scholarly journals Computing congruences of modular forms and Galois representations modulo prime powers

2010 ◽  
pp. 145-166 ◽  
Author(s):  
Xavier Taixés i Ventosa ◽  
Gabor Wiese
Keyword(s):  
Modular Forms ◽  
Galois Representations

Computations with modular forms and Galois representations

2011 ◽  
Author(s):  
Johan Bosman
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Cusp Forms ◽  
Space Of Cusp Forms

This chapter discusses several aspects of the practical side of computing with modular forms and Galois representations. It starts by discussing computations with modular forms, and from there work towards the computation of polynomials that give the Galois representations associated with modular forms. Throughout, the chapter denotes the space of cusp forms of weight k, group Γ‎₁(N), and character ε‎ by Sₖ(N, ε‎).

First description of the algorithms

2011 ◽  
Author(s):  
Jean-Marc Couveignes ◽  
Bas Edixhoven
Keyword(s):  
Finite Fields ◽  
Modular Forms ◽  
Minimal Polynomial ◽  
Galois Representations ◽  
Essential Step ◽  
Sufficient Precision ◽  
Informal Description

This chapter provides the first, informal description of the algorithms. It explains how the computation of the Galois representations V attached to modular forms over finite fields should proceed. The essential step is to approximate the minimal polynomial P of (3.1) with sufficient precision so that P itself can be obtained.

scholarly journals Potentially $GL_2$-type Galois representations associated to noncongruence modular forms

2018 ◽  
Vol 371 (8) ◽  
pp. 5341-5377
Author(s):  
Wen-Ching Winnie Li ◽  
Tong Liu ◽  
Ling Long
Keyword(s):  
Modular Forms ◽  
Galois Representations

scholarly journals UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ATTACHED TO HILBERT MODULAR FORMS MOD  OF WEIGHT 1

2018 ◽  
Vol 19 (2) ◽  
pp. 281-306 ◽  
Author(s):  
Mladen Dimitrov ◽  
Gabor Wiese
Keyword(s):  
Modular Forms ◽  
Galois Representations ◽  
Galois Representation ◽  
Important Case ◽  
Hilbert Modular Forms ◽  
Hilbert Modular

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over $\overline{\mathbb{F}}_{p}$ of parallel weight 1 and level prime to $p$ is unramified above $p$. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel weight 1 forms in characteristic $p$ embed into the ordinary part of parallel weight $p$ forms in two different ways per prime dividing $p$, namely via ‘partial’ Frobenius operators.

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