Related papers: Cohomological divisibility and point count divisib…

Continuing our work on group-theoretic generalizations of the prime Ax-Katz Theorem, we give a lower bound on the $p$-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\ldots,f_r)$ of $r$ maps…

Let $R$ be a discrete valuation ring of mixed characteristics $(0,p)$, with finite residue field $k$ and fraction field $K$, let $k'$ be a finite extension of $k$, and let $X$ be a regular, proper and flat $R$-scheme, with generic fibre…

Syntomic cohomology here defined yields a link between rigid cohomology and etale cohomology, viewing the last one as the fixed points under Frobenius of the former one. Let V be a complete discrete valuation ring, with perfect residue…

This is a survey of some recent developments concerning the p-adic cohomology of algebraic varieties over fields of positive characteristic and local fields of mixed characteristic, plus some related areas like p-adic Hodge theory.

It is known algebraically that any abelian group is a direct sum of a divisible group and a reduced group (See Theorem 21.3 of \cite{Fuchs:abelian-group}). In this paper, conditions to split off rational parts in homotopy types from a given…

The main purpose of this paper is to show that the mixed Hodge polynomial of the ``space of equations'' for smooth complete intersections of given multidegree in $\mathbb{C} P^n$ is divisible by the mixed Hodge polynomial of the group…

We prove that Frobenius eigenvalues of $\ell$-adic cohomology and $\ell$-adic intersection cohomology of rigid spaces over $p$-adic local fields are algebraic integers and we give bounds for their $p$-adic valuations. As an application, we…

We study a cohomology theory for rigid-analytic varieties over $\mathbb{C}_p$, without properness or smoothness assumptions, taking values in filtered quasi-coherent complexes over the Fargues-Fontaine curve, which compares to other…

This is meant to be a survey article for the Cubo Journal. We discuss the existence and number of rational points over a finite field, the Hodge type over the complex numbers, and the motivic conjectures which are controlling those…

We consider the interplay of point counts, singular cohomology, \'etale cohomology, eigenvalues of the Frobenius and the Grothendieck ring of varieties for two families of varieties: spaces of rational maps and moduli spaces of marked,…

We use the theory of Condensed Mathematics to build a condensed cohomology theory for the Weil group of a $p$-adic field. The cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This…

The isomonodromic tau function of the Fuchsian differential equations associated to Frobenius structures on Hurwitz spaces can be viewed as a section of a line bundle on the space of admissible covers. We study the asymptotic behavior of…

Let $L$ be a simply-connected simple connected algebraic group over a number field $F$, and $H$ be a semisimple absolutely maximal connected $F$-subgroup of $L$. Under a cohomological condition, we prove an asymptotic formula for the number…

We give a version of Ax-Katz's $p$-adic congruences and Moreno-Moreno's $p$-weight refinement that holds over any finite commutative ring of prime characteristic. We deduce this from a purely group-theoretic result that gives a lower bound…

We show that there is no simple congruence formula for the number of points of the mod p reduction of a regular model of a smooth proper variety defined over a local field with $\ell$-adic cohomology supported in codimension $\ge \kappa$.

We prove that the coherent cohomology of a proper morphism of noetherian schemes can be made arbitrarily p-divisible by passage to proper covers (for a fixed prime p). Under some extra conditions, we also show that p-torsion can be killed…

We obtain a rigidity phenomena of rational cohomology automorphisms of certain homogeneous spaces, in the presence of external cohomology classes arising from spaces with trivial cup product in rational cohomology algebra. We classify…

In the theory of configuration spaces, "splitting" usually refers to the phenomenon that the configuration spaces on a manifold and those on its punctured version are closely related cohomologically. We prove a splitting theorem that is…

Let G be a reductive algebraic group over a field of prime characteristic. One can associate to G (or subgroups thereof) its Lie algebra, its Frobenius kernels, and the finite Chevalley group of points over a finite field. The…

We study the problem of variation of Frobenius eigenvalues on the cohomology of families of local systems of algebraic curves over finite fields.