Local Fourier transform and epsilon factors

Abbes, Ahmed; Saito, Takeshi

doi:10.1112/s0010437x09004631

Cited by 13 publications

(39 citation statements)

References 20 publications

(26 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Shortly after the first version of the paper was finished, Abbes and Saito [1] were able to calculate the local Fourier transformations for a class of monomial Galois representations of Artin-Schreier-Witt type. Their results are more general than ours, and they used a blowing-up technique and the ramification theory of Kato.…”

Section: The Kloosterman Sheafmentioning

confidence: 99%

See 1 more Smart Citation

Calculation of ℓ-adic local Fourier transformations

2010

manuscripta math.

View full text Add to dashboard Cite

We calculate the local Fourier transformations for a class of Q ℓ -sheaves. In particular, we verify a conjecture of Laumon and Malgrange ([18] 2.6.3). As an application, we calculate the local monodromy of ℓ-adic hypergeometric sheaves introduced by Katz ([15]). We also discuss the characteristic p analogue of the Turrittin-Levelt Theorem for D-modules.We thus haveWe have seen that W is an irreducible representation of J ′ . Note that J ′ is a normal subgroup of J since P ⊂ J ′ ⊂ J and J/P is abelian. Using this fact, the fact that W ⊗ K is an irreducible representation of J ′ , and Proposition 22 inis an irreducible representation of J. This finishes the proof of (i).(ii) Keep the notations in the proof of (i). We have shownirreducible and the action of g N on Hom P (W, V W ) has a single Jordan block, Hom P (W, V W ) must have dimension 1. So after twisting W by a one dimensional representation, we may assume V ∼ = Ind J J ′ (W ), where W is a representation of J ′ which is irreducible as a representation of P . Let ρ : J ′ → GL(W ) be the homomorphism defined by the representation W . ρ(P ) is a finite group. For any h ∈ J ′ , let ρ(h) act on this finite group by conjugation. There exists a positive integer r such that the conjugation by ρ(h r )induces the identity map on ρ(P ). For example, we can take r to be the order of the automorphism group of ρ(P ). Thus we havefor any σ ∈ P , that is, ρ(g rN ) is a P -equivariant isomorphism of W . But W is an irreducible representation of P . So ρ(g rN ) is a scalar multiplication by Schur's Lemma. Choose c ∈ Q ℓ such that * ℓ by χ(1) = c. We claim that the representation V ⊗ K χ −1 factors through a finite discrete quotient group of J. We haveIt suffices to show W ⊗ Res J J ′ K χ −1 factors through a finite discrete quotient group of J ′ . By our construction, g rN acts trivially on W ⊗ Res J J ′ K χ −1 . Moreover, there exists an open subgroup P ′ of P such that P ′ acts trivially on W ⊗ Res J J ′ K χ −1 . Since the subgroup of J ′ generated by P ′ and g rN has finite index, our assertion follows.(iii) Let V = W ⊗ U (n). We can find a filtrationof V by J-invariant subspaces such that F i /F i+1 ∼ = W for any 0 ≤ i ≤ n − 1. By the Jordan-Hölder theorem, any irreducible subquotient of V is isomorphic to W . Write V = V 1 ⊕ · · · ⊕ V m so that each V i is indecomposable. By (i), we must have

show abstract

Section: The Kloosterman Sheafmentioning

confidence: 99%

“…such that the morphism on henselizations (1,0) and its image in the residue field is 1 2 ∂ 2 g ∂x 2 (1, 0), which is a nonzero square in k. Since char k = 2, this element in the residue field has two distinct square roots. By the Hensel lemma [14] 18.5.13, g(x,z ′ ) (1,0) . Let δ(x − 1, z ′ ) be one of the square roots.…”

Section: The Canonical Diagrammentioning

confidence: 99%

Calculation of ℓ-adic local Fourier transformations

2010

manuscripta math.

View full text Add to dashboard Cite

show abstract

“…An explicit stationary phase formula was obtained in [30,11] (see also [14]) for D-modules, and in [12,1] for ℓ-adic sheaves.…”

Section: On One Hand (12) Impliesmentioning

confidence: 99%

A microlocal approach to the enhanced Fourier–Sato transform in dimension one

D'Agnolo

Kashiwara

2018

Advances in Mathematics

View full text Add to dashboard Cite

Let M be a holonomic algebraic D-module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to M at irregular points. The stationary phase formula states that the exponential factors of the Fourier transform of M are obtained by Legendre transform from the exponential factors of M. We give a microlocal proof of this fact, by translating it in terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.

show abstract

“…Recently, Fu [9] and, independently, Abbes and Saito [1] have given an explicit description of the different local Fourier transforms for a wide class of ℓ-adic sheaves. We will mainly be using the description given in [1], which works over an arbitrary (not necessarily algebraically closed) perfect base field, and therefore gives an explicit formula for Ai f as a representation of the decomposition group D ∞ .…”

Section: Computation Of the Degree Of The L-functionmentioning

confidence: 99%