Papers by Jacques Tilouine
Compositio Mathematica, 1988
arXiv (Cornell University), Aug 7, 2015
arXiv (Cornell University), Jan 19, 2021
We prove-under certain conditions (local-global compatibility and vanishing of modulo p cohomolog... more We prove-under certain conditions (local-global compatibility and vanishing of modulo p cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of GL(N) over a CM field; we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as the first Tate-Shafarevich group of this motive. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. By a result of Y. Cai generalizing previous results by Galatius-Venkatesh on the graded cohomology group of a locally symmetric space, our maps relate the (non-Eisenstein) localization of the graded cohomology group for a locally symmetric space to the exterior algebra of the Selmer group of the Tate dual of the adjoint representation. We generalize this to Hida families as well. 1 2 J. TILOUINE AND E. URBAN (Gal A) There exists a continuous Galois representation ρ A : G F) → GL N (A) unramified outside np and such that for any v prime to np, char(ρ m (Frob v)) = Hecke v (X) where the Hecke polynomial at v is given by Hecke v (X) = X N − T v,1 X N −1 +. .. + (−1) i T v,i Nv i(i−1) 2 X N −i +. .. + (−1) N T v,N Nv N(N−1) 2. This conjecture is proven in [Sch15] for A = T/I where I is a nilpotent ideal of exponent bounded in terms of N and [F : Q]. The exponent of I is bounded by 4 in [NT16]. By [CGH+19], the ideal can be taken to be 0 if p splits totally in F. We assume (M IN) The level n is squarefree and ρ π is n-minimal. Write the weight λ as λ = (λ τ,i) where τ : F → Q and i = 1,. .. , N. Let S p = S F,p be the set of places of F above p. By [Ca14], it is known that the local Galois representations ρ π | G Fw is crystalline at all places w ∈ S p (see Section 2). We will assume either of the two following local conditions at p : (F L) p is unramified in F , p > N and λ τ,1 − λ τ,N < p − N for any τ ∈ I F. Under this assumption, it follows from the crystallinity of ρ π at places of S p that for any w ∈ S p , ρ π | G Fw and ρ π | G Fw are Fontaine-Laffaille. (ORD π) F contains an imaginary quadratic field in which p splits, π is unramified and ordinary at all places v ∈ S p. Recall that (ORD π) implies that ρ π | G Fv takes values in a Borel of GL N (O). More precisely, there exists g v ∈ GL N (O) such that g v •ρ π | G Fv •g −1 v is upper triangular. Let B = T N be the Levi decomposition of the Borel B of upper triangular matrices with T the subgroup of diagonal matrices. We denote by χ v : G Fv → T (O) the homomorphism g v • ρ π | G Fv • g −1 v modulo N (O). We put χ v = diag(χ v,i) i=1,...,N and we write χ v,i for the reduction of χ v,i modulo ̟. With these notations, we consider the so-called pdistinguished condition (DIST) At each v ∈ S p , the characters χ v,i (i = 1,. .. , N) are mutually distinct.
In this short note we partially answer a question of Fukaya and Kato by constructing a $q$-expans... more In this short note we partially answer a question of Fukaya and Kato by constructing a $q$-expansion with coefficients in a non-commutative Iwasawa algebra whose constant term is a non-commutative p-adic zeta function.
p-Adic Aspects of Modular Forms, 2016
Geometric Aspects of Dwork Theory, 2004
ABSTRACT Cet article a pour origine deux exposés donnés par le second auteur à Varenna. Son conte... more ABSTRACT Cet article a pour origine deux exposés donnés par le second auteur à Varenna. Son contenu est cependant assez différent de celui des exposés. Il traite de certains aspects arithmétiques du lien entre les variétés et les formes modulaires de Hilbert. Plusieurs points classiques ne sont pas abordés faute de temps : opérateurs de Hecke, théorie de Serre–Tate. Par contre, cet article donne des détails sur les compactifications toroïdales de la variété abélienne de Hilbert–Blumenthal universelle (voir la partie 6), ainsi que plusieurs applications ; la plupart, si elles sont peut-être connues des experts, ne semblent pas figurer dans les publications sur ce thème ; en particulier, celles relatives à la théorie de Hodge, aux formes de poids demi-entier et de Hilbert–Jacobi nous paraissent nouvelles. Nous avons grandement bénéficié d&#39;un séminaire sur ce sujet que nous avons organisé au premier semestre 2001-02 à Paris 13. Nous tenons à en remercier tous les participants et en particulier G. Chenevier, Y. Henrio, A. Mokrane, S. Morel et S. Rozensztajn. Nous voulons également remercier H. Hida, qui nous a éclairés sur plusieurs points de ce travail. Une partie de cet article a été rédigée alors que le second auteur séjournait à l&#39;Institut de Mathématiques de l&#39;Université de Münster dans le cadre de la SFB 478 sur l&#39;invitation de C. Deninger. Il a apprécié les excellentes conditions de travail et l&#39;atmosphère cordiale qui y règnent. Nous avons divisé notre travail en deux articles ; ainsi ce texte est muni d&#39;un compagnon [7] qui donne les détails sur les compactifications (toroïdales et minimale) des variétés de Hilbert–Blumenthal en niveau 1 (c, n), en particulier aux pointes ramifiées. L&#39;organisation du présent article est la suivante : Table des matières 1 Variétés modulaires de Hilbert analytiques 554 2 Variétés abéliennes de Hilbert–Blumenthal et formes de Hilbert 558 3 Compactifications toroïdales analytiques. 565 4 Variétés et formes de Hilbert arithmétiques. 569 5 Compactifications arithmétiques de la variété de Hilbert 573 6 Compactifications toroïdales des variétés de Kuga–Sato 575 554 Mladen Dimitrov and Jacques Tilouine 7 Applications des compactifications toroïdales arithmétiques 583 8 Autres formes de Hilbert arithmétiques 592 9 Tour d&#39;Igusa et formes modulaires de Hilbert p-adiques 599 1 Variétés modulaires de Hilbert analytiques Références : [12], [34]. Notations. Soit F un corps de nombres totalement réel de degré d = d F , d&#39;anneau des en-tiers o, de différente d et de discriminant F = N F/Q (d). On note J F = Hom Q−alg. (F, C) l&#39;ensemble de ses plongements (réels). On abrégera N = N F/Q . On se donne un groupe algébrique D / Q , intermédiaire entre G m/ Q et Res F Q G m , connexe : G m → D D→ Res F Q G m .
Contemporary Mathematics, 1998
Publications mathématiques
A “main conjecture” typically relates an algebraic object, usually an Iwasawa module, to an analy... more A “main conjecture” typically relates an algebraic object, usually an Iwasawa module, to an analytic object, for example a p-adic L- function. For example, see the recent papers of B. Mazur and A. Wiles [Invent. Math. 76, 179-330 (1984; Zbl 0545.12005)], A. Wiles [Ann. Math., II. Ser. 123, 407-456 (1986; Zbl 0613.12013)], and K. Rubin [Invent. Math. 103, 25-68 (1991; Zbl 0737.11030)]. The present article gives a fairly general setting for this type of result. Let ρ ¯:Gal(ℚ ¯/ℚ)→GL 2 (κ) be a representation over a finite field κ associated to a p-adic p- ordinary Hecke eigenform f. Let Np r be the exact conductor of f, with (N,p)=1. Let R 0 be the ring of universal co-ordinary (i.e., the coinvariants of inertia at p are free of rank 1) deformations of ρ ¯. Let h ∞ ord be the ordinary Hecke algebra of level Np ∞ constructed by H. Hida [Ann. Sci. Ec. Norm. Super., IV. Ser. 19, 231-273 (1986; Zbl 0607.10022)] and let R be the localization of h ∞ ord at the maximal ideal whose residue fi...
Inventiones Mathematicae, 1994
Modular Forms and Fermat’s Last Theorem, 1997
The goal of this paper is to show the importance of the Gorenstein property for the Hecke algebra... more The goal of this paper is to show the importance of the Gorenstein property for the Hecke algebra and its relation with the local freeness of the cohomology of modular curves as a module over the Hecke algebra.
International Mathematics Research Notices
We prove under certain conditions (local-global compatibility and vanishing of modulo $p$ cohomol... more We prove under certain conditions (local-global compatibility and vanishing of modulo $p$ cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of $\operatorname{\textsf{GL}}(N)$ over a CM field; we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as the first Tate-Shafarevich group of this motive. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. By a result of Y. Cai generalizing previous results by Galatius-Venkatesh on the graded cohomology group of a locally symmetric space, our maps relate the (non-Eisenstein) localization of the graded cohomology group for a locally symmetric space to the exterior algebra of the S...
In Guise of Introduction - GL1 and Class-Field Theory Deformation Theory Deformations of Galois R... more In Guise of Introduction - GL1 and Class-Field Theory Deformation Theory Deformations of Galois Representations The Universal Ring Functorialities Obstructions Estimates on Dimension Nearly Ordinary Galois Representations and their Deformations The Krull Dimension Rmo Universal Ring The Hida-Iwasawa Algebra Classical Groups Universal Nearly Ordinary Hecke Algebras.
Infosys Science Foundation Series
In [GV18], the authors showed the importance of studying simplicial generalizations of Galois def... more In [GV18], the authors showed the importance of studying simplicial generalizations of Galois deformation functors. They established a precise link between the simplicial universal deformation ring R prorepresenting such a deformation problem (with local conditions) and a derived Hecke algebra. Here we focus on the algebraic part of their study which we complete in two directions. First, we introduce the notion of simplicial pseudo-characters and prove relations between the (derived) deformation functors of simplicial pseudo-characters and that of simplicial Galois representations. Secondly, we define the relative cotangent complex of a simplicial deformation functor and, in the ordinary case, we relate it to the relative complex of ordinary Galois cochains. Finally, we recall how the latter can be used to relate the fundamental group of R to the ordinary dual adjoint Selmer group, by a homomorphism already introduced in [GV18] and studied in greater generality in [TU20].
T h e s e notes g r e w out of a s e r i e s of lectures g i v e n at H o k k a i d o U n i v e r... more T h e s e notes g r e w out of a s e r i e s of lectures g i v e n at H o k k a i d o U n i v e r s i t y , S a p p o r o , in J u n e 1987. T h i s is my p l e a s u r e to t h a n k profs. Doi K. and H i d a H. to h a v e m a d e p o s s i b l e this s t a y in S a p p o r o , and also the J.S.P.S. to h a v e supported i t financially. 1 1 Now , our aim is to show the link between the Iwasawa module S i X_(K 1 and the Congruence Module ; this link implies in particular that Hi(Tl divides (twisted) fi(T) in A. This is the main theorem in these notes (th.4.3 below). Note that i t is conjectured that H. (T) coincides with the specialisation to the 1 anticyclotomic variable of the Katz-Yager p-adic L function of i two variables corresponding to the branch K (in the Yager's terminology, i t corresponds to the pair (i,-ill. Therefore, according to the Main Conjecture of Iwasawa-Coates (presented in C31), i t is believed that those three power series coincide, up to a unit in A. We hope to prove partial results in this direction in a subsequent paper. Let us add that the idea of comparing the values at T=O of H.(T) and (twisted) fi(T) was already present in C91 which is 1 the origin o f our work. Contents 0. Introduction 1. Hida's theory of the ordinary Hecke algebra and of the Congruence Module ord 2. Component of h of C.M. type attached to a Grossencharacter of an imaginary quadratic field 3. Gorenstein-ness of the local component associated to a Grossencharacter 4. The anticyclotomic-extension and P its Iwasawa theory 5. The A-divisible group attached to the local S i component and the A-linear map from (twisted) Xm(#) to the module of differentials 6. The properties of the map am 7. Surjectivity of am 8. An exact sequence for the Congruence Module and module of differentials 9. A link between the Fitting ideals of C o and C1
© Groupe de travail d’analyse ultramétrique (Secrétariat mathématique, Paris), 1986-1987, tous dr... more © Groupe de travail d’analyse ultramétrique (Secrétariat mathématique, Paris), 1986-1987, tous droits réservés. L’accès aux archives de la collection « Groupe de travail d’analyse ultramétrique » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
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Papers by Jacques Tilouine