Serre’s modularity conjecture (II)

Khare, Chandrashekhar; Wintenberger, Jean-Pierre

doi:10.1007/s00222-009-0206-6

Cited by 208 publications

(194 citation statements)

References 60 publications

Supporting

Mentioning

187

Contrasting

Unclassified

Order By: Relevance

“…. 6 This is an allusion to Serre's conjecture on the modularity of mod p 2-dimensional representations of the absolute Galois group of Q, which he formulated in a letter to Tate, on May 1, 1973, but published only in 1987 (in an extended and more precise form which made it possible to deduce Fermat's last theorem from the Taniyama-Weil conjecture). This conjecture is now a theorem [6], and Tate's result is the starting point for a complicated induction over the set of prime numbers.…”

Section: Tate's Report On Elliptic Curvesmentioning

confidence: 97%

Tate’s work and the Serre–Tate correspondence

Colmez¹

2017

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The Serre-Tate correspondence contains a lot of Tate's work in a casual form. We present some excerpts that show how some of Tate's best known contributions came into being. Excuse all these letters. I find that writing you is an excellent method of organizing my thoughts. Some of this work was only published much later, some was never published. We give below some excerpts that show how some of Tate's best known contributions came into being. There are many other topics on which Tate worked that appear in the correspondence and that we will not consider, such as Galois cohomology, class field theory, p-adic Hodge theory, Honda-Tate theory, Serre-Tate theory, elliptic curves with everywhere good reduction, modular forms, Stark's conjectures. . . . Serre and The Tate curveElliptic curves over C are usually thought of as C modulo a lattice Λ. Now, two homothetic lattices give isomorphic elliptic curves, and so one can take the lattice Λ, corresponding to an elliptic curve E, of the form 2πi(Z + Zτ ), with Im τ > 0. Setting q = e 2iπτ and using the exponential map, one obtains an isomorphism E(C) ∼ = C * /q Z of complex Lie groups. If the equation of the elliptic curve is y 2 = x 3 − g 2 x − g 3 , classical formulas express g 2 , g 3 as power series in q and x(w), y(w), if w ∈ C * , as series in q, w, and w −1 . These series have rational coefficients, and Tate had the amazing idea that they could be used to give a description of (special) elliptic curves over a p-adic field analogous to the above description over C. This first shows up, in the correspondence, in a letter by Serre of July 31, 1959.Il paraît que vous faites des choses rupinantes avec les courbes elliptiques sur les p-adiques (j non entier), m'a raconté Lang; vous savez faire marcher ce que nos pères appelaient les "fonctions loxodromiques" sur les p-adiques. C'est bien sympathique, et j'aimerais beaucoup avoir des détails, siça ne vous ennuie pas d'écrire.Si j'ai bien compris ce que me racontait Lang, votre théorie montre de façon amusante qu'une courbe elliptiqueà multiplication

show abstract

Section: Tate's Report On Elliptic Curvesmentioning

confidence: 97%

Tate’s work and the Serre–Tate correspondence

Colmez¹

2017

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…En prenant la limite inductive, on obtient le lemme suivant. [31]) pour démontrer l'existence et la modularité de relevés avec comportement local prescrit de représentations ρ : Gal(Q/F ) → GL 2 (k E ) (continues, irréductibles, totalement impaires). Il généralise des résultats de [18] (pour v = p) et de [30] (pour v = p) dans le cas F = Q, eux-mêmes inspirés des résultats de changement de niveau de Ribet ([35], [36]).…”

Section: Fixons Des Plongementsunclassified

Formes modulaires de Hilbert modulo $p$ et valeurs d'extensions entre caractères galoisiens

Breuil¹,

Diamond²

2014

Ann. Sci. École Norm. Sup.

View full text Add to dashboard Cite

“…The conjecture of Serre ([59]), recently proved by Khare and Wintenberger (see [36], [33], [35]) asserts that every irreducible continuous two-dimensional G Q representation over a finite field in which complex conjugation involution does not act as a scalar comes, in this manner, from a cuspform. 10 4.5.…”

Section: Abelian Extensions and Two-dimensional Galois Representationsmentioning

confidence: 99%

Untitled

2011

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Irregular primes-37 being the first such prime-have played a great role in number theory. This article discusses Ken Ribet's constructionfor all irregular primes p-of specific abelian, unramified, degree p extensions of the number fields Q(e 2πi/p ). These extensions with explicit information about their Galois groups (they are Galois over Q) were predicted to exist ever since the work of Herbrand in the 1930s. Ribet's method involves a tour through the theory of modular forms; it demonstrates the usefulness of congruences between cuspforms and Eisenstein series, a fact that has inspired, and continues to inspire, much work in number theory.

show abstract

Serre’s modularity conjecture (II)

Abstract: Abstract. We provide proofs of Theorems 4.1 and 5.1 of [30].

Cited by 208 publications

References 60 publications

Tate’s work and the Serre–Tate correspondence

Tate’s work and the Serre–Tate correspondence

Formes modulaires de Hilbert modulo $p$ et valeurs d'extensions entre caractères galoisiens

Untitled

Contact Info

Product

Resources

About