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Big Galois representations and $p$-adic $L$-functions

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  13 November 2014

Haruzo Hida
Haruzo Hida*
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA email hida@math.ucla.edu
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Abstract

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Let $p\geqslant 5$ be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup ${\rm\Gamma}(L)$ of $\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$ for a principal ideal $(L)\neq 0$ of $\mathbb{Z}_{p}[[T]]$ for the canonical ‘weight’ variable $t=1+T$. If $L\notin {\rm\Lambda}^{\times }$, the power series $L$ is proven to be a factor of the Kubota–Leopoldt $p$-adic $L$-function or of the square of the anticyclotomic Katz $p$-adic $L$-function or a power of $(t^{p^{m}}-1)$.

Keywords

p-adic L-functionHecke algebraGalois representationGalois deformation

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight 11F25: Hecke-Petersson operators, differential operators (one variable) 11F33: Congruences for modular and $p$-adic modular forms 11F80: Galois representations
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 4 , April 2015 , pp. 603 - 664
Copyright
© The Author 2014 

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