2009
DOI: 10.4007/annals.2009.169.229
|View full text |Cite
|
Sign up to set email alerts
|

On Serre’s conjecture for 2-dimensional mod p representations of Gal(ℚ∕ℚ)

Abstract: We prove the existence in many cases of minimally ramified p-adic lifts of 2-dimensional 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… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1
1

Citation Types

1
89
0

Year Published

2009
2009
2019
2019

Publication Types

Select...
3
3
1

Relationship

1
6

Authors

Journals

citations
Cited by 85 publications
(90 citation statements)
references
References 72 publications
1
89
0
Order By: Relevance
“…The argument of Prop. 2.3 of [27] gives that the affine algebra of the moduli of such lifts with fixed determinant is formally smooth of relative dimension 3.3. The case of a finite place v not above p. Let q be the residue characteristic of v. We fix the determinant φ.…”
Section: Types Of Deformationsmentioning
confidence: 99%
See 1 more Smart Citation
“…The argument of Prop. 2.3 of [27] gives that the affine algebra of the moduli of such lifts with fixed determinant is formally smooth of relative dimension 3.3. The case of a finite place v not above p. Let q be the residue characteristic of v. We fix the determinant φ.…”
Section: Types Of Deformationsmentioning
confidence: 99%
“…Modularity lifting results when combined with presentation results for deformation rings due to Böckle [5] (see Proposition 4.4 below), and Taylor's potential version of Serre's conjecture, lead by the method of [27] and [31] to the existence of p-adic lifts asserted in Theorem 5.1 of [30] (see Corollary 4.6 below). These lifts are made part of compatible systems using arguments of Taylor (see 5.3.3 of [54]) and Dieulefait (see [17], [57]).…”
Section: Introductionmentioning
confidence: 99%
“…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%
“…Specifically, its reductionρ {w,691} : G Q → GL 2 (F 691 ) is a reducible representation and its semisimplication,ρ ss {w,691} , is equivalent to 1⊕ω 11 . These ρ {w,691} all have the same preferred indecomposable residual representation, 35 Greenberg tells me that one can check it just using the Hecke operators for the primes 2 and 3. Explicitly, since τ (2) = −24 and τ (3) = 252 one must check that there is no integer a simultaneously satisfying the following two congruences modulo 691 2 = 477481: −24 ≡ 2 a + 2 11−a and 252 ≡ 3 a + 3 11−a .…”
Section: Returning To P = 691mentioning
confidence: 99%
See 1 more Smart Citation