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