Journal of the Mathematical Society of Japan

Lifting puzzles and congruences of Ikeda and Ikeda–Miyawaki lifts

Neil DUMMIGAN

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show how many of the congruences between Ikeda lifts and non-Ikeda lifts, proved by Katsurada, can be reduced to congruences involving only forms of genus 1 and 2, using various liftings predicted by Arthur's multiplicity conjecture. Similarly, we show that conjectured congruences between Ikeda–Miyawaki lifts and non-lifts can often be reduced to congruences involving only forms of genus 1, 2 and 3.

Article information

Source
J. Math. Soc. Japan Volume 69, Number 2 (2017), 801-818.

Dates
First available in Project Euclid: 20 April 2017

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1492653647

Digital Object Identifier
doi:10.2969/jmsj/06920801

Subjects
Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

Keywords
congruences of automorphic forms Ikeda lift Ikeda–Miyawaki lift

Citation

DUMMIGAN, Neil. Lifting puzzles and congruences of Ikeda and Ikeda–Miyawaki lifts. J. Math. Soc. Japan 69 (2017), no. 2, 801--818. doi:10.2969/jmsj/06920801. http://projecteuclid.org/euclid.jmsj/1492653647.


Export citation

References

  • J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups, Amer. Math. Soc. Colloquium Publications, 61, Amer. Math. Soc., Providence, RI, 2013, xviii+590 pp.
  • M. Asgari and R. Schmidt, Siegel modular forms and representations, Manuscripta Math., 104 (2001), 173–200.
  • J. Bergström and N. Dummigan, Eisenstein congruences for split reductive groups, Selecta Math., 22 (2016), 1073–1115.
  • J. Bergström, C. Faber and G. van der Geer, Siegel modular forms of degree three and the cohomology of local systems, Selecta Math. (N.S.), 20 (2014), 83–124.
  • J. Brown, Saito–Kurokawa lifts and applications to the Bloch–Kato conjecture, Compos. Math., 143 (2007), 290–322.
  • G. Chenevier and C. Lannes, Formes automorphes et voisins de Kneser des réseaux de Niemeier, 388 pages, arXiv:1409.7616, 26/09/2014.
  • G. Chenevier and D. Renard, Level one algebraic cuspforms of classical groups of small rank, Mem. Amer. Math. Soc., 237 (2015), no. 1121.
  • P. Deligne, Valeurs de Fonctions $L$ et Périodes d'Intégrales, AMS Proc. Symp. Pure Math., 33 (1979), part 2, 313–346.
  • N. Dummigan, Symmetric square $L$-functions and Shafarevich–Tate groups, II, Int. J. Number Theory, 5 (2009), 1321–1345.
  • C. Faber and G. van der Geer, Sur la cohomologie des systèmes locaux sur les espaces de modules des courbes de genre $2$ et des surfaces abéliennes, I, II, C. R. Math. Acad. Sci. Paris, 338 (2004), 381–384 and 467–470.
  • G. Harder, A congruence between a Siegel and an elliptic modular form, manuscript, 2003, reproduced in The 1-2-3 of Modular Forms (ed. K. Ranestad), 247–262, Springer-Verlag, Berlin Heidelberg, 2008.
  • T. Ibukiyama, Lifting conjectures from vector valued Siegel modular forms of degree two, Comment. Math. Univ. St. Pauli, 61 (2012), 87–102.
  • T. Ibukiyama, H. Katsurada, C. Poor and D. Yuen, Congruences to Ikeda–Miyawaki lifts and triple $L$-values of elliptic modular forms, J. Number Theory, 134 (2014), 142–180.
  • T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math. (2), 154 (2001), 641–681.
  • T. Ikeda, Pullback of lifting of elliptic cusp forms and Miyawaki's conjecture, Duke Math. J., 131 (2006), 469–497.
  • H. Katsurada, Congruence between Duke–Imamoglu–Ikeda lifts and non–Duke–Imamoglu–Ikeda lifts, Comment. Math. Univ. St. Pauli, 64 (2015), 109–129.
  • H. Katsurada, Congruence of Siegel modular forms and special values of their standard zeta functions, Math. Z., 259 (2008), 97–111.
  • H. Katsurada and K. Kawamura, Ikeda's conjecture on the period of the Duke–Imamoglu–Ikeda lift, Proc. London Math. Soc. (3), 111 (2015), 445–483.
  • H. Katsurada and S. Mizumoto, Congruences for Hecke eigenvalues of Siegel modular forms, Abh. Math. Semin. Univ. Hambg., 82 (2012), 129–152.
  • N. Kurokawa, Congruences between Siegel modular forms of degree $2$, Proc. Japan Acad., 55A (1979), 417–422.
  • I. Miyawaki, Numerical examples of Siegel cusp forms of degree 3 and their zeta functions, Mem. Fac. Sci. Kyushu Univ., 46 (1992), 307–339.
  • S. Mizumoto, Congruences for eigenvalues of Hecke operators on Siegel modular forms of degree two, Math. Ann., 275 (1986), 149–161.
  • C. Poor, N. Ryan and D. S. Yuen, Lifting puzzles in degree four, Bull. Aust. Math. Soc., 80 (2009), 65–82.
  • T. Satoh, On certain vector valued Siegel modular forms of degree two, Math. Ann., 274 (1986), 335–352.
  • O. Taibi, Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula, preprint, 2014, http://arxiv.org/abs/1406.4247.
  • G. van der Geer, Siegel modular forms and their applications, in The 1-2-3 of Modular Forms (ed. K. Ranestad), 181–245, Springer-Verlag, Berlin Heidelberg, 2008.