We complete the proof that every elliptic curve over rational numbers is modular.

The authors would like to give special thanks N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this paper. They are also grateful A. Agboola, M. Bertolini, Edixhoven, J. Fearnley, R. Gross, L. Guo, F. Jarvis, H. Kisilevsky, E. Liverance, Manoharmayum, Ribet, D. Rohrlich, Rosen, Schoof, J.-P. Serre, C. Skinner, Thakur, Tilouine, Tunnell, Van der Poorten, Washington their helpful comments. Darmon the members CICMA Quebec-Vermont Number Theory...

We show that certain potentially semistable lifts of modular mod <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> representations are themselves modular. As a result we any elliptic curve over the rational numbers with conductor not divisible by 27 is

Let f be a newform of weight k⩾2, level N with coefficients in number field K, and A the adjoint motive M associated to f. We carefully discuss construction realisations A, as well natural integral structures these realisations. then use method Taylor Wiles verify λ-part Tamagawa conjecture Bloch Kato for L(A,0) L(A,1). Here λ is any prime K not dividing Nk!, so that mod representation absolutely irreducible when restricted Galois group over Q((−1)(ℓ−1)/2ℓ) where λ|ℓ. The also establishes...

In 1987 Serre conjectured that any mod l ("ell", not "1") two-dimensional irreducible odd representation of the absolute Galois group rationals came from a modular form in precise way. We present generalisation this conjecture to 2-dimensional representations totally real field where is unramified. The hard work formulating an analogue "weight" part Serre's conjecture. furthermore asked whether his could be rephrased terms "mod Langlands philosophy". Using ideas Emerton and Vigneras, we...

Abstract We consider integral models of Hilbert modular varieties with Iwahori level structure at primes over p , first proving a Kodaira–Spencer isomorphism that gives concise description their dualizing sheaves. then analyze fibres the degeneracy maps to prime and deduce vanishing higher direct images sheaves, generalizing prior work Kassaei Sasaki (for unramified in totally real field F ). apply results prove flatness finite morphisms resulting Stein factorizations, combine them simplify...

Soit F un corps totalement réel, v une place de non ramifiée divisant p et ρ : Gal(Q/F ) → GL 2 (F représentation continue irréductible dont la restriction ρ| Gal(Fv/Fv) est réductible suffisamment générique.Si modulaire (et satisfait quelques conditions techniques faibles), nous montrons comment retrouver l'extension correspondante entre les deux caractères Gal(F /F en terme l'action sur cohomologie modulo p.Abstract.-Let be a totally real field, an unramified of dividing and continuous...

Let p be a prime and F totally real field in which is unramified. We consider mod Hilbert modular forms for , defined as sections of automorphic line bundles on varieties level to characteristic . For Hecke eigenform arbitrary weight (without parity hypotheses), we associate two-dimensional representation the absolute Galois group give conjectural description set weights all eigenforms from it arises. This conjecture can viewed “geometric” variant “algebraic” Serre Buzzard–Diamond–Jarvis,...

We shall explain how the following is a corollary of results Wiles [W]:Theorem.Suppose that E an elliptic curve over Q all whose 2division points are rational, i.e., defined byfor some distinct rational numbers a, b and c.Then modular.

We consider the Hilbert modular varieties in characteristic p with Iwahori level at and construct a geometric Jacquet-Langlands relation showing that irreducible components are isomorphic to projective bundles over quaternionic Shimura of prime p.We use this establish between mod forms reflects representation theory GL 2 generalizes result Serre for classical forms.Finally we study fibres degeneracy map prove cohomological vanishing is used associate Galois representations forms.

We study minimal and toroidal compactifications of p-integral models Hilbert modular varieties. review the theory in setting Iwahori level at primes over p, extend it to certain finer structures. also prove extensions recent results on Iwahori-level Kodaira–Spencer isomorphisms cohomological vanishing for degeneracy maps. Finally we apply q-expansions forms, especially effect Hecke operators p general base rings.

We consider mod $p$ Hilbert modular forms associated to a totally real field of degree $d$ in which is unramified. prove that every such form arises by multiplication partial Hasse invariants from one whose weight (a -tuple integers) lies certain cone contained the set non-negative weights, answering question Andreatta and Goren. The proof based on properties Goren–Oort stratification varieties established Goren Oort, Tian Xiao.

Abstract Let K be a finite unramified extension of Q p . We parametrize the (φ,Γ)-modules corresponding to reducible two-dimensional $\overline {\F }_p$ -representations G and characterize those which have crystalline lifts with certain Hodge–Tate weights.

Let F be a totally real field, and v place of dividing an odd prime p. We study the weight part Serre's conjecture for continuous, odd, two-dimensional mod p representations rhobar absolute Galois group that are reducible locally at v. W set predicted Serre weights semisimplification restricted to decomposition prove when local representation is generic, in which modular exactly ones (assuming modular). also determine precisely subsets arise as varies with fixed generic semisimplification.

We prove that all mod $p$ Hilbert modular forms arise via multiplication by generalized partial Hasse invariants from whose weight falls within a certain minimal cone. This answers question posed Andreatta and Goren, generalizes our previous results which treated the case where is unramified in totally real field. Whereas work made use of deep Jacquet-Langlands type on Goren-Oort stratification (not yet available when ramified), here we instead properties at Iwahori level are more readily...

The Tamagawa number conjecture of Bloch and Kato describes the behavior at integers L-function associated to a motive over Q.Let f be newform weight k ≥ 2, level N with coefficients in field K. Let M let A adjoint .Let λ finite prime We verify λ-part Bloch-Kato for L(A, 0) 1) when Nk! mod representation is absolutely irreducible restricted Galois group Q (-1) ( -1)/2 where | .