The half-integral weight eigencurve

Ramsey, Nick

doi:10.2140/ant.2008.2.755

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The half-integral weight eigencurve

Nick Ramsey

Appendix: Brian Conrad

Vol. 2 (2008), No. 7, 755–808

DOI: 10.2140/ant.2008.2.755

Abstract

In this paper we define Banach spaces of overconvergent half-integral weight $p$ -adic modular forms and Banach modules of families of overconvergent half-integral weight $p$ -adic modular forms over admissible open subsets of weight space. Both spaces are equipped with a continuous Hecke action for which $U_{p^{2}}$ is moreover compact. The modules of families of forms are used to construct an eigencurve parameterizing all finite-slope systems of eigenvalues of Hecke operators acting on these spaces. We also prove an analog of Coleman’s theorem stating that overconvergent eigenforms of suitably low slope are classical.

Keywords

modular forms of half-integral weight, $p$-adic modular forms, eigenvarieties

Mathematical Subject Classification 2000

Primary: 11F33

Secondary: 14G22, 11F37

Milestones

Received: 25 September 2007

Revised: 3 July 2008

Accepted: 22 August 2008

Published: 23 November 2008

Authors

Nick Ramsey
Department of Mathematics University of Michigan 2074 East Hall 530 Church Street Ann Arbor, MI 48109-1043 United States

Brian Conrad
Department of Mathematics Stanford University Building 380, Sloan Hall Stanford, CA 94305 United States
http://math.stanford.edu/~conrad/

Vol. 2, No. 7, 2008