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CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  29 September 2020

JIM BROWN  and
HUIXI LI
JIM BROWN
Affiliation:
Department of Mathematics, Occidental College, Los Angeles, CA90041, USA, e-mail: [email protected]
HUIXI LI
Affiliation:
Department of Mathematics and Statistics, University of Nevada - Reno, Reno, NV89557, USA, e-mail: [email protected]
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Abstract

It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F32: Modular correspondences, etc.
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 3 , September 2021 , pp. 660 - 681
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Agarwal, M., p-adic L-functions for GSp(4)× GL (2), PhD Thesis (University of Michigan, Ann Arbor, MI, 2007).Google Scholar
Agarwal, M. and Brown, J., On the Bloch-Kato conjecture for elliptic modular forms of square-free level, Math. Z. 276(3–4) (2014), 889–924.CrossRefGoogle Scholar
Agarwal, M. and Klosin, K., Yoshida lifts and the Bloch-Kato conjecture for the convolution L-function, J. Number Theory 133(8) (2013), 2496–2537.CrossRefGoogle Scholar
Asgari, M. and Schmidt, R., Siegel modular forms and representations, Manuscripta Math. 104(2) (2001), 173–200.CrossRefGoogle Scholar
Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, Progress in Mathematics, vol. 86 (Birkhäuser Boston, Boston, MA, 1990), 333–400.Google Scholar
Böcherer, S., Dummigan, N. and Schulze-Pillot, R., Yoshida lifts and Selmer groups, J. Math. Soc. Japan 64(4) (2012), 1353–1405.CrossRefGoogle Scholar
Brown, J., Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture, Compos. Math. 143(2) (2007), 290–322.CrossRefGoogle Scholar
Brown, J., On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the Bloch-Kato conjecture, Int. Math. Res. Not. 2011(7) (2011), 1706–1756.Google Scholar
Brown, J. and Klosin, K., Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts, Kyoto J. Math. 60(1) (2020), 179–217.CrossRefGoogle Scholar
Brown, J. and Pitale, A., Special values of L-functions for Saito-Kurokawa lifts, Math. Proc. Cambridge Philos. Soc. 155(2) (2013), 237–255.CrossRefGoogle Scholar
Brumer, A. and Kramer, K., Paramodular abelian varieties of odd conductor, Trans. Amer. Math. Soc. 366(5) (2014), 2463–2516.CrossRefGoogle Scholar
Dasgupta, S., Darmon, H. and Pollack, R., Hilbert modular forms and the Gross-Stark conjecture, Ann. Math. (2) 174(1) (2011), 439–484.CrossRefGoogle Scholar
Fontaine, J.-M., Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. Math. (2) 115(3) (1982), 529–577.CrossRefGoogle Scholar
Garrett, P., On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients, Invent. Math. 107(3) (1992), 453–481.CrossRefGoogle Scholar
Gritsenko, V., Irrationality of the moduli spaces of polarized abelian surfaces, in Abelian varieties (Egloffstein, 1993) (de Gruyter, Berlin, 1995), 63–84. With an appendix by the author and K. Hulek.Google Scholar
Gross, B., Kohnen, W. and Zagier, D., Heegner points and derivatives of L-series. II, Math. Ann. 278(1–4) (1987), 497–562.CrossRefGoogle Scholar
Hida, H., Galois representations and the theory of P-adic Hecke algebras, Sūgaku 39(2) (1987), 124–139. Sugaku Expositions 2(1) (1989), 75–102.Google Scholar
Katsurada, H., Congruence of Siegel modular forms and special values of their standard zeta functions, Math. Z. 259(1) (2008), 97–111.CrossRefGoogle Scholar
Klosin, K., Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture, Ann. Inst. Fourier (Grenoble) 59(1) (2009), 81–166.CrossRefGoogle Scholar
Klosin, K., The Maass space for U(2,2) and the Bloch-Kato conjecture for the symmetric square motive of a modular form, J. Math. Soc. Japan 67(2) (2015), 797–860.CrossRefGoogle Scholar
Ribet, K., A modular construction of unramified p-extensions of Q(μp), Invent. Math. 34(3) (1976), 151–162.CrossRefGoogle Scholar
Roberts, B. and Schmidt, R., Local newforms for GSp(4), Lecture Notes in Mathematics, vol. 1918 (Springer, Berlin, 2007).Google Scholar
Rubin, K., Euler systems , Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000). Hermann Weyl Lectures. The Institute for Advanced Study.Google Scholar
Schmidt, R., On classical Saito-Kurokawa liftings, J. Reine Angew. Math. 604 (2007), 211–236.Google Scholar
Shimura, G., Confluent hypergeometric functions on tube domains, Math. Ann. 260(3) (1982), 269–302.CrossRefGoogle Scholar
Shimura, G., Nearly holomorphic functions on Hermitian symmetric spaces, Math. Ann. 278(1–4) (1987), 1–28.CrossRefGoogle Scholar
Shimura, G., Eisenstein series and zeta functions on symplectic groups, Invent. Math. 119(3) (1995), 539–584.CrossRefGoogle Scholar
Shimura, G., Euler products and Eisenstein series, CBMS Regional Conference Series in Mathematics, vol. 93 (Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997).CrossRefGoogle Scholar
Skinner, C. and Urban, E., The Iwasawa main conjectures for GL2, Invent. Math. 195(1) (2014), 1–277.CrossRefGoogle Scholar
Skoruppa, N. and Zagier, D., Jacobi forms and a certain space of modular forms, Invent. Math. 94(1) (1988), 113–146.CrossRefGoogle Scholar
Vatsal, V., Canonical periods and congruence formulae, Duke Math. J. 98(2) (1999), 397–419.CrossRefGoogle Scholar
Weissauer, R., Four dimensional Galois representations, Astérisque (302) (2005), 67–150. Formes automorphes. II. Le cas du groupe GSp (4).Google Scholar
Wiles, A., The Iwasawa conjecture for totally real fields, Ann. Math. (2) 131(3) (1990), 493–540.CrossRefGoogle Scholar