On fundamental Fourier coefficients of Siegel cusp forms of degree 2. (English) Zbl 1533.11087
Let \(F\) be a Siegel cusp form of degree \(2\), even weight \(k \geq 2\), and odd square-free level \(N\). The authors study properties of Fourier coefficients \(a(F, S)\) of \(F\) at fundamental matrices \(S\) (i.e. matrices \(S\) with \(\operatorname{disc}(S)=-4\det(S)\) equal to a fundamental discriminant). In particular they study sign changes of \(a(F, S)\) and large values of \(|a(F, S)|\). A new aspect compared to previous works is the restriction to fundamental matrices, which makes the problems more difficult.
Concerning sign changes, the authors show that when \(F\) has real Fourier coefficients, there is a sequence of \(\geq X^{1-\varepsilon}\) increasing integers \(n_j \asymp X\) and associated fundamental matrices \(S_j\) with \(|\operatorname{disc}(S_j)| = n_j\) such that \(a(F, S_j) a(F, S_{j+1}) < 0\) for every \(j\).
Concerning large values, the authors obtain that there are \(\geq X^{1-\varepsilon}\) odd square-free integers \(n \in [X, 2X]\) with associated fundamental matrices \(S_n\) with \(|\operatorname{disc}(S_n)| = n\) such that \[ |a(F, S_n)| \geq n^{k/4-3/4} \exp\left(\frac{1}{82}\sqrt{\frac{\log n}{\log \log n}}\right). \] Note that conjecturally \(|a(F, S_n)| \ll n^{k/4-3/4+\varepsilon}\).
It is worth pointing out that prior to the present work it was only known that there are \(\gg X^{5/8-\varepsilon}\) non-vanishing \(a(F, S_j)\) with \(|\operatorname{disc}(S_j)| \asymp X\) (see works of A. Saha [Math. Ann. 355, No. 1, 363–380 (2013; Zbl 1329.11043)] and A. Saha and R. Schmidt [J. Lond. Math. Soc., II. Ser. 88, No. 1, 251–270 (2013; Zbl 1341.11022)]). Of course either of the above results immediately improves \(5/8\) to \(1\).
The proofs start by reducing the questions to corresponding questions concerning half-integral weight forms.
Concerning sign changes, the authors show that when \(F\) has real Fourier coefficients, there is a sequence of \(\geq X^{1-\varepsilon}\) increasing integers \(n_j \asymp X\) and associated fundamental matrices \(S_j\) with \(|\operatorname{disc}(S_j)| = n_j\) such that \(a(F, S_j) a(F, S_{j+1}) < 0\) for every \(j\).
Concerning large values, the authors obtain that there are \(\geq X^{1-\varepsilon}\) odd square-free integers \(n \in [X, 2X]\) with associated fundamental matrices \(S_n\) with \(|\operatorname{disc}(S_n)| = n\) such that \[ |a(F, S_n)| \geq n^{k/4-3/4} \exp\left(\frac{1}{82}\sqrt{\frac{\log n}{\log \log n}}\right). \] Note that conjecturally \(|a(F, S_n)| \ll n^{k/4-3/4+\varepsilon}\).
It is worth pointing out that prior to the present work it was only known that there are \(\gg X^{5/8-\varepsilon}\) non-vanishing \(a(F, S_j)\) with \(|\operatorname{disc}(S_j)| \asymp X\) (see works of A. Saha [Math. Ann. 355, No. 1, 363–380 (2013; Zbl 1329.11043)] and A. Saha and R. Schmidt [J. Lond. Math. Soc., II. Ser. 88, No. 1, 251–270 (2013; Zbl 1341.11022)]). Of course either of the above results immediately improves \(5/8\) to \(1\).
The proofs start by reducing the questions to corresponding questions concerning half-integral weight forms.
Reviewer: Kaisa Matomäki (Turku)
MSC:
| 11F30 | Fourier coefficients of automorphic forms |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
References:
| [1] | Andrianov, A. N., Euler products that correspond to Siegel’s modular forms of genus \(2\), Uspekhi Mat. Nauk29(3) (1974), 43-110. · Zbl 0304.10020 |
| [2] | Arthur, J., The Endoscopic Classification of Representations, American Mathematical Society Colloquium Publications, 61 (American Mathematical Society, Providence, RI, 2013). · Zbl 1297.22023 |
| [3] | Atkin, A. O. L. and Lehner, J., Hecke operators on \({\varGamma}_0(m)\), Math. Ann.185 (1970), 134-160. · Zbl 0177.34901 |
| [4] | Berger, T., Dembélé, L., Pacetti, A. and Şengün, M. H., Theta lifts of Bianchi modular forms and applications to paramodularity, J. Lond. Math. Soc. (2)92(2) (2015), 353-370. · Zbl 1396.11074 |
| [5] | Blomer, V. and Brumley, F., ‘Simultaneous equidistribution of toric periods and fractional moments of \(L\) -functions’, Preprint, 2020, URL: arXiv:2009.07093. |
| [6] | Böcherer, S., Bemerkungen über die Dirichletreihen von Koecher und Maass, Mathematica Gottingensis68 (Georg-August University, Göttingen1986), 36 pp. · Zbl 0593.10025 |
| [7] | Böcherer, S. and Das, S., On fundamental Fourier coefficients of Siegel modular forms, J. Inst. Math. Jussieu (2021), 1-41. doi.org/10.1017/S1474748021000086 |
| [8] | Borel, A., Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math.35 (1976), 233-259. · Zbl 0334.22012 |
| [9] | Bump, D. and Ginzburg, D., Symmetric square \(L\) -functions on \(\mathsf{GL}(r)\), Ann. of Math. (2)136(1) (1992), 137-205. · Zbl 0753.11021 |
| [10] | Chan, P.-S. and Gan, W. T., The local Langlands conjecture for \(\text{GSp}(4)\) III: Stability and twisted endoscopy, J. Number Theory146 (2015), 69-133. · Zbl 1366.11074 |
| [11] | Chandee, V., Explicit upper bounds for \(L\) -functions on the critical line, Proc. Amer. Math. Soc.137(12) (2009), 4049-4063. · Zbl 1243.11088 |
| [12] | Choie, Y., Gun, S. and Kohnen, W., An explicit bound for the first sign change of the Fourier coefficients of a Siegel cusp form, Int. Math. Res. Not. IMRN2015(12) (2015), 3782-3792. · Zbl 1333.11041 |
| [13] | Das, S., Omega results for Fourier coefficients of half-integral weight and Siegel modular forms, in International Conference on Number Theory, pp. 59-72 (Springer, Singapore, 2020). · Zbl 1469.11112 |
| [14] | Das, S. and Kohnen, W., On sign changes of eigenvalues of Siegel cusp forms of genus 2 in prime powers, Acta Arith.183(2) (2018), 167-172. · Zbl 1434.11098 |
| [15] | Dickson, M., Pitale, A., Saha, A. and Schmidt, R., Explicit refinements of Böcherer’s conjecture for Siegel modular forms of squarefree level, J. Math. Soc. Japan72(1) (2020), 251-301. · Zbl 1476.11079 |
| [16] | Eichler, M. and Zagier, D., The Theory of Jacobi Forms, Progress in Mathematics, 55 (Birkhäuser Boston Inc., Boston, MA, 1985). · Zbl 0554.10018 |
| [17] | Furusawa, M., On \(L\) -functions for \(\mathsf{GSp}(4)\times \mathsf{GL}(2)\) and their special values, J. Reine Angew. Math.438 (1993), 187-218. · Zbl 0770.11025 |
| [18] | Furusawa, M. and Morimoto, K., Refined global Gross-Prasad conjecture on special Bessel periods and Böcherer’s conjecture, J. Eur. Math. Soc. (JEMS)23(4) (2021), 1295-1331. · Zbl 1486.11072 |
| [19] | Gan, W. T. and Takeda, S., The local Langlands conjecture for \(\mathsf{GSp}(4)\), Ann. of Math. (2)173(3) (2011), 1841-1882. · Zbl 1230.11063 |
| [20] | Gritsenko, V., Analytic continuation of symmetric squares, Math. USSR Sb.35(5) (1979), 593-614. · Zbl 0426.10025 |
| [21] | Gun, S., Kohnen, W. and Soundararajan, K., ‘Large Fourier coefficients of half-integer weight modular forms’, Preprint, 2020, arXiv:2004.14450. |
| [22] | Gun, S. and Sengupta, J., Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups, Int. J. Number Theory13(10) (2017), 2597-2625. · Zbl 1428.11079 |
| [23] | Harcos, G., An additive problem in the Fourier coefficients of cusp forms, Math. Ann.326(2) (2003), 347-365. · Zbl 1045.11028 |
| [24] | He, X. and Zhao, L., On the first sign change of Fourier coefficients of cusp forms, J. Number Theory190 (2018), 212-228. · Zbl 1441.11093 |
| [25] | Heath-Brown, D. R., A mean value estimate for real character sums, Acta Arith.72(3) (1995), 235-275. · Zbl 0828.11040 |
| [26] | Henniart, G., Sur la fonctorialité, pour \(\mathsf{G}L(4)\) , donnée par le carré extérieur, Mosc. Math. J.9(1) (2009), 33-45. · Zbl 1183.22009 |
| [27] | Huang, B. and Lester, S., ‘Quantum variance for dihedral Maass forms’, Preprint, 2020, arXiv:2007.02055. · Zbl 1506.11068 |
| [28] | Iwaniec, H., Primes represented by quadratic polynomials in two variables, Acta Arith.24 (1973/74), 435-459. · Zbl 0271.10043 |
| [29] | Johnson-Leung, J. and Roberts, B., Siegel modular forms of degree two attached to Hilbert modular forms, J. Number Theory132(4) (2012), 543-564. · Zbl 1272.11063 |
| [30] | Jutila, M., Tranformations of exponential sums, in Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, 1989), pp. 263-270 (University of Salerno, Salerno, 1992). · Zbl 0794.11033 |
| [31] | Kaczorowski, J. and Perelli, A., On the prime number theorem for the Selberg class, Arch. Math. (Basel)80(3) (2003), 255-263. · Zbl 1126.11334 |
| [32] | Kim, H., Functoriality for the exterior square of \({\mathsf{GL}}_4\) and the symmetric fourth of \({\mathsf{GL}}_2\), J. Amer. Math. Soc.16(1) (2003), 139-183, with appendices by D. Ramakrishnan, H. Kim and P. Sarnak. · Zbl 1018.11024 |
| [33] | Kohnen, W., Newforms of half-integral weight, J. Reine Angew. Math.333 (1982), 32-72. · Zbl 0475.10025 |
| [34] | Kohnen, W., Fourier coefficients of modular forms of half-integral weight, Math. Ann.271(2) (1985), 237-268. · Zbl 0542.10018 |
| [35] | Kohnen, W., Estimates for Fourier coefficients of Siegel cusp forms of degree two, Compos. Math.87(2) (1993), 231-240. · Zbl 0783.11023 |
| [36] | Kohnen, W., Sign changes of Fourier coefficients and eigenvalues of cusp forms, in Number Theory, Ser. Number Theory Appl., vol. 2, pp. 97-107 (World Scientifice Publishing, Hackensack, NJ, 2007). · Zbl 1128.11023 |
| [37] | Lapid, E. M., On the nonnegativity of Rankin-Selberg \(L\) -functions at the center of symmetry, Int. Math. Res. Not. IMRN2003(2) (2003), 65-75. · Zbl 1046.11032 |
| [38] | Lester, S. and Radziwiłł, M., Quantum unique ergodicity for half-integral weight automorphic forms, Duke Math. J.169(2) (2020), 279-351. · Zbl 1441.11107 |
| [39] | Lester, S. and Radziwiłł, M., Signs of Fourier coefficients of half-integral weight modular forms, Math. Ann.379(3-4) (2021), 1553-1604. · Zbl 1467.11047 |
| [40] | Liu, Y., Refined global Gan-Gross-Prasad conjecture for Bessel periods, J. Reine Angew. Math.717 (2016), 133-194. · Zbl 1404.11065 |
| [41] | Manickam, M. and Ramakrishnan, B., On Shimura, Shintani and Eichler-Zagier correspondences, Trans. Amer. Math. Soc.352(6) (2000), 2601-2617. · Zbl 0985.11020 |
| [42] | Matomäki, K. and Radziwiłł, M., Sign changes of Hecke eigenvalues, Geom. Funct. Anal.25(6) (2015), 1937-1955. · Zbl 1359.11040 |
| [43] | Milinovich, M. B. and Turnage-Butterbaugh, C. L., Moments of products of automorphic \(L\) -functions, J. Number Theory139 (2014), 175-204. · Zbl 1305.11037 |
| [44] | Pitale, A., Saha, A. and Schmidt, R., Transfer of Siegel cusp forms of degree 2, Mem. Amer. Math. Soc.232(1090) (2014). · Zbl 1303.11063 |
| [45] | Pitale, A. and Schmidt, R., Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2, Proc. Amer. Math. Soc.136(11) (2008), 3831-3838. · Zbl 1205.11058 |
| [46] | Pitale, A. and Schmidt, R., Integral representation for \(L\) -functions for \(\text{GSp}4\times \text{GL}2\), J. Number Theory129(10) (2009), 1272 - 1324. · Zbl 1258.11059 |
| [47] | Pitale, A. and Schmidt, R., Ramanujan-type results for Siegel cusp forms of degree 2, J. Ramanujan Math. Soc.24(1) (2009), 87-111. · Zbl 1258.11058 |
| [48] | Prasad, D. and Ramakrishnan, D., On the global root numbers of \(\mathsf{GL}(n)\times \mathsf{GL}(m)\) , in Automorphic Forms, Automorphic Representations, and Arithmetic (Fort Worth, TX, 1996), Proceedings of Symposia in Pure Mathematics, 66, pp. 311-330 (American Mathematical Societ, Providence, RI, 1999). · Zbl 0991.11025 |
| [49] | Qiu, Y., ‘The Bessel period functional on \(\mathsf{SO}(5)\) : The nontempered case, Preprint, 2013, URL: arXiv:1312.5793. |
| [50] | Radziwiłł, M. and Soundararajan, K., Moments and distribution of central \(L\) -values of quadratic twists of elliptic curves, Invent. Math.202(3) (2015), 1029-1068. · Zbl 1396.11098 |
| [51] | Resnikoff, H. and Saldana, R. L., Some properties of Fourier coefficients of Eisenstein series of degree two, J. Reine Angew. Math.265 (1974), 90-109. · Zbl 0278.10028 |
| [52] | Roberts, B. and Schmidt, R., Local Newforms for GSp(4), Lecture Notes in Mathematics, 1918 (Springer, Berlin, 2007). |
| [53] | Royer, E., Sengupta, J. and Wu, J., Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two, Ramanujan J. 39(1) (2016), 179-199, with an appendix by E. Kowalski and A. Saha. · Zbl 1402.11074 |
| [54] | Saha, A., \(L\) -functions for holomorphic forms on \(\mathsf{GSp}(4)\times \mathsf{GL}(2)\) and their special values, Int. Math. Res. Not. IMRN2009(10) (2009), 1773-1837. · Zbl 1252.11045 |
| [55] | Saha, A., Pullbacks of Eisenstein series from \(\text{GU}\left(3,3\right)\) and critical \(L\) -values for \(\text{GSp}4\times \text{GL}2\), Pacific J. Math.246(2) (2010), 435-486. · Zbl 1213.11106 |
| [56] | Saha, A., Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients, Math. Ann.355(1) (2013), 363-380. · Zbl 1329.11043 |
| [57] | Saha, A., On ratios of Petersson norms for Yoshida lifts, Forum Math.27(4) (2015), 2361-2412. · Zbl 1383.11060 |
| [58] | Saha, A. and Schmidt, R., Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular \(L\) -functions, J. Lond. Math. Soc. (2)88 (2013), 251-270. · Zbl 1341.11022 |
| [59] | Schmidt, R., Packet structure and paramodular forms, Trans. Amer. Math. Soc.370(5) (2018), 3085-3112. · Zbl 1440.11075 |
| [60] | Shahidi, F., On certain \(L\) -functions, Amer. J. Math.103(2) (1981), 297-355. · Zbl 0467.12013 |
| [61] | Shahidi, F., On non-vanishing of twisted symmetric and exterior square \(L\) -functions for \(\mathsf{GL}(n)\), Pacific J. Math., Special Issue, 1997, Olga Taussky-Todd: in memoriam, pp. 311-322. |
| [62] | Shimura, G., On modular forms of half integral weight, Ann. of Math. (2)97 (1973), 440-481. · Zbl 0266.10022 |
| [63] | Soundararajan, K., Extreme values of zeta and \(L\) -functions, Math. Ann.342(2) (2008), 467-486. · Zbl 1186.11049 |
| [64] | Soundararajan, K., Moments of the Riemann zeta function, Ann. of Math. (2)170(2) (2009), 981-993. · Zbl 1251.11058 |
| [65] | Soundararajan, K. and Young, M. P., The second moment of quadratic twists of modular \(L\) -functions, J. Eur. Math. Soc. (JEMS)12(5) (2010), 1097-1116. · Zbl 1213.11165 |
| [66] | Takeda, S., On a certain metaplectic Eisenstein series and the twisted symmetric square \(L\) -function, Math. Z.281(1-2) (2015), 103-157. · Zbl 1337.11032 |
| [67] | Waldspurger, J.-L., Sur les valeurs de certaines fonctions \(L\) automorphes en leur centre de symétrie, Compos. Math.54(2) (1985), 173-242. · Zbl 0567.10021 |
| [68] | Weissauer, R., Endoscopy for \(\mathsf{GSp}(4)\) and the Cohomology of Siegel Modular Threefolds, Lecture Notes in Mathematics, 1968 (Springer-Verlag, Berlin, 2009). |
| [69] | Wu, J. and Ye, Y., Hypothesis H and the prime number theorem for automorphic representations, Funct. Approx. Comment. Math.37(2) (2007), 461-471. · Zbl 1230.11065 |
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