Related papers: The Frobenius and monodromy operators for curves a…

We define Frobenius and monodromy operators on the de Rham cohomology of $K$-dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction $Y$, over a complete discrete valuation ring $K$ of mixed…

We study congruences relating Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues of elliptic curves corresponding to their poles. We develop a $p$-adic cohomological framework that interprets these congruences via…

We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in…

For varieties over a finite field $\mathbb F_q$ with "many" automorphisms, we study the $\ell$-adic properties of the eigenvalues of the Frobenius operator on their cohomology. The main goal of this paper is to consider towers such as $y^2…

In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining…

Grothendieck defined a group that represents the local obstruction for an abelian variety to have semi-stable reduction. These groups were studied by Silverberg and Zarhin and more recently by the author in order to give a group theoretic…

Let $k$ be an algebraically closed field of characteristic $0$. For a log curve $X/k^{\times}$ over the standard log point, we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part…

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture,…

In the 1960s, Dwork developed a p-adic cohomology theory of de Rham type for varieties over finite fields, based on a trace formula for the action of a Frobenius operator on certain spaces of p-adic analytic functions. One can consider a…

We give an explicit description of the minimal regular model of bihyperelliptic curves with semistable reduction over a local field of odd residue characteristic. We do this using a generalisation of the cluster picture; a completely…

We build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo-Kato cohomology without recourse…

The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class - so-called modular Frobenius manifolds - lie at the fixed points of this symmetry. In this paper a classification of semi-simple…

We give a categorical description of all abelian varieties with commutative endomorphism ring over a finite field with $q=p^a$ elements in a fixed isogeny class in terms of pairs consisting of a fractional $\mathbb Z[\pi,q/\pi]$-ideal and a…

We study the homotopy fixed points under the Frobenius endomorphism on the stable $\mathbb A^1$-homotopy category of schemes in characteristic $p>0$ and prove a rigidity result for cellular objects in these categories after inverting $p$.…

We prove results that, for a certain class of non-compact Calabi-Yau threefolds, relate the Frobenius action on their $p$-adic cohomology to the Frobenius action on the $p$-adic cohomology of the corresponding curves. In the appendix, we…

Nekovar and Niziol have introduced in [arxiv:1309.7620] a version of syntomic cohomology valid for arbitrary varieties over p-adic fields. This uses a mapping cone construction similar to the rigid syntomic cohomology of the first author in…

The paper deals with a class of periods, Frobenius constants, which describe monodromy of Frobenius solutions of differential equations arising in algebraic geometry. We represent Frobenius constants related to families of elliptic curves…

The purpose of this paper is to prove a basic $p$-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure $C$ of a $p$-adic field: $p$-adic pro-\'etale cohomology, in a stable range, can be…

We reformulate the theory of p-adic iterated integrals on semistable curves using the unipotent log rigid fundamental group. This fundamental group carries Frobenius and monodromy operators whose basic properties are established. By…

The Ekedahl-Oort type is a combinatorial invariant of a principally polarized abelian variety $A$ defined over an algebraically closed field of characteristic $p > 0$. It characterizes the $p$-torsion group scheme of $A$ up to isomorphism.…