For homogeneous polynomials $G_1,\ldots ,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork’s theory. In this article, we will construct an explicit cochain map from the Dwork complex of $G_1,\ldots ,G_k$ to the Monsky–Washnitzer complex associated with some affine bundle over the complement $\mathbb {P}^n\setminus X_G$ of the common zero $X_G$ of $G_1,\ldots ,G_k$, which computes the rigid cohomology of $\mathbb {P}^n\setminus X_G$. We verify that this cochain map realizes the rigid cohomology of $\mathbb {P}^n\setminus X_G$ as a direct summand of the Dwork cohomology of $G_1,\ldots ,G_k$. We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz’s comparison results in [19] for projective hypersurface complements to arbitrary projective complements.

We develop an effective version of the Chabauty–Kim method which gives explicit upper bounds on the number of $S$-integral points on a hyperbolic curve in terms of dimensions of certain Bloch–Kato Selmer groups. Using this, we give a new ‘motivic’ proof that the number of solutions to the $S$-unit equation is bounded uniformly in terms of $\#S$.

We establish a Harder–Narasimhan formalism for modifications of $G$-bundles on the Fargues–Fontaine curve. The semi-stable stratum of the associated stratification of the ${B^+_{{\rm dR}}}$-Grassmannian coincides with the variant of the weakly admissible locus defined by Viehmann, and its classical points agree with those of the basic Newton stratum. When restricted to minuscule affine Schubert cells, the stratification corresponds to the Harder–Narasimhan stratification of Dat, Orlik and Rapoport. We also study basic geometric properties of the strata, and the relation to the Hodge–Newton decomposition.

We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point $\alpha$ where the small orbit by a polynomial $f$ is given by

\begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*}

$\mathcal {S}_\alpha$

$d$

$f$

\begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*}

$\alpha$

$|f^{\circ n}(\alpha )|_v \rightarrow \infty$

$v$

$d$

We provide a new formalism of de Rham–Witt complexes in the logarithmic setting. This construction generalises a result of Bhatt–Lurie–Mathew and agrees with those of Hyodo–Kato and Matsuue for log-smooth schemes of log-Cartier type. We then use our construction to study the monodromy action and slopes of Frobenius on log crystalline cohomology.

We construct examples of smooth proper rigid-analytic varieties admitting formal models with projective special fibers and violating Hodge symmetry for cohomology in degrees ${\geq }3$. This answers negatively the question raised by Hansen and Li.

Let K be a complete discrete valuation field of characteristic $0$ , with not necessarily perfect residue field of characteristic $p>0$ . We define a Faltings extension of $\mathcal {O}_K$ over $\mathbb {Z}_p$ , and we construct a Hodge-Tate filtration for abelian varieties over K by generalizing Fontaine’s construction [Fon82] where he treated the perfect residue field case.

We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second ℓ-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle–Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero.

In this note, we prove the logarithmic $p$-adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\unicode[STIX]{x1D70B},1)$ (in a certain sense) with respect to $\mathbb{F}_{p}$-local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.

In this paper, we study some aspects of the local fields generated by division values of formal Drinfeld modules.

Given systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer–Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer–Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.

Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J\,\to \,E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map ${{\Phi }_{J}}\,\to \,{{\Phi }_{E}}$ on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which ${{\Phi }_{J}}\,\to \,{{\Phi }_{E}}$ is surjective and discuss when these criteria hold for the Jacobians of modular curves.

For a smooth proper variety over a p-adic field, its Brauer group and abelian fundamental group are related to higher Chow groups by the Brauer–Manin pairing and class field theory. We generalize this relation to smooth (possibly nonproper) varieties, using motivic homology and a variant of Wiesend’s ideal class group. Several examples are discussed.

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 K be a local field of equal characteristic p>2, let XK/K be a smooth proper relative curve, and let ℱ be a rank 1 smooth l-adic sheaf (l≠p) on a dense open subset UK⊂XK. In this paper, under some assumptions on the wild ramification of ℱ, we prove a conductor formula that computes the Swan conductor of the etale cohomology of the vanishing cycles of ℱ. Our conductor formula is a generalization of the conductor formula of Bloch, but for non-constant coefficients.

Let $V$ be a complete discrete valuation ring with residue field $k$ of characteristic $p>0$ and fraction field $K$ of characteristic zero. Let ${\cal S}$ be a formal scheme over $V$ and let $\mathfrak{X}\to {\cal S}$ be a locally projective formal abelian scheme. In this paper we prove that, under suitable natural conditions on the Hasse–Witt matrix of $\mathfrak{X}\otimes_V V/\mathit{pV}$, the kernel of the Frobenius morphism on $\mathfrak{X}_k$ can be canonically lifted to a finite and flat subgroup scheme of $\mathfrak{X}$ over an admissible blow-up of ${\cal S}$, called the ‘canonical subgroup of $\mathfrak{X}$’. This is done by a careful study of torsors under group schemes of order $p$ over $\mathfrak{X}$. We also present a filtration on ${\rm H}^1(\mathfrak{X},\mu_p)$ in the spirit of the Hodge–Tate decomposition.

We investigate the action of the Weil group on the compactly supported $\ell$-adic étale cohomology groups of rigid spaces over a local field. We prove that the alternating sum of the traces of the action is an integer and is independent of $\ell$ when either the rigid space is smooth or the characteristic of the base field is equal to 0. We modify the argument of T. Saito to prove a result on $\ell$-independence for nearby cycle cohomology, which leads to our $\ell$-independence result for smooth rigid spaces. In the general case, we use the finiteness theorem of Huber, which requires the restriction on the characteristic of the base field.

Let K be the function field of a connected regular scheme S of dimension 1, and let $f : X\to Y$ be a finite cover of projective smooth and geometrically connected curves over K with $g(X)\ge 2$. Suppose that f can be extended to a finite cover ${\mathcal X} \to {\mathcal Y}$ of semi-stable models over S (it is known that this is always possible up to finite separable extension of K). Then there exists a unique minimal such cover. This gives a canonical way to extend $X\to Y$ to a finite cover of semi-stable models over S.

We compute the Chow group of 0-cycles on a rational surface defined over a finite extension K of the field $\mathbb{Q}_p$ of p-adic numbers (p a prime) when it is split by an unramified extension of K. We use intersection theory to define a specialisation map so we need to assume that the surface admits a regular proper integral model. A family of examples is worked out to illustrate the method.

This paper shows that a nontrivial uniform decay estimate for complete exponential sums modulo pr, determined by a polynomial map ${\bf P} = (P_1, P_2)$ follows from the existence of a ‘good P decomposition’ of ${\mathbb Z}_p^n$, a property that can be proved with geometric methods, and which was introduced in an earlier article by the present author.