Journal of the Mathematical Society of Japan

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

Martin DICKSON, Ameya PITALE, Abhishek SAHA, and Ralf SCHMIDT

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Abstract

We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of $L$-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan–Gross–Prasad conjecture for Bessel periods as proposed by Liu. We note several consequences of our conjecture to arithmetic and analytic properties of $L$-functions and Fourier coefficients of Siegel modular forms.

Note

The third author acknowledges the support of the EPSRC grant EP/L025515/1.

Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 51 pages.

Dates
Received: 18 August 2017
Revised: 9 October 2018
First available in Project Euclid: 28 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1572249776

Digital Object Identifier
doi:10.2969/jmsj/78657865

Subjects
Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11F30: Fourier coefficients of automorphic forms 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Keywords
Siegel modular forms Bessel periods Fourier coefficients $L$-functions

Citation

DICKSON, Martin; PITALE, Ameya; SAHA, Abhishek; SCHMIDT, Ralf. Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level. J. Math. Soc. Japan, advance publication, 28 October 2019. doi:10.2969/jmsj/78657865. https://projecteuclid.org/euclid.jmsj/1572249776


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References

  • [1] M. Asgari and R. Schmidt, Siegel modular forms and representations, Manuscripta Math., 104 (2001), 173–200.
  • [2] M. Asgari and R. Schmidt, On the adjoint $L$-function of the $p$-adic $\mathrm{GSp}(4)$, J. Number Theory, 128 (2008), 2340–2358.
  • [3] S. Böcherer, Bemerkungen über die Dirichletreihen von Koecher und Maass, Mathematica Gottingensis, 68 (1986), 1–36.
  • [4] S. Böcherer, N. Dummigan and R. Schulze-Pillot, Yoshida lifts and Selmer groups, J. Math. Soc. Japan, 64 (2012), 1353–1405.
  • [5] A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math., 35 (1976), 233–259.
  • [6] J. Brown, An inner product relation on Saito–Kurokawa lifts, Ramanujan J., 14 (2007), 89–105.
  • [7] J. H. Bruinier, Nonvanishing modulo $l$ of Fourier coefficients of half-integral weight modular forms, Duke Math. J., 98 (1999), 595–611.
  • [8] W. Casselman, The unramified principal series of $\mathfrak{p}$-adic groups. I. The spherical function, Compositio Math., 40 (1980), 387–406.
  • [9] A. J. Corbett, A proof of the refined Gan–Gross–Prasad conjecture for non-endoscopic Yoshida lifts, Forum Math., 29 (2017), 59–90.
  • [10] M. Dickson, Local spectral equidistribution for degree two Siegel modular forms in level and weight aspects, Int. J. Number Theory, 11 (2015), 341–396.
  • [11] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progr. Math., 55, Birkhäuser Boston Inc., Boston, MA, 1985.
  • [12] M. Furusawa, On $L$-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and their special values, J. Reine Angew. Math., 438 (1993), 187–218.
  • [13] M. Furusawa and K. Morimoto, Refined global Gross–Prasad conjecture on special Bessel periods and Böcherer's conjecture, to appear in J. Eur. Math. Soc. (JEMS), arXiv:1611.05567.
  • [14] W. T. Gan, B. Gross and D. Prasad, Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups, In: Sur les conjectures de Gross et Prasad. I, Astérisque, 346, Soc. Math. France, Paris, 2012, 1–109.
  • [15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, seventh edition, Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian, translation edited and with a preface by A. Jeffrey and D. Zwillinger, with one CD-ROM (Windows, Macintosh and UNIX).
  • [16] M. Harris and S. Kudla, The central critical value of a triple product $L$-function, Ann. of Math. (2), 133 (1991), 605–672.
  • [17] M. L. Hsieh and S. Yamana, Bessel periods and anticyclotomic $p$-adic spinor $L$-functions, arXiv:1810.03231.
  • [18] A. Ichino, Trilinear forms and the central values of triple product $L$-functions, Duke Math. J., 145 (2008), 281–307.
  • [19] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture, Geom. Funct. Anal., 19 (2010), 1378–1425.
  • [20] R. Keaton and A. Pitale, Restrictions of Eisenstein series and Rankin–Selberg convolution, Doc. Math., 24 (2019), 1–45.
  • [21] H. Klingen, Introductory Lectures on Siegel Modular Forms, Cambridge Stud. Adv. Math., 20, Cambridge Univ. Press, Cambridge, 1990.
  • [22] A. Knightly and C. Li, On the distribution of Satake parameters for Siegel modular forms, Doc. Math., 24 (2019), 677–747.
  • [23] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann., 271 (1985), 237–268.
  • [24] E. Kowalski, A. Saha and J. Tsimerman, Local spectral equidistribution for Siegel modular forms and applications, Compositio Math., 148 (2012), 335–384.
  • [25] Y. Liu, Refined global Gan–Gross–Prasad conjecture for Bessel periods, J. Reine Angew. Math., 717 (2016), 133–194.
  • [26] H. Narita, A. Pitale and R. Schmidt, Irreducibility criteria for local and global representations, Proc. Amer. Math. Soc., 141 (2013), 55–63.
  • [27] P. Nelson, A. Pitale and A. Saha, Bounds for Rankin–Selberg integrals and quantum unique ergodicity for powerful levels, J. Amer. Math. Soc., 27 (2014), 147–191.
  • [28] J. Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by N. Schappacher, with a foreword by G. Harder.
  • [29] A. Pitale, A. Saha and R. Schmidt, Lowest weight modules of $\mathrm{Sp}_4(\mathbb{R})$ and nearly holomorphic Siegel modular forms, to appear in Kyoto J. Math., arXiv:1501.00524.
  • [30] A. Pitale, A. Saha and R. Schmidt, Transfer of Siegel cusp forms of degree 2, Mem. Amer. Math. Soc., 232 (2014), no. 1090, vi+107pp.
  • [31] A. Pitale and R. Schmidt, Ramanujan-type results for Siegel cusp forms of degree 2, J. Ramanujan Math. Soc., 24 (2009), 87–111.
  • [32] A. Pitale and R. Schmidt, Bessel models for $\mathrm{GSp}(4)$: Siegel vectors of square-free level, J. Number Theory, 136 (2014), 134–164.
  • [33] D. Prasad and R. Takloo-Bighash, Bessel models for $\mathrm{GSp}(4)$, J. Reine Angew. Math., 655 (2011), 189–243.
  • [34] Y Qiu, The Bessel period functional on $\mathrm{SO}(5)$: the nontempered case, arXiv:1312.5793.
  • [35] H. Resnikoff and R. L. Saldana, Some properties of Fourier coefficients of Eisenstein series of degree two, J. Reine Angew. Math., 265 (1974), 90–109.
  • [36] B. Roberts and R. Schmidt, Local Newforms for $\mathrm{GSp}(4)$, Lecture Notes in Math., 1918, Springer-Verlag, Berlin, 2007.
  • [37] B. Roberts and R. Schmidt, Some results on Bessel functionals for $\mathrm{GSp}(4)$, Doc. Math., 21 (2016), 467–553.
  • [38] A. Saha, On ratios of Petersson norms for Yoshida lifts, Forum Math., 27 (2015), 2361–2412.
  • [39] A. Saha and R. Schmidt, Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular $L$-functions, J. London Math. Soc., 88 (2013), 251–270.
  • [40] R. Schmidt, Iwahori-spherical representations of $\mathrm{GSp}(4)$ and Siegel modular forms of degree 2 with square-free level, J. Math. Soc. Japan, 57 (2005), 259–293.
  • [41] R. Schmidt, On classical Saito–Kurokawa liftings, J. Reine Angew. Math., 604 (2007), 211–236.
  • [42] G. Shimura, On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 17 (1975), 261–268.
  • [43] J. L. Waldspurger, Sur les valeurs de certaines fonctions $L$ automorphes en leur centre de symétrie, Compositio Math., 54 (1985), 173–242.
  • [44] T. Watson, Rankin triple products and quantum chaos, arXiv:0810.0425, (2008).
  • [45] R. Weissauer, Endoscopy for $\mathrm{GSp}(4)$ and the Cohomology of Siegel Modular Threefolds, Lecture Notes in Math., 1968, Springer-Verlag, Berlin, 2009.