Search

In this paper, we improve the algorithms of Lauder-Wan \cite{LW} and Harvey\cite{Ha} to compute the zeta function of a system of $m$ polynomial equationsin $n$ variables over the finite field $\FF_q$ of $q$ elements, for $m$ large.The dependence on $m$ in the original algorithms was exponential in $m$. Ourmain result is a reduction of the exponential dependence on $m$ to a polynomialdependence on $m$. As an application, we speed up a doubly exponential timealgorithm from a software verification paper \cite{BJK} (on universalequivalence of programs over finite fields) to singly exponential time. One keynew ingredient is an effective version of the classical Kronecker theorem which(set-theoretically) reduces the number of defining equations for a "large"polynomial system over $\FF_q$ when $q$ is suitably large.

We describe an algorithm for computing, for all primes $p \leq X$, themod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometricmotive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellittrace formula with average polynomial time techniques of Harvey et al.

We deduce Katz's theorems for $(A,B)$-exponential sums over finite fieldsusing $\ell$-adic cohomology and a theorem of Denef-Loeser, removing thehypothesis that $A+B$ is relatively prime to the characteristic $p$. In somedegenerate cases, the Betti number estimate is improved using toricdecomposition and Adolphson-Sperber's bound for the degree of $L$-functions.Applying the facial decomposition theorem in \cite{W1}, we prove that theuniversal family of $(A,B)$-polynomials is generically ordinary for its$L$-function when $p$ is in certain arithmetic progression.

The ring of Witt vectors over a perfect valuation ring of characteristic p,often denoted A_inf, plays a pivotal role in p-adic Hodge theory; for instance,Bhatt, Morrow, and Scholze have recently reinterpreted and refined thecrystalline comparison isomorphism by relating it to a certain A_inf-valuedcohomology theory. We address some basic ring-theoretic questions about A_infmotivated by analogies with two-dimensional regular local rings. For example,we show that in most cases A_inf, which is manifestly not noetherian, is alsonot coherent. On the other hand, it does have the property that vector bundlesover the complement of the closed point in Spec A_inf do extend uniquely overthe puncture; moreover, a similar statement holds in Huber's category of adicspaces.

Suppose $p$ is a prime, $t$ is a positive integer, and$f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ withcoefficients of absolute value $

A CM-order is a reduced order equipped with an involution that mimics complex conjugation. The Witt-Picard group of such an order is a certain group of ideal classes that is closely related to the "minus part" of the class group. We present a deterministic polynomial-time algorithm for the following problem, which may be viewed as a special case of the principal ideal testing problem: given a CM-order, decide whether two given elements of its Witt - Picard group are equal. In order to prevent coefficient blow-up, the algorithm operates with lattices rather than with ideals. An important ingredient is a technique introduced by Gentry and Szydlo in a cryptographic context. Our application of it to lattices over CM-orders hinges upon a novel existence theorem for auxiliary ideals, which we deduce from a result of Konyagin and Pomerance in elementary number theory.

For commutative rings, we introduce the notion of a universal grading, which can be viewed as the “largest possible grading”. While not every commutative ring (or order) has a universal grading, we prove that every reduced order has a universal grading, and this grading is by a finite group. Examples of graded orders are provided by group rings of finite abelian groups over rings of integers in number fields. We also generalize known properties of nilpotents, idempotents, and roots of unity in such group rings to the case of graded orders; this has applications to cryptography. Lattices play an important role in this paper; a novel aspect is that our proofs use that the additive group of any reduced order can in a natural way be equipped with a lattice structure.

Let R be a perfect ring of characteristic p. We show that the group ofcontinuous R-linear automorphisms of the perfect power series ring over R isgenerated by the automorphisms of the ordinary power series ring together withFrobenius; this answers a question of Jared Weinstein.

In the late 1990's, R. Coleman and R. Greenberg (independently) asked for aglobal property characterizing those $p$-ordinary cuspidal eigenforms whoseassociated Galois representation becomes decomposable upon restriction to adecomposition group at $p$. It is expected that such $p$-ordinary eigenformsare precisely those with complex multiplication. In this paper, we studyColeman-Greenberg's question using Galois deformation theory. In particular,for $p$-ordinary eigenforms which are congruent to one with complexmultiplication, we prove that the conjectured answer follows from the$p$-indivisibility of a certain class group.

For all odd primes N up to 500000, we compute the action of the Heckeoperator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not thereduction mod 2 (with respect to a suitable basis) has 0 and/or 1 aseigenvalues. We then partially explain the results in terms of class fieldtheory and modular mod-2 Galois representations. As a byproduct, we obtain somenonexistence results on elliptic curves and modular forms with certain mod-2reductions, extending prior results of Setzer, Hadano, and Kida.

We give an interim report on some improvements and generalizations of theAbbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate amplehypersurface in a projectively normal toric variety over $\mathbb{F}_p$ inlinear time in $p$. These are illustrated with a number of examples includingK3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example isa non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisoron moduli space; this verifies that the coplanar divisor is not aNoether-Lefschetz divisor in the sense of Hassett.

To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinskyintroduce a system of differential equations, which are now called the GKZhypergeometric system. Its solutions are GKZ hypergeometric functions. We studythe $p$-adic counterpart of the GKZ hypergeometric system. The $p$-adic GKZhypergeometric complex is a twisted relative de Rham complex of over-convergentdifferential forms with logarithmic poles. It is an over-holonomic object inthe derived category of arithmetic $\mathcal D$-modules with Frobeniusstructures. Traces of Frobenius on fibers at Techm\"uller points of the GKZhypergeometric complex define the hypergeometric function over the finite fieldintroduced by Gelfand and Graev. Over the non-degenerate locus, the GKZhypergeometric complex defines an over-convergent $F$-isocrystal. It is thecrystalline companion of the $\ell$-adic GKZ hypergeometric sheaf that weconstructed before. Our method is a combination of Dwork's theory and thetheory of arithmetic $\mathcal D$-modules of Berthelot.

We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal to 2 and any non-negative real b be less than or equal to 2 / a, we show that there is a Cantor-like set with Hausdorff dimension equal to b such that, with respect to its uniform measure, almost all real numbers have irrationality exponent equal to a. We give an analogous result relating the irrationality exponent and the effective Hausdorff dimension of individual real numbers. We prove that there is a Cantor-like set such that, with respect to its uniform measure, almost all elements in the set have effective Hausdorff dimension equal to b and irrationality exponent equal to a. In each case, we obtain the desired set as a distinguished path in a tree of Cantor sets.

We show that for k k a perfect field of characteristic p p , there exist endomorphisms of the completed algebraic closure of k ( ( t ) ) k((t)) which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for p p a prime and C p \mathbb {C}_p a completed algebraic closure of Q p \mathbb {Q}_p , there exist closed points of the Fargues-Fontaine curve associated to C p \mathbb {C}_p whose residue fields are not (even abstractly) isomorphic to C p \mathbb {C}_p as topological fields.

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel–Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.

We study the problem of whether a commutative nonarchimedean Banach ringwhich is algebraically a field can be topologized by a multiplicative norm.This can fail in general, but it holds for uniform Banach rings under some mildextra conditions. Notably, any perfectoid ring whose underlying ring is a fieldis a perfectoid field.

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be an odd prime and$\iota: \overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ an embedding. Let$K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert classfield. For a class group character $\chi$ over $K$, let $\mathbb{Q}(\chi)$ bethe field generated by the image of $\chi$ and $\mathfrak{p}_{\chi}$ the primeof $\mathbb{Q}(\chi)$ above $p$ determined via $\iota_p$. Under mildhypotheses, we show that the number of class group characters $\chi$ such thatthe $\chi$-isotypic Tate--Shafarevich group of $E$ over $H_{K}$ is finite withtrivial $\mathfrak{p}_{\chi}$-part increases with the absolute value of thediscriminant of $K$.

We give a sharp divisibility bound, in terms of g, for the degree of thefield extension required to realize the endomorphisms of an abelian variety ofdimension g over an arbitrary number field; this refines a result ofSilverberg. This follows from a stronger result giving the same bound for theorder of the component group of the Sato-Tate group of the abelian variety,which had been proved for abelian surfaces by Fite-Kedlaya-Rotger-Sutherland.The proof uses Minkowski's reduction method, but with some care required in theextremal cases when p equals 2 or a Fermat prime.

Suppose $p$ is a prime, $t$ is a positive integer, and$f\!\in\!\mathbb{Z}[x]$ is a univariate polynomial of degree $d$ withcoefficients of absolute value $

Suppose $q$ is a prime power and $f\in\mathbb{F}_q[x]$ is a univariatepolynomial with exactly $t$ monomial terms and degree $

For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish this, based on the work of Gentry and Szydlo. The techniques involve algorithmic algebraic number theory, analytic number theory, commutative algebra, and lattice basis reduction.

We prove that the eigencurve associated to a definite quaternion algebra over$\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur andBuzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurveis a disjoint union of (countably) infinitely many connected components eachfinite and flat over the weight annuli, (b) the $U_p$-slopes of points on eachfixed connected component are proportional to the $p$-adic valuations of theparameter on weight space, and (c) the sequence of the slope ratios form aunion of finitely many arithmetic progressions with the same common difference.In particular, as a point moves towards the boundary on an irreducibleconnected component of the eigencurve, the slope converges to zero.

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of roots of unity can be solved in polynomial time. As an auxiliary result, we solve the discrete logarithm problem for certain unit groups in finite rings. Our techniques, which are taken from commutative algebra, may have further potential in the context of cryptology and computer algebra.

The purpose of this paper is to modify the notion of the Nevanlinna constant$\operatorname{Nev}(D)$, recently introduced by the first author, for aneffective Cartier divisor on a projective variety $X$. The modified notion iscalled the birational Nevanlinna constant and is denoted by$\operatorname{Nev}_{\text{bir}}(D)$. By computing$\operatorname{Nev}_{\text{bir}}(D)$ using the filtration constructed byAutissier in 2011, we establish a general result (see the General Theorem inthe Introduction), in both the arithmetic and complex cases, which extends togeneral divisors the 2008 results of Evertse and Ferretti and the 2009 resultsof the first author. The notion $\operatorname{Nev}_{\text{bir}}(D)$ isoriginally defined in terms of Weil functions for use in applications, and itis proved later in this paper that it can be defined in terms of localeffectivity of Cartier divisors after taking a proper birational lifting. Inthe last two sections, we use the notion $\operatorname{Nev}_{\text{bir}}(D)$to recover the proof of an example of Faltings from his 2002 Baker's Gardenarticle.

Let $H \subseteq G$ be an inclusion of $p$-adic Lie groups. When $H$ isnormal or even subnormal in $G$, the Hochschild-Serre spectral sequence impliesthat any continuous $G$-module whose $H$-cohomology vanishes in all degreesalso has vanishing $G$-cohomology. With an eye towards applications in $p$-adicHodge theory, we extend this to some cases where $H$ is not subnormal, assumingthat the $G$-action is analytic in the sense of Lazard.

We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite fieldof characteristic $p$, and consider the$\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower defined by $\bar f(x)$; thisis a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0=\mathbb{A}^1$, whose Galois group is canonically isomorphic to$\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of$\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as atopological group. We study the Newton slopes of zeta functions of this towerof curves. This reduces to the study of the Newton slopes of L-functionsassociated to characters of the Galois group of this tower. We prove that, whenthe conductor of the character is large enough, the Newton slopes of theL-function asymptotically form a finite union of arithmetic progressions. As acorollary, we prove the spectral halo property of the spectral varietyassociated to the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower. Thisextends the main result in [DWX] from rank one case $\ell=1$ to the higher rankcase $\ell\geq 1$.

In this paper we consider integers in base 10 like $abc$, where $a$, $b$, $c$ are digits of the integer, such that $abc^2 - (abc \cdot cba) \; = \; \pm n^2$, where $n$ is a positive integer, as well as equations $abc^2 - (abc \cdot cba) \; = \; \pm n^3$, and $abc^3 - (abc \cdot cba) \; = \; \pm n^2$ We consider asymptotic density of solutions. We also compare the results with ones with bases different from 10.

For a global function field K of positive characteristic p, we show thatArtin conjecture for L-functions of geometric p-adic Galois representations ofK is true in a non-trivial p-adic disk but is false in the full p-adic plane.In particular, we prove the non-rationality of the geometric unit rootL-functions.

© 2015 The Author(s) We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let (Formula presented.) be a reductive group over a number field (Formula presented.) which admits discrete series representations at infinity. Let (Formula presented.) be the associated (Formula presented.)-group and (Formula presented.) a continuous homomorphism which is irreducible and does not factor through (Formula presented.). The families under consideration consist of discrete automorphic representations of (Formula presented.) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato–Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321–331, 1987) and Serre (J Am Math Soc 10(1):75–102, 1997). As an application we study the distribution of the low-lying zeros of the associated family of (Formula presented.)-functions (Formula presented.), assuming from the principle of functoriality that these (Formula presented.)-functions are automorphic. We find that the distribution of the (Formula presented.)-level densities coincides with the distribution of the (Formula presented.)-level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz–Sarnak heuristics. We provide a criterion based on the Frobenius–Schur indicator to determine this symmetry type. If (Formula presented.) is not isomorphic to its dual (Formula presented.) then the symmetry type is unitary. Otherwise there is a bilinear form on (Formula presented.) which realizes the isomorphism between (Formula presented.) and (Formula presented.). If the bilinear form is symmetric (resp. alternating) then (Formula presented.) is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

This is a survey article on ordinary differential equations overnonarchimedean fields based on the author's lecture at the 2015 SimonsSymposium on nonarchimedean and tropical geometry. Topics include: theconvergence polygon associated to a differential equation (or a connection on acurve); links to the formal classification of differential equations(Turrittin-Levelt); index formulas for de Rham cohomology of connections;ramification of finite morphisms; relations with the Oort lifting problem onautomorphisms of curves. The appendices include some new technical results andan extensive thematic bibliography.

We compute the complete set of candidates for the zeta function of a K$3$surface over$\mathbb{F}_{2}$consistent with the Weil and Tate conjectures, as well as the complete set of zeta functions of smooth quartic surfaces over$\mathbb{F}_{2}$. These sets differ substantially, but we do identify natural subsets which coincide. This gives some numerical evidence towards a Honda–Tate theorem for transcendental zeta functions of K$3$surfaces; such a result would refine a recent theorem of Taelman, in which one must allow an uncontrolled base field extension.

We establish the group-theoretic classification of Sato-Tate groups ofself-dual motives of weight 3 with rational coefficients and Hodge numbersh^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motivesthat realize some of these Sato-Tate groups, and provide numerical evidencesupporting equidistribution. One of these families arises in the middlecohomology of certain Calabi-Yau threefolds appearing in the Dwork quinticpencil; for motives in this family, our evidence suggests that the Sato-Tategroup is always equal to the full unitary symplectic group USp(4).

We make explicit Serre's generalization of the Sato-Tate conjecture formotives, by expressing the construction in terms of fiber functors from themotivic category of absolute Hodge cycles into a suitable category of Hodgestructures of odd weight. This extends the case of abelian varietes, which wetreated in a previous paper. That description was used byFite--Kedlaya--Rotger--Sutherland to classify Sato-Tate groups of abeliansurfaces; the present description is used by Fite--Kedlaya--Sutherland to makea similar classification for certain motives of weight 3. We also giveconditions under which verification of the Sato-Tate conjecture reduces to theidentity connected component of the corresponding Sato-Tate group.

Based on the Lagarias-Odlyzko effectivization of the Chebotarev densitytheorem, Kumar Murty gave an effective version of the Sato-Tate conjecture foran elliptic curve conditional on analytic continuation and Riemann hypothesisfor the symmetric power $L$-functions. We use Murty's analysis to give asimilar conditional effectivization of the generalized Sato-Tate conjecture foran arbitrary motive. As an application, we give a conditional upper bound ofthe form $O((\log N)^2 (\log \log 2N)^2)$ for the smallest prime at which twogiven rational elliptic curves with conductor at most $N$ have Frobenius tracesof opposite sign.

We use the theory of congruences between modular forms to prove the existence of newforms with square-free level having a fixed number of prime factors such that the degree of their coefficient fields is arbitrarily large. We also prove a similar result for certain almost square-free levels.

We revisit Huber's theory of continuous valuations, which give rise to theadic spectra used in his theory of adic spaces. We instead consider valuationswhich have been reified, i.e., whose value groups have been forced to containthe real numbers. This yields reified adic spectra which provide a frameworkfor an analogue of Huber's theory compatible with Berkovich's construction ofnonarchimedean analytic spaces. As an example, we extend the theory ofperfectoid spaces to this setting.

We provide new proofs of two key results of p-adic Hodge theory: theFontaine-Wintenberger isomorphism between Galois groups in characteristic 0 andcharacteristic p, and the Cherbonnier-Colmez theorem on decompletion of (phi,Gamma)-modules. These proofs are derived from joint work with Liu on relativep-adic Hodge theory, and are closely related to Scholze's study of perfectoidalgebras and spaces.

We show that if a closed, oriented 3-manifold M is secretly homeomorphic to a lens space L(n,k), then we can compute n and k in randomized polynomial time (in the size of the triangulation of M) with a discrete logarithm oracle. Using Shor's algorithm, a quantum computer can thus identify lens spaces in quantum polynomial time. In addition, k can be computed in functional NP. A given value of k can be certified in randomized polynomial time, specifically in coRP. The idea of the algorithm is to calculate Reidemeister torsion over a prime field that has nth roots of unity.

Consider a vector bundle with connection on a p-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author's 2010 book and on subsequent improvements by Baldassarri and by Poineau and Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point.

We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field$\mathbb{F}_p(t)$, when $p$ is prime and $r\ge 2$ is an integer prime to $p$.When $q$ is a power of $p$ and $d$ is a positive integer, we compute the$L$-function of $J$ over $\mathbb{F}_q(t^{1/d})$ and show that the Birch andSwinnerton-Dyer conjecture holds for $J$ over $\mathbb{F}_q(t^{1/d})$. When $d$is divisible by $r$ and of the form $p^\nu +1$, and $K_d :=\mathbb{F}_p(\mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, showthat they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$is finite and a power of $p$, and show that the order of the Tate-Shafarevichgroup of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the "new"part of $J$ is isogenous over $\overline{\mathbb{F}_p(t)}$ to the square of asimple abelian variety of dimension $\phi(r)/2$ with endomorphism algebra$\mathbb{Z}[\mu_r]^+$. For a prime $\ell$ with $\ell \nmid pr$, we prove that$J[\ell](L)=\{0\}$ for any abelian extension $L$ of$\overline{\mathbb{F}}_p(t)$.

How many rational points are there on a random algebraic curve of large genus g over a given finite field Fq? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q+1+1/(q-1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g.

© 2015 The Author(s) We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let (Formula presented.) be a reductive group over a number field (Formula presented.) which admits discrete series representations at infinity. Let (Formula presented.) be the associated (Formula presented.)-group and (Formula presented.) a continuous homomorphism which is irreducible and does not factor through (Formula presented.). The families under consideration consist of discrete automorphic representations of (Formula presented.) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato–Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321–331, 1987) and Serre (J Am Math Soc 10(1):75–102, 1997). As an application we study the distribution of the low-lying zeros of the associated family of (Formula presented.)-functions (Formula presented.), assuming from the principle of functoriality that these (Formula presented.)-functions are automorphic. We find that the distribution of the (Formula presented.)-level densities coincides with the distribution of the (Formula presented.)-level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz–Sarnak heuristics. We provide a criterion based on the Frobenius–Schur indicator to determine this symmetry type. If (Formula presented.) is not isomorphic to its dual (Formula presented.) then the symmetry type is unitary. Otherwise there is a bilinear form on (Formula presented.) which realizes the isomorphism between (Formula presented.) and (Formula presented.). If the bilinear form is symmetric (resp. alternating) then (Formula presented.) is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

We derive new results about properties of the Witten zeta function associated with the group SU(3), and use them to prove an asymptotic formula for the number of n-dimensional representations of SU(3) counted up to equivalence. Our analysis also relates the Witten zeta function of SU(3) to a summation identity for Bernoulli numbers discovered in 2008 by Agoh and Dilcher. We give a new proof of that identity and show that it is a special case of a stronger identity involving the Eisenstein series.

Using maximal isotropic submodules in a quadratic module over Z_p, we provethe existence of a natural discrete probability distribution on the set ofisomorphism classes of short exact sequences of co-finite type Z_p-modules, andthen conjecture that as E varies over elliptic curves over a fixed global fieldk, the distribution of 0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --> 0 is thatone. We show that this single conjecture would explain many of the knowntheorems and conjectures on ranks, Selmer groups, and Shafarevich-Tate groupsof elliptic curves. We also prove the existence of a discrete probabilitydistribution of the set of isomorphism classes of finite abelian p-groupsequipped with a nondegenerate alternating pairing, defined in terms of thecokernel of a random alternating matrix over Z_p, and we prove that the twoprobability distributions are compatible with each other and with Delaunay'spredicted distribution for Sha. Finally, we prove new theorems on the fppfcohomology of elliptic curves in order to give further evidence for ourconjecture.

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p-1] is "almost" finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p-1] to a finite étale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p-adic Hodge theory. As an illustration, we construct (ϕ, Γ)-modules associated with Artin Motives over Q. The (ϕ, Γ)-modules we construct are defined over a base ring which seems well-suited to generalization to a more global setting; we plan to pursue such generalizations in later work.

We describe the analogue of the Sato-Tate conjecture for an abelian varietyover a number field; this predicts that the zeta functions of the reductionsover various finite fields, when properly normalized, have a limitingdistribution predicted by a certain group-theoretic construction related toHodge theory, Galois images, and endomorphisms. After making precise thedefinition of the "Sato-Tate group" appearing in this conjecture, we describethe classification of Sato-Tate groups of abelian surfaces due toFite-Kedlaya-Rotger-Sutherland. (These are notes from a three-lecture seriespresented at the NATO Advanced Study Institute "Arithmetic of HyperellipticCurves" held in Ohrid (Macedonia) August 25-September 5, 2014, and are expectedto appear in a proceedings volume.)

We make explicit a construction of Serre giving a definition of an algebraic Sato-Tate group associated to an abelian variety over a number field, which is conjecturally linked to the distribution of normalized L-factors as in the usual Sato-Tate conjecture for elliptic curves. The connected part of the algebraic Sato-Tate group is closely related to theMumford-Tate group, but the group of components carries additional arithmetic information. We then check that, in many cases where the Mumford-Tate group is completely determined by the endomorphisms of the abelian variety, the algebraic Sato-Tate group can also be described explicitly in terms of endomorphisms. In particular, we cover all abelian varieties (not necessarily absolutely simple) of dimension at most 3; this result figures prominently in the analysis of Sato-Tate groups for abelian surfaces given recently by Fité, Kedlaya, Rotger, and Sutherland.

We describe a new approach to relative p-adic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry in the style of Berkovich. In this paper, we give a thorough development of ?-modules over a relative Robba ring associated to a perfect Banach ring of characteristic p, including the relationship between these objects and étale Zp-local systems and Qp-local systems on the algebraic and analytic spaces associated to the base ring, and the relationship between étale cohomology and φ-cohomology. We also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite étale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic p and the finite étale algebras over a corresponding Banach Qp-algebra. This recovers the homeomorphism between the absolute Galois groups of Fp((π)) and Qp(μp∞)given by the field of norms construction of Fontaine and Wintenberger, as well as generalizations considered by Andreatta, Brinon, Faltings, Gabber, Ramero, Scholl, and most recently Scholze. Applications to the description of étale local systems on nonarchimedean analytic spaces over p-adic fields will be described in subsequent papers.