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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 (fabrizio.andreatta@unimi.it)
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 (iovita@mathstat.concordia.ca)
Glenn Stevens
Affiliation:
Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (ghs@math.bu.edu)

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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