Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T07:08:27.708Z Has data issue: false hasContentIssue false
  • English
  • Français

Congruences for Modular Forms mod 2 and Quaternionic S-ideal Classes

Published online by Cambridge University Press:  20 November 2018

Kimball Martin
Kimball Martin*
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin–Lehner eigenvalues. The proofs involve the notion of quaternionic $S$-ideal classes and the distribution of Atkin–Lehner signs among newforms.

Keywords

11F3311R52modular formscongruencesquaternion algebras
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 5 , 01 October 2018 , pp. 1076 - 1095
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Deligne, P. amd Serre, J.-P., Formes modulaires de poids 1. Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530. http://dx.doi.org/10.24033/asens.1277Google Scholar
[2] Hasegawa, Y. and Hashimoto, K., On type numbers of split orders of definite quaternion algebras. ManuscriptaMath. 88 (1995), no. 4, 525–534. http://dx.doi.Org/10.1007/BF02567839Google Scholar
[3] Hida, H., Congruence of cusp forms and special values of their zeta functions. Invent. Math. 63 (1981), no. 2, 225–261. http://dx.doi.org/10.1007/BF01393877Google Scholar
[4] Le Hung, B. V. and Li, C., Level raising mod 2 and arbitrary 2-Selmer ranks. Compos. Math. 152 (2016), no. 8, 1576–1608. http://dx.doi.Org/10.1112/S0010437X16007454Google Scholar
[5] Martin, K., The Jacquet-Langlands correspondence, Eisenstein congruences, and integral L-values in weight 2. Math. Res. Let., to appear.Google Scholar
[6] Martin, K., Refined dimensions of cusp forms, and equidistribution and bias of signs. arxiv:1 609.05386Google Scholar
[7] Mazur, B., Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186.Google Scholar
[8] Pizer, A., The action of the canonical involution on modular forms of weight 2 on Fo(M). Math. Ann. 226 (1977), no. 2, 99–116. http://dx.doi.Org/10.1007/BF01360861Google Scholar
[9] Shemanske, T. R. and Walling, L. H., Twists of Hilbert modular forms. Trans. Amer. Math. Soc. 338 (1993), no. 1, 375–403, http://dx.doi.org/10.1090/S0002-9947-1993-1102225-XGoogle Scholar
[10] Yoo, H., Non-optimal levels of a reducible mod I modular representation. arxiv:1409.8342v3Google Scholar