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Let $\rho_\ell$ be a semisimple $\ell$-adic representation of a number field $K$ that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $\rho_\ell$ and completely characterize them, for example, if the algebraic monodromy of $\rho_\ell$ is connected. If $\rho_\ell$ is in addition $E$-rational for some number field $E$, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system. Using our results on weak abelian direct summands, when $K$ is totally real and $\rho_\ell$ is the three-dimensional $\ell$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $\pi$ of $\mathrm{GL}_3(\mathbb{A}_K)$ together with an isomorphism $\mathbb{C}\simeq \overline{\mathbb{Q}}_\ell$, we prove that $\rho_\ell$ is irreducible. We deduce in this case also some $\ell$-adic Hodge theoretic properties of $\rho_\ell$ if $\ell$ belongs to a Dirichlet density one set of primes., Comment: Title changed

We show that crystalline points are Zariski dense in the deformation space of a representation of the absolute Galois group of a $p$-adic field. We also show that these points are dense in the subspace parameterizing deformations with determinant equal to a fixed crystalline character. Our proof is purely local and works for all $p$-adic fields and all residual Galois representations., Comment: 12 pages, comments welcome!

Let $G$ be a split semi-simple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on modularity lifting theorems, together with a Smith theory argument to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb{Z}/\ell\mathbb{Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of representations called toral supercuspidal representations., Comment: Minor revisions

We show that framed deformation rings of mod $p$ representations of the absolute Galois group of a $p$-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application we prove density results of loci with prescribed $p$-adic Hodge theoretic properties., Comment: Revised version after a referee report. Lemma 3.21 has a better proof following a suggestion of the referee, and the appendix on Kummer-irreducible points has been rewritten

We continue our study of the Wiles defect of deformation rings $R$ and Hecke rings $T$ (at a newform $f$) acting on the cohomology of Shimura curves. The Wiles defect at an augmentation $\lambda_f:T \to O$ measures the failure of $R,T$ to be complete intersections locally at $\lambda_f$. In situations we study here the Taylor-Wiles-Kisin patching method gives an isomorphism $R=T$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of our earlier results that compute the Wiles defect at $\lambda_f: R=T \to O$, and explain in an a priori manner why the answer is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet--Takahashi that computes change of degrees of optimal parametrizations of elliptic curves by Shimura curves as we vary the Shimura curve., Comment: with an appendix by Najmuddin Fakhruddin and Chandrashekhar Khare

Let $\mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(\mathbb{F})$ for $\mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R'$ be complete noetherian local $W(\mathbb{F})$ -algebras with residue field $\mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $\mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $\mathcal{G}(R)$ with full residual image $\mathcal{G}(\mathbb{F})$ is a conjugate of a group $\mathcal{G}(A)$ for $A\subset R$ a closed subring that is local and has residue field $\mathbb{F}$ . (2) Every surjective homomorphism $\mathcal{G}(R)\to\mathcal{G}(R')$ is, up to conjugation, induced from a ring homomorphism $R\to R'$. (3) The identity map on $\mathcal{G}(R)$ represents the universal deformation of the representation of the profinite group $\mathcal{G}(R)$ given by the reduction map $\mathcal{G}(R)\to\mathcal{G}(\mathbb{F})$. This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $\mathcal{G}(R)$ with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general $\mathcal{G}$, and we study in the case at hand in great detail what conditions on $\mathbb{F}$ or on $p$ in relation to $\mathcal{G}$ are necessary for the above results to hold., Comment: 39 pages. Change in the numbering of the theorems and other latex environments

There are various reasons why a naive analog of the Maeda conjecture has to fail for Drinfeld cusp forms. Focussing on double cusp forms and using the link found by Teitelbaum between Drinfeld cusp forms and certain harmonic cochains, we observed a while ago that all obvious counterexamples disappear for certain Hecke-invariant subquotients of spaces of Drinfeld cusp forms of fixed weight, which can be defined naturally via representation theory. The present work extends Teitelbaum's isomorphism to an adelic setting and to arbitrary levels, it makes precise the impact of representation theory, it relates certain intertwining maps to hyperderivatives of Bosser-Pellarin, and it begins an investigation into dimension formulas for the subquotients mentioned above. We end with some numerical data for $A=\mathbb{F}_3[t]$ that displays a new obstruction to an analog of a Maeda conjecture by discovering a conjecturally infinite supply of $\mathbb{F}_3(t)$-rational eigenforms with combinatorially given (conjectural) Hecke eigenvalues at the prime $t$., Comment: 48 pages; to appear in Research in Number Theory

By combining theorems of Drinfeld and Strauch, we show that the monodromy representation on the special fibre of a Drinfeld modular variety, with level not divisible by the characteristic, is surjective. We illustrate this result in the special case of Drinfeld $\mathbb{F}_q[t]$-modules in level $t$, and apply this to show that the Kronecker factors of a Drinfeld modular polynomial in rank $r$ are irreducible., Comment: 12 pages

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level were isomorphic and complete intersections, provided the same was true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\overline \rho$. If $\overline \rho$ is scalar at some primes dividing the discriminant of the quaternion algebra, then Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight 2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda_f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda_f$ by computing the associated {\it Wiles defect} purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho_f$ which $f$ gives rise to by the Eichler--Shimura construction. One of the main tools used in the proof is Taylor--Wiles--Kisin patching., Comment: revision after comments of referees, to appear in Compositio Mathematica

Let $X$ be a normal geometrically connected variety over a finite field $\kappa$ of characteristic~$p$. Let $E$ be a number field. Using automorphic methods over global function fields, we derive properties of the geometric monodromy groups of arbitrary connected $E$-rational semisimple compatible systems $(\rho_\lambda)$ of $n$-dimensional representations of the arithmetic fundamental group $\pi_1(X)$, where $\lambda$ ranges over the finite places of $E$ not above $p$: Let $\Lambda_\lambda$ be any $\pi_1(X)$-stable lattice in $E_\lambda^n$ under $\rho_\lambda$. Then for almost all $\lambda$, the schematic closure of the geometric monodromy $\rho_\lambda(\pi_1(X_{\overline{\kappa}}))$ in $\mathrm{Aut}_{\mathcal{O}_\lambda}(\Lambda_\lambda)$ is a semisimple $\mathcal{O}_\lambda$-group scheme, and its special fiber agrees with the Nori envelope of the geometric monodromy of the mod-$\lambda$ reduction of $\rho_\lambda$. A comparable result under different hypotheses was recently proved by Cadoret, Hui and Tamagawa by other methods. We also provide natural criteria for the image of $\pi_1(X_{\overline{\kappa}})$ under $\prod_\lambda\rho_\lambda$ to have adelic open image in an appropriate sense., Comment: Accepted for publication in Transactions of the American Mathematical Society, 66 pages

In this article, we describe how to compute slopes of $p$-adic $\mathcal{L}$-invariants of arbitrary weight and level by means of the Greenberg-Stevens formula. Our method is based on work of Lauder and Vonk on computing the reverse characteristic series of the $U_p$ operator on overconvergent modular forms. Using higher derivatives of this characteristic series, we construct a polynomial whose zeros are precisely the $\mathcal{L}$-invariants appearing in the corresponding space of modular forms with fixed sign of the Atkin-Lehner involution at $p$. In addition, we describe how to compute this polynomial efficiently. In the final section, we give computational evidence for relations between slopes of $\mathcal{L}$-invariants for small primes., Comment: 17 pages, 8 tables, improved exposition, including more details

We consider questions in Galois cohomology which arise by considering mod $p$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich primes, i.e. prime $p$ such that $2^{p-1}$ is 1 mod $p^2$. Our analog relates to asking if in a compatible system of Galois representations, for almost all primes $p$, the residual mod $p$ representation arising from it has unobstructed deformation theory. This analog leads in particular to formulating a mod $p$ analog for almost all primes $p$ of the classical Leopoldt conjecture, which has been considered previously by G. Gras. Leopoldt conjectured that for a number field $F$, and a prime $p$, the $p$-adic regulator $R_{F,p}$ is non-zero. The mod $p$ analog is that for a fixed number field $F$, for almost all primes $p$, the $p$-adic regulator $R_{F,p}$ is a unit at $p$., Comment: 31 pages, 5 tables

In this note we prove a variational open adelic image theorem for the Galois action on the cohomology of smooth proper $S$-schemes where $S$ is a smooth variety over a finitely generated field of positive characteristic. A central tool is a recent result of Cadoret, Hui and Tamagawa., Comment: 8 pages

Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\hat{G}$ is a split reductive group over $\mathbb{Z}$. Conjecturally, any $l$-adic $\hat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $\pi_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$) should be associated to an everywhere unramified automorphic representation of the group $G$. We show that for any homomorphism $\pi_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$ of Zariski dense image, there exists a finite Galois cover $Y \to X$ over which the associated local system becomes automorphic., Comment: Accepted manuscript. To appear in Acta Mathematica. With two appendices by Dennis Gaitsgory

Let X be a smooth projective curve of genus g over a finite field F_q of characteristic p. Consider primes l different from p. We formulate some questions related to a well known counting formula of Drinfeld. Drinfeld counts rank 2, irreducible l-adic sheaves on the base change X_n of X to F_{q^n} as n varies. We would like to count rank 2, irreducible mod l sheaves on X_n as n varies. Drinfeld's l-adic count gives an upper bound for the mod l count. We conjecture that Drinfeld's count is the correct asymptotic for the count of rank 2, irreducible mod l sheaves on X_n as n varies with (n,\ell)=1.

Building on our previous work "Cartier modules: finiteness results" we start in this manuscript an in depth study of the derived category of Cartier modules and the cohomological operations which are defined on them. After localizing at the sub-category of locally nilpotent objects we show that for a morphism essentially of finite type $f$ the operations $Rf_*$ and $f^!$ are defined for Cartier crystals. We show that, if $f$ is of finite type (but not necessarily proper) $Rf_*$ preserves coherent cohomology (up to nilpotence) and that $f^!$ has bounded cohomological dimension. In a sequel we will explain how Grothendieck-Serre Duality relates our theory of Cartier Crystals to the theory of $\tau$-crystals as developed by Pink and the second author., Comment: 53 pages

Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the ell-adic etale cohomology modules of X. We prove that this family of representations varying over ell is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ell become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting facts on the images and ramification of this family of representations., Comment: 27 pages

Let $\Lambda$ be a maximal $\mathbb{F}_q[T]$-order in a division quaternion algebra over $\mathbb{F}_q(T)$ which is split at the place $\infty$. The present article gives an algorithm to compute a fundamental domain for the action of the group of units $\Lambda^*$ on the Bruhat-Tits tree $\mathcal{T}$ associated to $PGL_2(\mathbb{F}_q((1/T)))$. This action is a function field analog of the action of a co-compact Fuchsian group on the upper half plane. The algorithm also yields an explicit presentation of the group $\Lambda^*$ in terms of generators and relations. Moreover we determine an upper bound for its running time using that $\Lambda^*\backslash\mathcal{T}$ is {\em almost} Ramanujan., Comment: 30 pages, 2 figures

On a locally Noetherian scheme X over a field of positive characteristic p we study the category of coherent O_X-modules M equipped with a p^{-e}-linear map, i.e. an additive map C: O_X \to O_X satisfying rC(m)=C(r^{p^e}m) for all m in M, r in O_X. The notion of nilpotence, meaning that some power of the map C is zero, is used to rigidify this category. The resulting quotient category, called Cartier crystals, satisfies some strong finiteness conditions. The main reasult in this paper states that, if the Frobenius morphism on X is a finite map, i.e. if X is F-finite, then all Cartier crystals have finite length. We further show how this and related results can be used to recover and generalize other finiteness results of Hartshorne-Speiser, Lyubeznik, Sharp, Enescu-Hochster, and Hochster about the structure of modules with a left action of the Frobenius. For example, we show that over any regular F-finite scheme X Lyubeznkik's F-finite modules have finite length., Comment: 44 pages

Abelian $t$-modules and the dual notion of $t$-motives were introduced by Anderson as a generalization of Drinfeld modules. For such Anderson defined and studied the important concept of uniformizability. It is an interesting question, and the main objective of the present article to see how uniformizability behaves in families. Since uniformizability is an analytic notion, we have to work with families over a rigid analytic base. We provide many basic results, and in fact a large part of this article concentrates on laying foundations for studying the above question. Building on these, we obtain a generalization of a uniformizability criterion of Anderson and, among other things, we establish that the locus of uniformizability is Berkovich open., Comment: 40 pages, v2: Section 7 rewritten; to appear in Trans. Amer. Math. Soc

As a sequel to our proof of the analog of Serre's conjecture for function fields in Part I of this work, we study in this paper the deformation rings of $n$-dimensional mod $\ell$ representations $\rho$ of the arithmetic fundamental group $\pi_1(X)$ where $X$ is a geometrically irreducible, smooth curve over a finite field $k$ of characteristic $p$ ($\neq \ell$). We are able to show in many cases that the resulting rings are finite flat over $\BZ_\ell$. The proof principally uses a lifting result of the authors in Part I of this two-part work, Taylor-Wiles systems and the result of Lafforgue. This implies a conjecture of A.J. ~de Jong for representations with coefficients in power series rings over finite fields of characteristic $\ell$, that have this mod $\ell$ representation as their reduction., Comment: This revised version is cleaner, although not substantially different. We check that our arguments work for \ell=2

We formulate for function fields an analog of Serre's conjecture on the modularity of 2-dimensional irreducible mod l representations of the absolute Galois group of Q: our analog is not restricted to 2-dimensional represntations. While the original conjecture of Serre is wide open at the moment, in this paper we prove the function field analog of Serre's conjecture only assuming largeness of image of the representation and some assumptions on the ramification of the representation (for example if the mod l representation is unramified and has full image, l is odd and the dimension and characteristic of the representation are coprime)., Comment: This revised version is cleaner although not substantially different. We check that our arguments work in many cases for \ell=2