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Overconvergent Eichler–Shimura isomorphisms

Published online by Cambridge University Press:  02 January 2014

Fabrizio Andreatta
Affiliation:
Dipartimento di Matematica ‘Federigo Enriques’, Università degli Studi di Milano, Via C. Saldini 50, Milano 20133, Italia ([email protected])
Adrian Iovita
Affiliation:
Dipartimento di Matematica dell’Universita di Padova, via Trieste 63, Padova, Italia Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve West Blvd., Montreal, QC, H3G 1M8, Canada ([email protected])
Glenn Stevens
Affiliation:
Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA ([email protected])

Abstract

Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.

Type
Research Article
Copyright
©Cambridge University Press 2013 

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