We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the Fontaine-Mazur conjectures, specifically, the modularity of certain potentially Barsotti-Tate Galois representations. The proof follows the template of Wiles, Taylor-Wiles, and Breuil-Conrad-Diamond-Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the integers of a local field. This makes crucial use of the filtered ϕ_1-modules of C. Breuil.
Let $R$ be a complete discrete valuation ring with perfect residue field $k$ of positive characteristic $p$ and field of fractions $K$ of characteristic 0. In this paper we consider a $K$-1-motive $M_K$ as in [Ra] and its associated Barsotti-Tate group. This last does not in general extend to a Barsotti-Tate group over $R$. However, with some assumptions, it extends to a logarithmic Barsotti-Tate group over $R$. This follows from [Ra] and Kato's results on finite logarithmic group schemes. Once chosen a uniformizing parameter $π$ of $R$, any logarithmic Barsotti-Tate group over $R$ is describe…
We study stacks of truncated Barsotti-Tate groups and the $G$-zips defined by Pink, Wedhorn & Ziegler. The latter occur naturally when studying truncated Barsotti-Tate groups of height 1 with additional structure. By studying objects over finite fields and their automorphisms we determine the zeta functions of these stacks. These zeta functions can be expressed in terms of the Weyl group of the reductive group $G$ and its action on the root system. The main ingredients are the classification of $G$-zips over algebraically closed fields and their automorphism groups by Pink, Wedhorn & Ziegler, …
We develop a descent criterion for $K$-linear abelian categories. Using recent advances in the Langlands correspondence due to Abe, we build a correspondence between certain rank 2 local systems and certain Barsotti-Tate groups on complete curves over a finite field. We conjecture that such Barsotti-Tate groups "come from" a family of fake elliptic curves. As an application of these ideas, we provide a criterion for being a Shimura curve over $\mathbb{F}_q$. Along the way, we formulate a conjecture on the field-of-coefficients of certain compatible systems.
Let $S$ be the spectrum of a complete discrete valuation ring with fraction field of characteristic 0 and perfect residue field of characteristic $p\geq 3$. Let $G$ be a truncated Barsotti-Tate group of level 1 over $S$. If ``$G$ is not too supersingular'', a condition that will be explicitly expressed in terms of the valuation of a certain determinant, we prove that we can canonically lift the kernel of the Frobenius endomorphism of its special fibre to a subgroup scheme of $G$, finite and flat over $S$. We call it the canonical subgroup of $G$.
Previous work of Kisin and Gee proves potential diagonalisability of two dimensional Barsotti-Tate representations of the Galois group of a finite extension $K/\mathbb{Q}_p$. In this paper we build upon their work by relaxing the Barsotti-Tate condition to one we call pseudo-Barsotti-Tate (which means that for certain embeddings $κ:K \rightarrow \overline{\mathbb{Q}}_p$ we allow the $κ$-Hodge-Tate weights to be contained in $[0,p]$ rather than $[0,1]$).
Let R be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti-Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K' of K. We prove that there exists a constant c which depends on the absolute ramification index e(K'/Q_p) and the height of G such that G has good reduction over K if and only if G[p^c] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's p-adic Ner…
Let $p$ be a prime. Let $R$ be a regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ of $k[[x_1,\ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $\text{Frac}(R)$ which extends to every local ring of $\text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group ove…
Let F be a unramified finite extension of Qp and rhobar be an irreducible mod p two-dimensional representation of the absolute Galois group of F. The aim of this article is the explicit computation of the Kisin variety parameterizing the Breuil-Kisin modules associated to certain families of potentially Barsotti-Tate deformations of rhobar. We prove that this variety is a finite union of products of P^1. Moreover, it appears as an explicit closed subvariety of P^1^[F:\Qp]. We define a stratification of the Kisin variety by locally closed subschemes and explain how the Kisin variety equipped wi…
Let $ \CO $ be the ring of integers of a non-Archimedean local field and $ π$ a fixed uniformizer of $ \CO $. We establish three main results. The first one states that the exterior powers of a $ π$-divisible $ \CO $-module scheme of dimension at most 1 over a field exist and commute with algebraic field extensions. The second one states that the exterior powers of a $p$-divisible group of dimension at most 1 over arbitrary base exist and commute with arbitrary base change. The third one states that when $ \CO $ has characteristic zero, then the exterior powers of $ π$-divisible groups with sc…
Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that $D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over $k$ of the same codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers $γ_D(i):=\dim(\pmb{Aut}(D[p^i]))$. We show that $n_D\le cd$, $(γ_D(i+1)-γ_D(i))_{i\in\Bbb N}$ is a decreasing sequence in $\Bbb N$, for $cd>0$ we have $γ_D(1)<γ_D(2)<...<γ_D(n_D)$, and for $m\in\{1,…
Let F be an unramified extension of Qp. The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformations rings with tame Galois type of irreducible two-dimensional representations of the absolute Galois group of F. We then apply our method in the particular case where F has degree 2 over Q_p and determine this way almost all these deformations rings. In this particular case, we observe a close relationship between the structure of these deformations rings and the geometry of the associated Kisin variety. As a corollary and still assuming that…
We explicitly construct and investigate a number of examples of $\mathbb{Z}/p^r$-equivariant formal group laws and complex-oriented spectra, including those coming from elliptic curves and $p$-divisible groups, as well as some other related examples.
Let $F$ be a finite unramified extension of $\mathbb Q\_p$ and $\barρ$ be an absolutely irreducible mod~$p$ $2$-dimensional representation of the absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We conjecture that the deformation space parametrizing the potentially Barsotti--Tate liftings of $\barρ$ having type $t$ depends only on the Kisin variety attached to the situation, enriched with its canonical embedding into $(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this conjecture by proving that the Kisin variety determines the cardinality of the …
The formal deformation space of a supersingular Barsotti-Tate group over of dimension two equipped with an action of Z_{p^2} is known to be isomorphic to the formal spectrum of a power series ring in two variables. If one chooses an extra Z_{p^2}-linear endomorphism of the p-divisible group then the locus in the formal deformation space formed by those deformations for which the extra endomorphism lifts is a closed formal subscheme of codimension two. We give a complete description of the irreducible components of this formal subscheme, compute the multiplicities of these components, and compu…
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d\in \mathbb{N}$ be such that $h=c+d>0$. Let $H$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. For $m\in\mathbb{N}^\ast$ let $H[p^m]=\ker([p^m]:H\rightarrow H)$. It is a finite commutative group scheme over $k$ of $p$ power order, called a Barsotti-Tate group of level $m$. We study a particular type of $p$-divisible groups $H_π$, where $π$ is a permutation on the set $\{1,2,\dots,h\}$. Let $(M,φ_π)$ be the Dieudonné module of $H_π$. Each $H_π$ is uniquely determined by $H_π[p]$ and by the fact th…
Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension of $W(k)[\frac{1}{p}]$ of ramification degree $e$. We consider an unramified base ring $R_0$ over $W(k)$ satisfying certain conditions, and let $R = R_0\otimes_{W(k)}\mathcal{O}_K$. Examples of such $R$ include $R = \mathcal{O}_K[\![s_1, \ldots, s_d]\!]$ and $R = \mathcal{O}_K\langle t_1^{\pm 1}, \ldots, t_d^{\pm 1}\rangle$. We show that the generalization of Raynaud's theorem on extending $p$-divisible groups holds over the base ring $R$ when $e < p-1$, whereas it does not hold when $R = \…
We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Galois representations. To achieve this, we extend results of Breuil and Mezard (classifying Galois lattices in semistable representations in terms of "strongly divisible modules") to the potentially crystalline case in Hodge-Tate weights (0,1). We then use these strongly divisible modules to compute the desired deformation rings. As a corollary, we obtain new results on the modularity of potentially Barsotti-Tate representations.
In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate $λ$-adic 2-dimensional Galois representation unramified at 3 and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p^2 \}) = \Q $ always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of …
Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove the following result related to the Tate Conjecture: for a smooth, projective, geometrically-connected variety defined over a global function field, the algebraic rank is less than or equal to the analytic rank. Also discussed is the analogous (open) question for number fields and an easy extension of Lafforgue's theorem to remove the "finite-order character" assumption. All results are likely "known to the experts", but don't appear to be wri…
Let p > 2 be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call "pseudo-Barsotti-Tate representations", over arbitrary finite extensions of the p-adics. As a consequence, we establish (under the usual Taylor-Wiles hypothesis) the weight part of Serre's conjecture for GL(2) over arbitrary totally real fields.
We prove a modularity lifting theorem for potentially Barostti-Tate representations over totally real fields, generalising recent results of Kisin. Unfortunately, there was an error in the original version of this paper, meaning that we can only obtain a slightly weaker result in the case where the representations are potentially ordinary; an erratum has been added explaining this error.
We construct some admissible Banach representations of GL_2(Q_p) that conjecturally should correspond to some 2-dimensional tamely ramified, potentially Barsotti-Tate representations of G_{Q_p} via the p-adic local Langlands correspondence. To achieve this, we generalize Breuil's work in the semi-stable case and work on the first covering of Drinfel'd upper half plane. Our main tool is an explicit semi-stable model of the first covering.
The moduli space of (1,3)-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common third étale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group Γ_1(6). By Tate's conjecture, this should imply that Y, …
We prove modularity for any irreducible crystalline $\ell$-adic odd 2-dimensional Galois representation (with finite ramification set) unramified at 3 verifying an "ordinarity at 3" easy to check condition, with Hodge-Tate weights $\{0, w \}$ such that $2 w < \ell$ (and $\ell > 3$) and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p \}) = \Q $. This result applies in particular to any motivic compatible family of odd two-dimensional Galois representations of $\Gal(\bar{\Q}/\Q)$ if the motive has rational coefficients, good reduction at 3, and the "ordinarity at 3" conditi…
Let $k$ be a field of characteristic $p>0$. Let $D_m$ be a $\BT_m$ over $k$ (i.e., an $m$-truncated Barsotti--Tate group over $k$). Let $S$ be a\break $k$-scheme and let $X$ be a $\BT_m$ over $S$. Let $S_{D_m}(X)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}(X)$ is pure in $S$ i.e., the immersion $S_{D_m}(X) \hookrightarrow S$ is affine. For $p\in\{2,3\}$, we prove purity if $D_m$ satisfies a certain property depending only on its $p$-torsion $D_m[p]$. For $p\ge 5$, we apply the developed te…
Let p > 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil-Mézard conjecture for 2-dimensional mod p representations of the absolute Galois group of Qp. We also state a conjectural generalisation to n-dimensional representations of the absolute Galois group of an arbitrary finite extension of Qp, and give a conditional proof of this conjecture, subject to a certain R = T-type theorem together with a strong version of the weight part of Serre's conjecture for rank n unitary groups. We deduce an unconditional result in the c…
In the studies on the modularity conjecture for rigid Calabi-Yau threefolds several examples with the unique level 8 cusp form were constructed. According to the Tate Conjecture correspondences inducing isomorphisms on the middle cohomologies should exist between these varieties. In the paper we construct several examples of such correspondences. In the constructions elliptic fibrations play a crucial role. In fact we show that all but three examples are in some sense built upon two modular curves from the Beauville list.
In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura varieties, in particular, to contexts in which the relevant automorphic forms contributed to cohomology in exactly two degrees. In this sequel, we extend our method to the general case in which Galois representations are expected to occur in cohomology, contingent on the (as yet unproven) existence of certain Galois representations with the expected properties. A…
Let $K$ be a finite extension of $\mathbb{Q}_p$, and choose a uniformizer $π\in K$, and put $K_\infty:=K(\sqrt[p^\infty]π)$. We introduce a new technique using restriction to $\Gal(\ol K/K_\infty)$ to study flat deformation rings. We show the existence of deformation rings for $\Gal(\ol K/K_\infty)$-representations ``of height $\leqslant h$'' for any positive integer $h$, and we use them to give a variant of Kisin's proof of connected component analysis of a certain flat deformation rings, which was used to prove Kisin's modularity lifting theorem for potentially Barsotti-Tate representations.…
We prove conjectures of Breuil and Breuil-Dembele (C. Breuil, "Sur un probleme de compatibilite local-global modulo p pour GL(2)"), including a generalisation from the principal series to the cuspidal case, subject to a mild global hypothesis that we make in order to apply certain R=T theorems. More precisely, we prove a multiplicity one result for the mod p cohomology of a Shimura curve at Iwahori level, and we show that certain apparently globally defined lattices in the cohomology of Shimura curves are determined by the corresponding local p-adic Galois representations. We also indicate a n…
Let $F$ be a totally real field in which $p$ is unramified. Let $\overline{r}: G_F \rightarrow \mathrm{GL}_2(\overline{\mathbb{F}}_p)$ be a modular Galois representation which satisfies the Taylor-Wiles hypotheses and is generic at a place $v$ above $p$. Let $\mathfrak{m}$ be the corresponding Hecke eigensystem. Then the $\mathfrak{m}$-torsion in the mod $p$ cohomology of Shimura curves with full congruence level at $v$ coincides with the $\mathrm{GL}_2(k_v)$-representation $D_0(\overline{r}|_{G_{F_v}})$ constructed by Breuil and Paškūnas. In particular, it depends only on the local representa…
We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on…
We say that a two dimensional p-adic Galois representation of a number field F is weight two if it is de Rham with Hodge-Tate weights 0 and -1 equally distributed at each place above p; for example, the Tate module of an elliptic curve has this property. The purpose of this paper is to establish a variety of results concerning odd weight two representations of totally real fields in as great a generality as we are able. Most of these results are improvements upon existing results. Three of our main results are as follows. (1) We prove a modularity lifting theorem for odd weight two representat…
A key ingredient in the Taylor-Wiles proof of Fermat last theorem is the classical Ihara's lemma which is used to rise the modularity property between some congruent galoisian representations. In their work on Sato-Tate, Clozel-Harris-Taylor proposed a generalization of the Ihara's lemma in higher dimension for some similitude groups. The main aim of this paper is then to prove some new instances of this generalized Ihara's lemma by considering some particular non pseudo Eisenstein maximal ideals of unramified Hecke algebras. As a consequence, we prove a level rising statement.
Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbf{A}_F)$.
We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur-Tate $L$-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we r…
Let F be a number field with adele ring A_F, and πan isobaric, algebraic automorphic representation of GL_4(A_F) of a fixed archimedean weight, which is quasi-regular, meaning that at every archimedean place v of F, the 4-dimensional representation σ_v of the Weil group W_{F_v} attached to π_v is multiplicity free. Suppose there is an associated 4-dimensional, Hodge-Tate p-adic representation ρof the absolute Galois group G_F, whose local L-factors agree with those of π(up to a shift) at almost all primes P of F. Then our first result is that the semisimplification of ρdoes not contain any irr…
