The Taylor-Wiles construction and multiplicity one

Diamond, Fred

doi:10.1007/s002220050144

Cited by 104 publications

(107 citation statements)

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“…Note that Lemma 5.8 may provide an alternate route to the conclusion of the previous lemma (sometimes one can prove multiplicity one for a maximal ideal without relying on multiplicity one for differentials, e.g., see [Dia97]). Observe that in the proofs of Lemmas 5.11 and 5.8, all we needed was (locally) a non-zero free T-module (of finite rank, say) that is attached functorially to J.…”

Section: Multiplicity One For Differentialsmentioning

confidence: 99%

The modular degree, congruence primes, and multiplicity one

Agashe¹,

Ribet

Stein

2011

Number Theory, Analysis and Geometry

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Abstract.The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and Müller have asked whether these invariants coincide. We find that the question has a negative answer, and show that in the counterexamples, multiplicity one (defined below) does not hold. At the same time, we prove a theorem about the relation between the two invariants: the modular degree divides the congruence number, and the ratio is divisible only by primes whose squares divide the conductor of the elliptic curve. We discuss the ratio even in the case where the square of a prime does divide the conductor, and we study analogues of the two invariants for modular abelian varieties of arbitrary dimension.

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Section: Multiplicity One For Differentialsmentioning

confidence: 99%

The modular degree, congruence primes, and multiplicity one

Agashe¹,

Ribet

Stein

2011

Number Theory, Analysis and Geometry

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“…In a lot of cases one even knows that (F p ⊗ T(N, k)) m is a complete intersection, see [16], and also [15].…”

Section: Remark 53 We Do Not Know If Hmentioning

confidence: 99%

Comparison of Integral Structures on Spaces of Modular Forms of Weight Two, and Computation of Spaces of Forms Mod 2 of Weight One. With Appendices by Jean-François Mestre and Gabor Wiese

Edixhoven¹,

Mestre²,

Wiese³

2005

JMJ

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Suppose that φ(f, g) = 0. Then we have 0 = θ(Af + F (g)) = θ(Af ), hence 0 = θ(f ). But then f = 0, as the weight of f is not divisible by p. It follows that F (g) = 0, hence that g = 0. 4.5Now that we know how to characterize the image ofKatz , it becomes time to investigate how to compute this ambient vector space. We want to avoid the problems related to the lifting of elements of S p (Γ 1 (N), ε, F) ′ Katz to characteristic zero with a given character (see [22, §1] and [22, Prop. 1.10], they have to do with what is called Carayol's Lemma). So we describeKatz of characteristic zero forms of weight p with no prescribed character.Proposition 4.6 Let p be a prime number, and N ≥ 1 an integer not divisible by p. Suppose that N = 1 or that p ≥ 5.

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“…Since θ π,S (µ p ) is a unit times the determinant of Frob p − 1 on ad 0 V π (1) Ip , we can combine this with Theorem 2.4 of [13] and Corollary 1.4.3 to conclude that Theorem 5.4.2 for all S follows from the special case S = ∅.…”

Section: If S ⊂ S and ∈ S Then There Exists A T S -Module Homomorphismmentioning

confidence: 99%

“…As in [15] and [12], the identification is made by matching local behavior of automorphic representations and Galois representations via the local Langlands correspondence (together with Fontaine's theory at the prime ). We work directly with cohomology of modular curves instead of using the Jacquet-Langlands correspondence, and we use the simplification of [46] provided by [13] and Fujiwara [23] independently.…”

Section: Introductionmentioning

confidence: 99%