#### Vol. 2, No. 7, 2008

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The half-integral weight eigencurve

### Nick Ramsey

Vol. 2 (2008), No. 7, 755–808
##### 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
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/