منابع مشابه
Serre’s Modularity Conjecture
These notes are based on lectures given by the author at the winter school on Galois theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre’s modularity conjecture and of its proof by Khare, Wintenberger, and Kisin [36] [37] [39], as well as of the results of other mathematicians that played an important role in the proof. Along the way we will r...
متن کاملRemarks on Serre’s modularity conjecture
In this article we give a proof of Serre’s conjecture for the case of odd level and arbitrary weight. Our proof will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations in any step of our proof). The key tool in the proof is Kisin’s recent modularity lifting result, which ...
متن کاملThe Modularity Conjecture for Calabi-Yau Threefolds
A Calabi-Yau variety, X, of dimension n, is defined to be a smooth projective variety over a field k that satisfies ωX := ∧n Ω ' OX and H(X,OX) = 0 for 0 < j < n. This can be seen as a higher dimensional “cohomological” analogue of an elliptic curve. A Calabi-Yau threefold is a Calabi-Yau variety of dimension three. The main goal of this paper is to introduce two equivalent notions of modularit...
متن کاملThe Modularity Conjecture for Rigid Calabi – Yau Threefolds over Q
: We formulate the modularity conjecture for rigid Calabi–Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi–Yau threefold arising from the root lattice A3. Our proof is based on geometric analysis. 1. The L–series of Calabi–Yau threefolds Let Q be the field of rational numbers, and let Q̄ be its algebraic closure with Galois group G := ...
متن کاملProving Serre’s modularity conjecture via Sophie Germain primes
In this article we give a proof of Serre’s conjecture for the cases of odd conductor and even conductor semistable at 2, and arbitrary weight. Our proof in both cases will be unconditional: in particular, it will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations in any s...
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ژورنال
عنوان ژورنال: Inventiones mathematicae
سال: 2009
ISSN: 0020-9910,1432-1297
DOI: 10.1007/s00222-009-0205-7