We compute the center of the ring of PD differential operators on a smooth variety over ℤ/pnℤ, confirming a conjecture of Kaledin (private communication). More generally, given an associative algebra A0 over ℤp and its flat deformation An over ℤ/pn+1ℤ, we prove that under a certain non-degeneracy condition, the center of An is isomorphic to the ring of length-(n+1) Witt vectors over the center of A0.

Given $f(x,y)\in \mathbb Z[x,y]$ with no common components with $x^a-y^b$ and $x^ay^b-1$, we prove that for $p$ sufficiently large, with $C(f)$ exceptions, the solutions $(x,y)\in \overline {\mathbb F}_p\times \overline {\mathbb F}_p$ of $f(x,y)=0$ satisfy $ {\rm ord}(x)+{\rm ord}(y)\gt c (\log p/\log \log p)^{1/2},$ where $c$ is a constant and ${\rm ord}(r)$ is the order of $r$ in the multiplicative group $\overline {\mathbb F}_p^*$. Moreover, for most $p\lt N$, $N$ being a large number, we prove that, with $C(f)$ exceptions, ${\rm ord}(x)+{\rm ord}(y)\gt p^{1/4+\epsilon (p)},$ where $\epsilon (p)$ is an arbitrary function tending to $0$ when $p$ goes to $\infty $.

We consider stacks of filtered $\varphi $-modules over rigid analytic spaces and adic spaces. We show that these modules parameterize $p$-adic Galois representations of the absolute Galois group of a $p$-adic field with varying coefficients over an open substack containing all classical points. Further, we study a period morphism (defined by Pappas and Rapoport) from a stack parameterizing integral data, and determine the image of this morphism.

From power series expansions of functions on curves over finite fields, one can obtain sequences with perfect or almost perfect linear complexity profile. It has been suggested by various authors to use such sequences as key streams for stream ciphers. In this work, we show how long parts of such sequences can be computed efficiently from short ones. Such sequences should therefore be considered to be cryptographically weak. Our attack leads in a natural way to a new measure of the complexity of sequences which we call expansion complexity.

For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of (the Galois type), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k, of which 34 can occur for k=ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.

We prove that the Brauer–Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global fields of positive characteristic $p$. More precisely, we show that the only obstructions come from étale covers of exponent $p$ or, alternatively, from flat covers coming from torsors under connected group schemes of exponent $p$.

The tempered fundamental group of a p-adic variety classifies analytic étale covers that become topological covers for Berkovich topology after pullback by some finite étale cover. This paper constructs cospecialization homomorphisms between the (p′) versions of the tempered fundamental group of the fibers of a smooth morphism with polystable reduction. We study the question for families of curves in another paper. To construct them, we will start by describing the pro-(p′) tempered fundamental group of a smooth and proper variety with polystable reduction in terms of the reduction endowed with its log structure, thus defining tempered fundamental groups for log polystable varieties.

Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of ℚ(ζ) into ℂ7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

We study the equation a2−2b6=cp and its specialization a2−2=cp, where p is a prime, using the modular method. In particular, we use a ℚ-curve defined over for which the solution (a,b,c)=(±1,±1,−1) gives rise to a CM-form. This allows us to apply the modular method to resolve the equation a2−2b6=cp for p in certain congruence classes. For the specialization a2−2=cp, we use the multi-Frey technique of Siksek to obtain further refined results.

We define and study a Lefschetz operator on the equivariant cohomology complex of the Drinfeld and Lubin–Tate towers. For ℓ-adic coefficients we show how this operator induces a geometric realization of the Langlands correspondence composed with the Zelevinski involution for elliptic representations. Combined with our previous study of the monodromy operator, this suggests a possible extension of Arthur’s philosophy for unitary representations occurring in the intersection cohomology of Shimura varieties to the possibly non-unitary representations occurring in the cohomology of Rapoport–Zink spaces. However, our motivation for studying the Lefschetz operator comes from the hope that its geometric nature will enable us to realize the mod-ℓ Langlands correspondence due to Vignéras. We discuss this problem and propose a conjecture.

We describe an algorithm to prove the Birch and Swinnerton-Dyer conjectural formula for any given elliptic curve defined over the rational numbers of analytic rank zero or one. With computer assistance we rigorously prove the formula for 16714 of the 16725 such curves of conductor less than 5000.

According to the André–Oort conjecture, an algebraic curve in Y (1)n that is not equal to a special subvariety contains only finitely many points which correspond to ann-tuple of elliptic curves with complex multiplication. Pink’s conjecture generalizes the André–Oort conjecture to the extent that if the curve is not contained in a special subvariety of positive codimension, then it is expected to meet the union of all special subvarieties of codimension two in only finitely many points. We prove this for a large class of curves in Y (1)n. When restricting to special subvarieties of codimension two that are not strongly special we obtain finiteness for all curves defined over . Finally, we formulate and prove a variant of the Mordell–Lang conjecture for subvarieties of Y (1)n.

Let X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

The main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p]) (which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p] and relate it with unramified cohomology.

We associate to certain filtrations of a graded linear series of a big line bundle a concave function on its Okounkov body, whose law with respect to the Lebesgue measure describes the asymptotic distribution of the jumps of the filtration. As a consequence, we obtain a Fujita-type approximation theorem in this general filtered setting. We then specialize these results to the filtrations by minima in the usual context of Arakelov geometry (and for more general adelically normed graded linear series), thereby obtaining in a simple way a natural construction of an arithmetic Okounkov body, the existence of the arithmetic volume as a limit and an arithmetic Fujita approximation theorem for adelically normed graded linear series. We also obtain an easy proof of the existence of the sectional capacity previously obtained by Lau, Rumely and Varley.

We prove that an algebraic subvariety of a Shimura variety is weakly special if and only if analytic components of its preimage in the symmetric space are algebraic. We also prove an analogous result in the case of abelian varieties.

Let E/ℚ be an elliptic curve and let D<0 be a sufficiently large fundamental discriminant. If contains Heegner points of discriminant D, those points generate a subgroup of rank at least |D|δ, where δ>0 is an absolute constant. This result is compatible with the Birch and Swinnerton-Dyer conjecture.

Following some remarks made by O'Grady and Oguiso, the potential density of rational points on the second punctual Hilbert scheme of certain K3 surfaces is proved.

We study the elliptic curve discrete logarithm problem over finite extension fields. We show that for any sequences of prime powers (qi)i∈ℕ and natural numbers (ni)i∈ℕ with ni⟶∞ and ni/log (qi)⟶0 for i⟶∞, the elliptic curve discrete logarithm problem restricted to curves over the fields 𝔽qnii can be solved in subexponential expected time (qnii)o(1). We also show that there exists a sequence of prime powers (qi)i∈ℕ such that the problem restricted to curves over 𝔽qi can be solved in an expected time of e𝒪(log (qi)2/3).

We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency.

Supplementary materials are available with this article.