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Consider the algebraic function $\Phi_{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Kronecker and Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi_{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties., Comment: 26 pages. Minor revisions. Comments welcome!

We show that the mod p points on a Shimura variety of abelian type with hyperspecial level, have the form predicted by the conjectures of Kottwitz and Langlands-Rapoport. Along the way we show that the isogeny class of a mod p point contains the reduction of a special point.

We correct an error in our paper ‘Connected components of affine Deligne–Lusztig varieties in mixed characteristic’.

We investigate an analogue of the Grothendieck $p$-curvature conjecture, where the vanishing of the $p$-curvature is replaced by the stronger condition, that the module with connection mod $p$ underlies a $\mathcal{D}_X$-module structure. We show that this weaker conjecture holds in various situations, for example if the underlying vector bundle is finite in the sense of Nori, or if the connection underlies a $\mathbb{Z}$-variation of Hodge structure. We also show isotriviality assuming a coprimality condition on certain mod $p$ Tannakian fundmental groups, which in particular resolves in the projective case a conjecture of Matzat-van der Put. v2: the well known 4.2 has been added to make the note self-contained., Comment: 9 pages

We construct smooth integral models of Shimura varieties of abelian type at primes where the level structure is hyperspecial.

We prove that, under some mild conditions, a two dimensional p-adic Galois representation which is residually modular and potentially Barsotti-Tate at p is modular. This provides a more conceptual way of establishing the Shimura-Taniyama-Weil conjecture, especially for elliptic curves which acquire good reduction over a wildly ramified extension of ℚ 3 . The main ingredient is a new technique for analyzing flat deformation rings. It involves resolving them by spaces which parametrize finite flat group scheme models of Galois representations. Introduction.

We prove a modularity lifting theorem for two dimensional, 2-adic, potentially Barsotti-Tate representations. This proves hypothesis (H) of Khare-Wintenberger, and completes the proof of Serre’s conjecture. The main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of ℚ2.

Introduction 641 1. Breuil-Mezard conjecture and the p-adic local Langlands 644 (1.1) The Breuil-Mezard conjecture 644 (1.2) Review of Colmez’s functor 647 (1.3) Hilbert-Samuel multiplicities 650 (1.4) Representations and pseudo-representations 653 (1.5) GL2(Qp)-representations mod p. 657 (1.6) Local patching and multiplicities 661 (1.7) From pseudo-representations to representations 669 2. Modularity via the Breuil-Mezard conjecture 675 (2.1) Quaternionic forms 675 (2.2) Global patching and multiplicities 677 (2.3) The Breuil-Mezard conjecture 686 Addendum to [Ki 1] 687 References 689

Le but de cette note est de fournir une introduction a la theorie des modeles entiers canoniques des varietes de Shimura, et de donner une esquisse de la preuve d'existence de tels modeles pour les varietes de Shimura de type Hodge, et plus generalement, de type abelien. Pour plus de details, le lecteur est renvoye a [Ki 3].

Let K/Qp be a finite extension and Gk = Gal(K/K) the Galois group of an alge braic closure K. Let F be a finite field of characteristic p, and Vf a finite dimensional F-vector space equipped with a continuous action of GkThe study of the defor mation theory of Galois representations was initiated by Mazur [Ma], who showed that if Vf has no non-trivial endomorphisms, then it admits a universal deformation ring Ryw. After the work of Wiles [Wi] and the conjectures of Fontaine-Mazur [FM] it became clear that for arithmetic applications it was important to understand cer tain quotients of Ry? corresponding to deformations satisfying certain conditions. For example Wiles uses deformations which arise from finite flat group schemes, and the corresponding quotient of Ryw was constructed by Ramakrishna [Ra]. Suppose that L/K is a finite extension and let a ^ b be integers. It seems to be a kind of folklore conjecture that there should be a quotient of Ry? whose points in finite extensions of W(F)[l/p] correspond to deformations of Vf which become semi-stable over L and have Hodge-Tate weights in the interval [a, b]. This is closely related to the final conjecture of [Fo 3]. Special cases of this, when Vf is 2-dimensional, are conjectured in the papers of Fontaine-Mazur [FM, p. 191], Breuil Conrad-Diamond-Taylor [BCDT, Conj. 1.1.1] and Breuil-M?zard [BM, 2.2.2.4]. The purpose of this paper is to prove such a result.

We generalize the construction of the eigencurve by Coleman-Mazur to the setting of totally real fields, and show that a finite slope Hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms. The key point is to show the overconvergence of the canonical subgroup and the complete continuity of the U p operator. We deduce this form some general considerations in rigid analytic geometry.

Let f be a newform on Γ1(N), and V f the 2-dimensional p-adic Galois representation attached to f. Let S be a finite set of primes containing the primes divisors of Np, and denote by adV f the adjoint of V f . Under some mild conditions on f, we show that H1 g (Gℚ,S,adV f )=0. Using this result, we show that the universal deformation space of the residual representation attached to f is smooth and 3-dimensional at the point corresponding to f. When f has finite slope, one can also use this result to give a deformation theoretic description of the “eigencurve” of Coleman-Mazur at the point corresponding to f.

We prove a conjecture of Fontaine and Mazur on modularity of representations of G ℚ which are potentially semi-stable at p, for representations coming from finite slope, overconvergent eigenforms. We also give an application of our technique to a question of Gouvea on overconvergence of certain modular forms. If ρ is a suitable representation of G ℚ,S on a two dimensional vector space over a finite field product of 𝔾 m and the S a finite set of primes – we construct a rigid analytic subspace of the universal deformation space of ρ, which is defined by a purely representation theoretic condition, and contains the eigencurve of Coleman-Mazur. Conjecturally, it is equal to this curve.

We extend the methods of Faltings and Tsuji, and prove that ifK is a field of characteristic 0 with a complete, discrete valuation, and a perfect residue field of characteristicp, then thep-adic etale cohomology of a finite typeK-scheme is potentially semi-stable. We prove a similar result for cohomology with compact support, and for cohomology with support in a closed subspace ofX. We establish a relationship between these cohomology groups, and the de Rham cohomology ofX.

Suppose a finite group acts as a group of automorphisms of a smooth complex algebraic variety which is defined over a number field. We show how, in certain circumstances, an equivariant comparison theorem in l -adic cohomology may be used to convert the computation of the graded character of the induced action on cohomology into questions about numbers of rational points of varieties over finite fields. This is carried through in three applications: first, for the symmetric group acting on the moduli space of n points on a genus zero curve; second, for a unitary reflection group acting on the complement of its reflecting hyperplanes; and third, for the symmetric group action on the space of configurations of points in any smooth variety which satisfies certain strong purity conditions.

We prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of the absolute Galois group $G_{K}$ , $K$ a finite extension of $\mathbb{Q}_{p}$ , for any $p>2$ (up to the question of determining precise values for the multiplicities that occur). In the case that $K/\mathbb{Q}_{p}$ is unramified, we also determine most of the multiplicities. We then apply these results to the weight part of Serre’s conjecture, proving a variety of results including the Buzzard–Diamond–Jarvis conjecture.

If $\mathcal S$ is a a scheme of finite type over a local field F, and $\mathcal X \longrightarrow \mathcal S$ is a proper smooth family, then to each rational point $s \in \mathcal S$ one can assign an extension of the absolute Galois group of F by the geometric fundamental group G of the fibre $\mathcal{X}_s$ . If F has uniformiser $\pi$ , and residue characteristic p, we show that the corresponding extension of the absolute Galois group of $\mathcal S$ by the maximal prime to p quotient of G is locally constant in the $\pi$ -adic topology of $\mathcal S$ . We give a similar result in the case of non-proper families, and families over $\pi$ -adic analytic spaces.

Soit F un corps ayant une valuation complete et discrete, et de caracteristique residuelle p. Si U est une variete sur F, notons π (p') 1,geom (u) le quotient maximal du groupe etale fondamental de U qui est premier a p. Nous considerons l'application ρ: Gal(F sep /F) → Out(π (p') 1,geom (u)) au groupe des automorphismes exterieurs, et nous montrons qu'elle applique le groupe de ramification sauvage sur un groupe fini. Nous montrons que sous certaines conditions p depend seulement de la reduction de U modulo une puissance de l'ideal maximal de F. Les preuves utilisent la theorie des schemas logarithmiques.

We give a new proof of a theorem of Siu and Trautmann on the vanishing of cohomology with support on complex or rigid analytic spaces. We indicate how our methods might be used to prove various comparison theorems in algebraic and analytic geometry.

Inside the universal deformation space of a local Galois representation one has the set of deformations which are potentially semi-stable of given p-adic Hodge and Galois type. It turns out these points cut out a closed subspace of the deformation space. A deep conjecture due to Breuil-Mezard predicts that part of the structure of this space can be described in terms of the local Langlands correspondence. For 2-dimensional representations the conjecture can be made precise. We explain some of the progress in this case, which reveals that the conjecture is intimately connected to the p-adic local Langlands correspondence, as well as to the Fontaine-Mazur conjecture.

Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck-Katz p-curvature conjecture for many locally symmetric varieties, including Hilbert-Blumenthal modular varieties and the moduli space of abelian varieties when g > 1.

Following ideas of Berger and Breuil, we give a new classification of crystalline representations. The objects involved may be viewed as local, characteristic 0 analogues of the “shtukas” introduced by Drinfeld. We apply our results to give a classification of p-divisible groups and finite flat group schemes, conjectured by Breuil, and to show that a crystalline representation with Hodge-Tate weights 0, 1 arises from a p-divisible group, a result conjectured by Fontaine.

If f is a polynomial over a field E, and σ a permutation of the roots of f, we show that σ is given by a polynomial p with coefficients in E (that is, p(a) = σ(a) for each root a of f) if and only if σ commutes with all the Galois automorphisms.

Let f: X -> Y be a separated morphism of schemes of finite type over a finite field of characteristic p, let Lambda be an artinian local Z_p-algebra with finite residue field, let m be the maximal ideal of Lambda, and let L^\bullet be a bounded constructible complex of sheaves of finite free Lambda-modules on the ��tale site of Y. We show that the ratio of L-functions L(X,L^\bullet)/L(Y,f_! L^\bullet), which is a priori an element of 1+T Lambda[[T]], in fact lies in 1+ m T Lambda [T]. This implies a conjecture of Katz predicting the location of the zeroes and poles of the L-function of a p-adic ��tale lisse sheaf on the closed unit disk in terms of ��tale cohomology with compact support., 26 pages, published version