Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T00:51:09.750Z Has data issue: false hasContentIssue false
  • English
  • Français

CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINOR L-VALUES

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  14 October 2021

NEIL DUMMIGAN Open the ORCID record for NEIL DUMMIGAN [Opens in a new window]
NEIL DUMMIGAN*
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, UK e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Following Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinor L-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

MSC classification

Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F33: Congruences for modular and $p$-adic modular forms 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 2 , May 2022 , pp. 504 - 525
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agarwal, M. and Brown, J., On the Bloch-Kato conjecture for elliptic modular forms of square-free level, Math. Z. 276 (2014), 889–924.CrossRefGoogle Scholar
Agarwal, M. and Brown, J., Saito-Kurokawa lifts of square-free level, Kyoto J. Math. 55 (2015), 641–662.CrossRefGoogle Scholar
Almkvist, G., van Enckevort, C., van Straten, D. and Zudilin, W., Tables of Calabi-Yau equations, arXiv:math/0507430v2.Google Scholar
Baily, W. L., Jr., Automorphic forms with integral Fourier coefficients, in 1970 Several Complex Variables, I (Proceedings of Conference, University of Maryland, College Park, MD, 1970) (Springer, Berlin), 1–8.Google Scholar
Bergström, J. and Dummigan, N., Eisenstein congruences for split reductive groups, Selecta Mathematica 22 (2016), 1073–1115.CrossRefGoogle Scholar
Berger, T. and Klosin, K., Deformations of Saito-Kurokawa type and the paramodular conjecture, with an appendix by Chris Poor, Jerry Shurman and David S. Yuen, Am. J. Math. 142 (2020), 1821–1875.CrossRefGoogle Scholar
Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift Volume I, Progress in Mathematics, vol. 86 (Birkhäuser, Boston, 1990), 333–400.Google Scholar
Böcherer, S., Bemerkungen über die Dirichletreihen von Koecher und Maass, Mathematica Gottingensis 68 (1986), 36 pp.Google Scholar
Bosma, W., Cannon, J. J. and Playoust, C., The Magma algebra system. I. The user language, J. Symb. Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
Brown, J., Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture, Compos. Math. 143 (2007), 290–322.CrossRefGoogle Scholar
Brown, J. and Li, H., Congruence primes for automorphic forms on symplectic groups, Glasgow Math. J. 63 (2021), 660–681.CrossRefGoogle Scholar
Brumer, A., Pacetti, A., Poor, C., Tornaría, G., Voight, J. and Yuen, D. S., Paramodularity of typical abelian surfaces, Algebra Number Theory 13 (2019), 1145–1195.CrossRefGoogle Scholar
Deligne, P., Valeurs, de Fonctions L et Périodes d’Intégrales, AMS Proceedings of Symposia in Pure Mathematics, vol. 33, part 2 (1979), 313–346.Google Scholar
Deligne, P., Formes modulaires et représentations l-adiques, Sém. Bourbaki, éxp. 355, Lecture Notes in Mathematics, vol. 179 (Springer, Berlin, 1969), 139–172.CrossRefGoogle Scholar
Diamond, F., Flach, M. and Guo, L., The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. école Norm. Sup. (4) 37 (2004), 663–727.CrossRefGoogle Scholar
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. Japan 72 (2020), 251–301.CrossRefGoogle Scholar
Dokchitser, T., Computing special values of motivic L-functions, Exp. Math. 13 (2004), 137–149.CrossRefGoogle Scholar
Dummigan, N., Eisenstein primes, critical values and global torsion, Pac. J. Math. 233 (2007), 291–308.CrossRefGoogle Scholar
Dummigan, N., Ibukiyama, T. and Katsurada, H., Some Siegel modular standard L-values, and Shafarevich-Tate groups, J. Number Theory 131(2011), 1296–1330.Google Scholar
Furusawa, M., On L-functions for $\textrm{GSp}(4)\times\textrm{GL}(2)$ and their special values, J. Reine Angew. Math. 438 (1993), 187–218.Google Scholar
Furusawa, M. and Morimoto, K., Refined global Gross-Prasad conjecture on special Bessel periods and Böcherer’s conjecture, J. Eur. Math. Soc., electronically published on December 24, 2020, doi: 10.4171/JEMS/1034 (to appear in print), arXiv:1611.05567v5.Google Scholar
Gross, B., Kohnen, W. and Zagier, D., Heegner points and derivatives of L-series. II, Math. Ann. 278 (1987), 497–562.CrossRefGoogle Scholar
Hein, J., Orthogonal modular forms: an application to a conjecture of Birch, algorithms and computations, PhD Thesis (Dartmouth College, 2016).Google Scholar
Hida, H., Congruences for cusp forms and special values of their zeta functions, Invent. Math. 63 (1981), 225–261.CrossRefGoogle Scholar
Katsurada, H., Congruence of Siegel modular forms and special values of their standard zeta functions, Math. Z. 259 (2008), 97–111.CrossRefGoogle Scholar
Katsurada, H., Exact standard zeta values of Siegel modular forms, Exp. Math. 19 (2010), 65–77.Google Scholar
Kohnen, W. and Skoruppa, N.-P., A certain Dirichlet series attached to Siegel modular forms of degree 2, Invent. Math. 95 (1989), 541–558.CrossRefGoogle Scholar
Kohnen, W. and Zagier, D., Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175–198.CrossRefGoogle Scholar
Kohnen, W., Lifting modular forms of half-integral weight to Siegel modular forms of even genus, Math. Ann. 322 (2002), 787–809.CrossRefGoogle Scholar
Kohnen, W., Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237–268.CrossRefGoogle Scholar
Liu, Y., Refined global Gan-Gross-Prasad conjecture for Bessel periods, J. Reine Angew. Math 717 (2016), 133–194.Google Scholar
Nekovář, J., Some consequences of a formula of Mazur and Rubin for arithmetic local constants, Algebra Number Theory 7 (2013), 1101–1120.CrossRefGoogle Scholar
Poor, C., Shurman, J. and Yuen, D. S., Appendix to T. Berger, K. Klosin, Deformations of Saito-Kurokawa type and the paramodular conjecture, Am. J. Math. 142 (2020), 1821–1875.CrossRefGoogle Scholar
Poor, C. and Yuen, D. S., Paramodular cusp forms, Math. Comp. 84 (2015), 1401–1438.Google Scholar
Rama, G. and Tornaría, G., Quinary orthogonal modular forms http://www.cmat.edu.uy/cnt/ (2020).Google Scholar
Ryan, N. C. and Tornaría, G., Formulas for central values of twisted spin L-functions attached to paramodular forms, Math. Comp. 85 (2016), 907–929.Google Scholar
Roberts, B. and Schmidt, R., Local newforms for $\textrm{GSp}(4)$ , Lecture Notes in Mathematics, vol. 1918 (Springer, Berlin, 2007).Google Scholar
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.CrossRefGoogle Scholar
Shimura, G., On the Fourier coefficients of modular forms in several variables, Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II (1975), 261–268.Google Scholar
Skoruppa, N.-P. and Zagier, D., Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988), 113–146.CrossRefGoogle Scholar
Sugano, T., On holomorphic cusp forms on quaternion unitary groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1985), 521–568.Google Scholar
Weissauer, R., Four dimensional Galois representations, Astérisque 302 (2005), 67–150.Google Scholar