We present an efficient algorithm to compute the Hasse–Witt matrix of a hyperelliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C/\mathbb{Q}$ modulo all primes of good reduction up to a given bound $N$, based on the average polynomial-time algorithm recently proposed by the first author. An implementation for hyperelliptic curves of genus 2 and 3 is more than an order of magnitude faster than alternative methods for $N = 2^{26}$.

In this paper we consider ordinary elliptic curves over global function fields of characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$. We present a method for performing a descent by using powers of the Frobenius and the Verschiebung. An examination of the local images of the descent maps together with a duality theorem yields information about the global Selmer groups. Explicit models for the homogeneous spaces representing the elements of the Selmer groups are given and used to construct independent points on the elliptic curve. As an application we use descent maps to prove an upper bound for the naive height of an $S$-integral point on $A$. To illustrate our methods, a detailed example is presented.

This paper introduces ‘hyper-and-elliptic-curve cryptography’, in which a single high-security group supports fast genus-2-hyperelliptic-curve formulas for variable-base-point single-scalar multiplication (for example, Diffie–Hellman shared-secret computation) and at the same time supports fast elliptic-curve formulas for fixed-base-point scalar multiplication (for example, key generation) and multi-scalar multiplication (for example, signature verification).

A conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields $K$ for the quartic del Pezzo surface $S$ of singularity type ${\boldsymbol{A}}_{3}$ with five lines given in ${\mathbb{P}}_{K}^{4}$ by the equations ${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

Weinvestigate the Hasse principle for complete intersections cut out by a quadric hypersurface and a cubic hypersurface defined over the rational numbers.

The $p$-cohomology of an algebraic variety in characteristic $p$ lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl, Illusie, Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc., and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for $p$-cohomology in characteristic $p$.

We give a separability criterion for the polynomials of the form

$$\begin{equation*} ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e). \end{equation*}$$

$$\begin{equation*} z^2 = ax^{2n + 2} + (bx^{2m} + c)(d x^{2k} + e) \end{equation*}$$

Soit $X=\mathcal{M}(\mathscr{A})$ un espace affinoïde et soient $f,g\in \mathscr{A}$ . Nous étudions les ensembles de composantes connexes des espaces définis par une inégalité de la forme $|f|\le r\, |g|$ , avec $r\ge 0$ . Nous montrons qu’il existe une partition finie de $\mathbf{R}_{+}$ en intervalles sur lesquels ces ensembles sont canoniquement en bijection et que les bornes de ces intervalles appartiennent à $\sqrt{\rho (\mathscr{A})}$ .

We study the automorphic Green function $\mathop{\rm gr}\nolimits _\Gamma $ on quotients of the hyperbolic plane by cofinite Fuchsian groups $\Gamma $, and the canonical Green function $\mathop{\rm gr}\nolimits ^{\rm can}_X$ on the standard compactification $X$ of such a quotient. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of $\Gamma $ on the hyperbolic plane to prove an “approximate spectral representation” for $\mathop{\rm gr}\nolimits _\Gamma $. Combining this with bounds on Maaß forms and Eisenstein series for $\Gamma $, we prove explicit bounds on $\mathop{\rm gr}\nolimits _\Gamma $. From these results on $\mathop{\rm gr}\nolimits _\Gamma $ and new explicit bounds on the canonical $(1,1)$-form of $X$, we deduce explicit bounds on $\mathop{\rm gr}\nolimits ^{\rm can}_X$.

We construct new indecomposable elements in the higher Chow group $CH^2(A,1)$ of a principally polarized Abelian surface over a $p$-adic local field, which generalize an element constructed by Collino [Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, J. Algebraic Geom. 6 (1997), 393–415]. These elements are constructed using a generalization, due to Birkenhake and Wilhelm [Humbert surfaces and the Kummer plane, Trans. Amer. Math. Soc. 355 (2003), 1819–1841 (electronic)], of a classical construction of Humbert. They can be used to prove a non-Archimedean analogue of the Hodge-${\mathcal{D}}$-conjecture – namely, the surjectivity of the boundary map in the localization sequence – in the case where the Abelian surface has good and ordinary reduction.

Given a prime $p\gt 2$ , an integer $h\geq 0$ , and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $ , we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$ , to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $ -coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$ -adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope  $\leq h$ .

We explain how the André–Oort conjecture for a general Shimura variety can be deduced from the hyperbolic Ax–Lindemann conjecture, a good lower bound for Galois orbits of special points and the definability, in the $o$ -minimal structure ${ \mathbb{R} }_{\mathrm{an} , \mathrm{exp} } $ , of the restriction to a fundamental set of the uniformizing map of a Shimura variety. These ingredients are known in some important cases. As a consequence a proof of the André–Oort conjecture for projective special subvarieties of ${ \mathcal{A} }_{6}^{N} $ for an arbitrary integer $N$ is given.

Let $X$ be a curve over a number field $K$ with genus $g\geq 2$, $\mathfrak{p}$ a prime of ${ \mathcal{O} }_{K} $ over an unramified rational prime $p\gt 2r$, $J$ the Jacobian of $X$, $r= \mathrm{rank} \hspace{0.167em} J(K)$, and $\mathscr{X}$ a regular proper model of $X$ at $\mathfrak{p}$. Suppose $r\lt g$. We prove that $\# X(K)\leq \# \mathscr{X}({ \mathbb{F} }_{\mathfrak{p}} )+ 2r$, extending the refined version of the Chabauty–Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Clifford’s theorem.

Let $k$ be a number field with algebraic closure $ \overline{k} $, and let $S$ be a finite set of primes of $k$ containing all the infinite ones. Let $E/ k$ be an elliptic curve, ${\mit{\Gamma} }_{0} $ be a finitely generated subgroup of $E( \overline{k} )$, and $\mit{\Gamma} \subseteq E( \overline{k} )$ the division group attached to ${\mit{\Gamma} }_{0} $. Fix an effective divisor $D$ of $E$ with support containing either: (i) at least two points whose difference is not torsion; or (ii) at least one point not in $\mit{\Gamma} $. We prove that the set of ‘integral division points on $E( \overline{k} )$’, i.e., the set of points of $\mit{\Gamma} $ which are $S$-integral on $E$ relative to $D, $ is finite. We also prove the ${ \mathbb{G} }_{\mathrm{m} } $-analogue of this theorem, thereby establishing the 1-dimensional case of a general conjecture we pose on integral division points on semi-abelian varieties.

In this paper, we introduce variants of formal nearby cycles for a locally noetherian formal scheme over a complete discrete valuation ring. If the formal scheme is locally algebraizable, then our nearby cycle gives a generalization of Berkovich’s formal nearby cycle. Our construction is entirely scheme theoretic, and does not require rigid geometry. Our theory is intended for applications to the local study of the cohomology of Rapoport–Zink spaces.

Local models are schemes, defined in terms of linear-algebraic moduli problems, which are used to model the étale-local structure of integral models of certain $p$ -adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at ${ \mathbb{Q} }_{p} $ is ramified, quasi-split $G{U}_{n} $ , Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper, we proved that their new local models are topologically flat when $n$ is odd. In the present paper, we prove topological flatness when $n$ is even. Along the way, we characterize the $\mu $ -admissible set for certain cocharacters $\mu $ in types $B$ and $D$ , and we show that for these cocharacters admissibility can be characterized in a vertexwise way, confirming a conjecture of Pappas and Rapoport.

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture. We prove this conjecture in the minuscule case.

Given an intersection of two quadrics $X\subset { \mathbb{P} }^{m- 1} $, with $m\geq 9$, the quantitative arithmetic of the set $X( \mathbb{Q} )$ is investigated under the assumption that the singular locus of $X$ consists of a pair of conjugate singular points defined over $ \mathbb{Q} (i)$.

We give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-étale site, which makes all constructions completely functorial.

Let $ \mathcal{X} $ be a curve over ${ \mathbb{F} }_{q} $ and let $N( \mathcal{X} )$, $g( \mathcal{X} )$ be its number of rational points and genus respectively. The Ihara constant $A(q)$ is defined by $A(q)= {\mathrm{lim~sup} }_{g( \mathcal{X} )\rightarrow \infty } N( \mathcal{X} )/ g( \mathcal{X} )$. In this paper, we employ a variant of Serre’s class field tower method to obtain an improvement of the best known lower bounds on $A(2)$ and $A(3)$.