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A combinatorial interpretation of Serre's conjecture on modular Galois representations

Adriaan Herremans (2003)

Annales de l’institut Fourier

We state a conjecture concerning modular absolutely irreducible odd 2-dimensional representations of the absolute Galois group over finite fields which is purely combinatorial (without using modular forms) and proof that it is equivalent to Serre’s strong conjecture. The main idea is to replace modular forms with coefficients in a finite field of characteristic p , by their counterparts in the theory of modular symbols.

Critical and ramification points of the modular parametrization of an elliptic curve

Christophe Delaunay (2005)

Journal de Théorie des Nombres de Bordeaux

Let E be an elliptic curve defined over with conductor N and denote by ϕ the modular parametrization: ϕ : X 0 ( N ) E ( ) . In this paper, we are concerned with the critical and ramification points of ϕ . In particular, we explain how we can obtain a more or less experimental study of these points.

Hecke operators in half-integral weight

Soma Purkait (2014)

Journal de Théorie des Nombres de Bordeaux

In [6], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators T p with p prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators T p ...

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