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Overconvergent Eichler–Shimura isomorphisms

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  02 January 2014

Fabrizio Andreatta ,
Adrian Iovita  and
Glenn Stevens
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)
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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 $\leq h$.

Keywords

overconvergent modular symbols; overconvergent modular forms; p-adic comparison isomorphisms; Galois representations; Hodge–Tate decomposition

MSC classification

Secondary: 11F85: $p$-adic theory, local fields 14G20: Local ground fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 14 , Issue 2 , April 2015 , pp. 221 - 274
Copyright
©Cambridge University Press 2013 

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