprocity I, Int. Math. Res. Not. IMRN 2021(3) (2021), 2191–2209. DOI: 10.1093/imrn/rnz100
D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, in: “Modular Functions of One Variable, VI” (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math. 627, Springer, Berlin, 1977, pp. 105–169. DOI: 10.1007/BFb0065299
O. Balkanova, Spectral decomposition formula and moments of symmetric square L-functions, Preprint (2020). arXiv:2011.05952
O. Balkanova, The first moment of Maaß form symmetric square L-functions, Ramanujan J. 55(2) (2021), 761–781. DOI: 10.1007/s11139-020-00272-z
O. Balkanova and D. Frolenkov, The mean value of symmetric square L-functions, Algebra Number Theory 12(1) (2018), 35–59. DOI: 10.2140/ant.2018.12.35
O. Balkanova and D. Frolenkov, Non-vanishing of Maass form symmetric square L-functions, J. Math. Anal. Appl. 500(2) (2021), Paper No. 125148,
23 pp. DOI: 10.1016/j.jmaa.2021.125148
O. Balkanova, D. Frolenkov, and H. Wu, On Weyl’s subconvex bound for cube-free Hecke characters: Totally real case, Preprint (2021). arXiv:2108.
12283
S. Bettin, On the reciprocity law for the twisted second moment of Dirichlet L-functions, Trans. Amer. Math. Soc. 368(10) (2016), 6887–6914.
DOI: 10.1090/tran/6771
A. Biro´, A duality relation for certain triple products of automorphic forms, Israel J. Math. 192(2) (2012), 587–636. DOI: 10.1007/s11856-012-0041-0
V. Blomer, P. Humphries, R. Khan, and M. B. Milinovich, Motohashi’s fourth moment identity for non-archimedean test functions and applications,
Compos. Math. 156(5) (2020), 1004–1038. DOI: 10.1112/s0010437x20007101
V. Blomer and R. Khan, Twisted moments of L-functions and spectral reciprocity, Duke Math. J. 168(6) (2019), 1109–1177.
DOI: 10.1215/00127094-2018-0060
V. Blomer and R. Khan, Uniform subconvexity and symmetry breaking reciprocity, J. Funct. Anal. 276(7) (2019), 2315–2358. DOI: 10.1016/j.jfa.2018.11.009
V. Blomer, X. Li, and S. D. Miller, A spectral reciprocity formula and nonvanishing for L-functions on GL(4) × GL(2), J. Number Theory 205 (2019), 1–43. DOI: 10.1016/j.jnt.2019.05.011
R. W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math. 45(1) (1978), 1–18. DOI: 10.1007/BF01406220
V. A. Bykovskiï, Density theorems and the mean value of arithmetic functions on short intervals (Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 212(12) (1994), 56–70, 196; translation in: J. Math. Sci. (New York) 83(6) (1997), 720–730. DOI: 10.1007/BF02439199
J. B. Conrey, The mean-square of Dirichlet L-functions, Preprint (2007). arXiv:0708.2699
S. Drappeau, Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method, Proc. Lond. Math. Soc. (3) 114(4) (2017), 684–732. DOI: 10.1112/plms.12022
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi ´ , “Tables of Integral Transforms”, Vol. I, Based, in part, on notes left by Harry Bateman,
McGraw-Hill Book Co., Inc., New York-Toronto-London, 1954.
D. Frolenkov, The cubic moment of automorphic L-functions in the weight aspect, J. Number Theory 207 (2020), 247–281.
DOI: 10.1016/j.jnt.2019.07.009
I. S. Gradshteyn and I. M. Ryzhik, “Table of Integrals, Series, and Products”, Translated from the Russian, Translation edited and with a preface by Alan
Jeffrey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX), Seventh edition, Elsevier/Academic Press, Amsterdam, 2007.
P. Humphries, Density theorems for exceptional eigenvalues for congruence subgroups, Algebra Number Theory 12(7) (2018), 1581–1610. DOI: 10.2140/ant.2018.12.1581
P. Humphries and R. Khan, On the random wave conjecture for dihedral Maaß forms, Geom. Funct. Anal. 30(1) (2020), 34–125.
DOI: 10.1007/s00039-020-00526-4
H. Iwaniec, “Topics in Classical Automorphic Forms”, Graduate Studies in Mathematics 17, American Mathematical Society, Providence, RI, 1997.
DOI: 10.1090/gsm/017
S. Jana and R. Nunes, Spectral reciprocity for GL(n) and simultaneous nonvanishing of central L-values, Preprint (2021). arXiv:2111.02297
I. Kaneko, Motohashi’s formula for the fourth moment of individual Dirichlet L-functions and applications, Forum Math. Sigma 10 (2022), Paper No. e41, 43 pp. DOI: 10.1017/fms.2022.33
R. Khan and M. P. Young, Moments and hybrid subconvexity for symmetricsquare L-functions, J. Inst. Math. Jussieu, First view, 1–45. DOI: 10.1017/S1474748021000566
E. M. Kıral and M. P. Young, Kloosterman sums and Fourier coefficients of Eisenstein series, Ramanujan J. 49(2) (2019), 391–409.
DOI: 10.1007/s11139-018-0031-x
E. M. Kıral and M. P. Young, The fifth moment of modular L-functions, J. Eur. Math. Soc. (JEMS) 23(1) (2021), 237–314. DOI: 10.4171/jems/1011
N. V. Kuznecov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums (Russian), Mat. Sb. (N.S.) 111(153) (1980), no. 3, 334–383, 479; English translation: Math. USSR-Sb 39(3) (1981), 299–342. DOI: 10.1070/SM1981v039n03ABEH001518
N. V. Kuznetsov, Sums of Kloosterman sums and the eighth power moment of the Riemann zeta-function, in: “Number Theory and Related Topics” (Bombay, 1988), Tata Inst. Fund. Res. Stud. Math. 12, Tata Inst. Fund. Res., Bombay, 1989, pp. 57–117.
N. V. Kuznetsov, Hecke series at the center of the critical strip (Russian), Preprint FEB RAS Khabarovsk Division of the Institute for Applied Mathematics, No. 21, Vladivostok: Dalnauka (1999), 1–27
N. V. Kuznetsov, Trace formulas and average value of arithmetic functions in short intervals (Russian), Vladivostok: Dalnauka (2003), 1–160.
C.-H. Kwan, Spectral moment formulae for GL(3)×GL(2) L-functions, Preprint (2021). arXiv:2112.08568
X. Miao, Spectral reciprocity for the product of Rankin–Selberg L-functions, Preprint (2021). arXiv:2110.11529
Y. Motohashi, Kuznetsov’s paper on the eighth power moment of the Riemann zeta-function (revised), Part I, Manuscript, June 22, 1991.
Y. Motohashi, Spectral mean values of Maass waveform L-functions, J. Number Theory 42(3) (1992), 258–284. DOI: 10.1016/0022-314X(92)90092-4
Y. Motohashi, An explicit formula for the fourth power mean of the Riemann zeta-function, Acta Math. 170(2) (1993), 181–220. DOI: 10.1007/BF0239278
Y. Motohashi, “Spectral Theory of the Riemann Zeta-Function”, Cambridge Tracts in Mathematics 127, Cambridge University Press, Cambridge, 1997.
DOI: 10.1017/CBO9780511983399
Y. Motohashi, A functional equation for the spectral fourth moment of modular Hecke L-functions, in: “Proceedings of the Session in Analytic Number Theory and Diophantine Equations”, Bonner Math. Schriften 360, Univ. Bonn, Bonn, 2003, 19 pp.
M. R. Murty, Oscillations of Fourier coefficients of modular forms, Math. Ann. 262(4) (1983), 431–446. DOI: 10.1007/BF01456059
P. D. Nelson, Eisenstein series and the cubic moment for PGL2, Preprint (2019). arXiv:1911.06310
R. M. Nunes, Spectral reciprocity via integral representations, Preprint (2020). arXiv:2002.01993
F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clarke, ed., “NIST Handbook of Mathematical Functions”, With 1 CD-ROM (Windows, Macintosh and UNIX), U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010
I. N. Petrow, A twisted Motohashi formula and Weyl-subconvexity for L-functions of weight two cusp forms, Math. Ann. 363(1-2) (2015), 175–216. DOI: 10.1007/s00208-014-1166-8
I. Petrow, Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field, Algebra Number Theory 12(10) (2018), 2471–2498. DOI: 10.2140/ant.2018.12.2471
I. Petrow and M. P. Young, The fourth moment of Dirichlet L-functions along a coset and the Weyl bound, Preprint (2019). arXiv:1908.10346
R. Schulze-Pillot and A. Yenirce, Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients, Int. J. Number Theory 14(8) (2018), 2277–2290. DOI: 10.1142/S1793042118501385
G. Shimura, On the holomorphy of certain Dirichlet series, Proc. London Math. Soc. (3) 31(1) (1975), 79–98. DOI: 10.1112/plms/s3-31.1.79
K. Soundararajan and M. P. Young, The prime geodesic theorem, J. Reine Angew. Math. 2013(676) (2013), 105–120. DOI: 10.1515/crelle.2012.002
H. Wu, On Motohashi’s formula, Preprint (2020). arXiv:2001.09733.
M. P. Young, The reciprocity law for the twisted second moment of Dirichlet L-functions, Forum Math. 23(6) (2011), 1323–1337. DOI: 10.1515/FORM.2011.053
R. Zacharias, Periods and reciprocity I, Int. Math. Res. Not. IMRN 2021(3) (2021), 2191–2209. DOI: 10.1093/imrn/rnz100
D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, in: “Modular Functions of One Variable, VI” (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math. 627, Springer, Berlin, 1977, pp. 105–169. DOI: 10.1007/BFb0065299