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43
On Serre’s conjecture for 2dimensional mod p representations of Gal(Q̄/Q)
"... We prove the existence in many cases of minimally ramified padic lifts of 2dimensional continuous, odd, absolutely irreducible, mod p representations ¯ρ of the absolute Galois group of Q. It is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weig ..."
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Cited by 27 (1 self)
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We prove the existence in many cases of minimally ramified padic lifts of 2dimensional continuous, odd, absolutely irreducible, mod p representations ¯ρ of the absolute Galois group of Q. It is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre’s conjectures in low levels and weights. For instance we prove that there are no irreducible (p, p) type group schemes over Z. We prove that a ¯ρ as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Deltafunction. In the last part of the paper we present arguments that reduce Serre’s conjecture to proving generalisations of modularity lifting theorems of the type pioneered by Wiles.
On Serre’s conjecture for mod ℓ Galois representations over totally real fields
, 2009
"... In 1987 Serre conjectured that any mod ℓ twodimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2dimensional representations of the absolute Galois group of a totally ..."
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Cited by 19 (2 self)
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In 1987 Serre conjectured that any mod ℓ twodimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2dimensional representations of the absolute Galois group of a totally real field where ℓ is unramified. The hard work is in formulating an analogue of the “weight ” part of Serre’s conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod ℓ Langlands philosophy”. Using ideas of Emerton and Vigneras, we formulate a mod ℓ localglobal principle for the group D ∗ , where D is a quaternion algebra over a totally real field, split above ℓ and at 0 or 1 infinite places, and show how it implies the conjecture.
THE MODULARITY CONJECTURE FOR RIGID CALABI–YAU Threefolds Over Q
, 2000
"... 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. ..."
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Cited by 15 (1 self)
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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.
Potential automorphy of odddimensional symmetric powers of elliptic curves and applications
 BIRKHÄUSER BOSTON INC
, 2009
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ON SERRE’S MODULARITY CONJECTURE FOR 2DIMENSIONAL MOD p REPRESENTATIONS OF ... Unramified Outside p
, 2005
"... We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ¯ρ: Gal ( ¯ Q/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in S k(¯ρ)(SL2(Z)). The proof relies on the methods introduced in an earlier joint wor ..."
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Cited by 13 (0 self)
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We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ¯ρ: Gal ( ¯ Q/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in S k(¯ρ)(SL2(Z)). The proof relies on the methods introduced in an earlier joint work with JP. Wintenberger, together with a new method of “weight reduction”.
THE ARITHMETIC OF QMABELIAN SURFACES THROUGH THEIR GALOIS REPRESENTATIONS
, 2003
"... This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our ..."
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Cited by 11 (8 self)
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This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with several explicit examples.
Modularity of some potentially BarsottiTate Galois representations
 Compos. Math
"... We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the FontaineMazur conjectures, specifically, the modularity of certain potentially BarsottiTate Galois representations. The proof follows the template of Wiles, TaylorWiles, and BreuilConradDi ..."
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Cited by 11 (6 self)
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We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the FontaineMazur conjectures, specifically, the modularity of certain potentially BarsottiTate Galois representations. The proof follows the template of Wiles, TaylorWiles, and BreuilConradDiamondTaylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered φ1modules of C. Breuil. 1. Notation, terminology, and results Throughout this article, we let l be an odd prime, and we fix an algebraic closure Ql of Ql with residue field Fl. The fields K, L, and E will always be finite extensions of Ql inside Ql. We denote by GK the Galois group Gal(Ql/K), by WK the Weil group of K, and by IK the inertia group of K. The group IQl will be abbreviated Il. The character ωn: GQl → Fln ⊂ Fl is defined via u ωn: u ↦→
Serre’s modularity conjecture (II)
, 2007
"... We provide proofs of Theorems 4.1 and 5.1 of [30]. ..."
The level 1 weight 2 case of Serre’s conjecture
, 2004
"... (24 de Marzo de 1917 8 de Julio de 2004): In memoriam We prove Serre’s conjecture for the case of Galois representations of Serre’s weight 2 and level 1. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argumen ..."
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(24 de Marzo de 1917 8 de Julio de 2004): In memoriam We prove Serre’s conjecture for the case of Galois representations of Serre’s weight 2 and level 1. We do this by combining the potential modularity results of Taylor and lowering the level for Hilbert modular forms with a Galois descent argument, properties of universal deformation rings, and the nonexistence of padic BarsottiTate conductor 1 Galois representations proved in [Di3]. 1