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On p -adic L -functions of G L ( 2 ) × G L ( 2 ) over totally real fields

Haruzo Hida (1991)

Annales de l'institut Fourier

Let D ( s , f , g ) be the Rankin product L -function for two Hilbert cusp forms f and g . This L -function is in fact the standard L -function of an automorphic representation of the algebraic group G L ( 2 ) × G L ( 2 ) defined over a totally real field. Under the ordinarity assumption at a given prime p for f and g , we shall construct a p -adic analytic function of several variables which interpolates the algebraic part of D ( m , f , g ) for critical integers m , regarding all the ingredients m , f and g as variables.

On the dynamics of ϕ : x x p + a in a local field

David Adam, Youssef Fares (2010)

Actes des rencontres du CIRM

Let K be a local field, a K and ϕ : x x p + a where p denotes the characteristic of the residue field. We prove that the minimal subsets of the dynamical system ( K , ϕ ) are cycles and describe the cycles of this system.

On the image of Λ -adic Galois representations

Ami Fischman (2002)

Annales de l’institut Fourier

We explore the question of how big the image of a Galois representation attached to a Λ -adic modular form with no complex multiplication is and show that for a “generic” set of Λ -adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all have a large image.

On the infinite fern of Galois representations of unitary type

Gaëtan Chenevier (2011)

Annales scientifiques de l'École Normale Supérieure

Let E be a CM number field, p an odd prime totally split in  E , and let  X be the p -adic analytic space parameterizing the isomorphism classes of  3 -dimensional semisimple p -adic representations of  Gal ( E ¯ / E ) satisfying a selfduality condition “of type U ( 3 ) ”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in  X has dimension at least 3 [ E : ] . As important steps, and in any rank, we prove that any first order...

On the slopes of the  U 5 operator acting on overconvergent modular forms

L. J. P Kilford (2008)

Journal de Théorie des Nombres de Bordeaux

We show that the slopes of the  U 5 operator acting on 5-adic overconvergent modular forms of weight  k with primitive Dirichlet character  χ of conductor 25 are given by either 1 4 · 8 i 5 : i or 1 4 · 8 i + 4 5 : i , depending on  k and  χ .We also prove that the space of classical cusp forms of weight  k and character  χ has a basis of eigenforms for the Hecke operators  T p and  U 5 which is defined over  Q 5 ( 5 4 , 3 ) .

Overconvergent modular symbols and p -adic L -functions

Robert Pollack, Glenn Stevens (2011)

Annales scientifiques de l'École Normale Supérieure

This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent p -adic modular symbols. Specifically, we give a constructive proof of acontrol theorem (Theorem 1.1) due to the second author [19] proving existence and uniqueness of overconvergent eigenliftings of classical modular eigensymbols of non-critical slope. As an application we describe a polynomial-time algorithm for explicit computation of associated p -adic L -functions in this case. In...

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