Related papers: Special Values without Semi-Simplicity Via K-Theor…

Let $k$ be a perfect field of characteristic $p>0$, $\mathcal{V}$ a complete discrete valuation ring with residue field $k$ and field of fractions $K$ of characteristic 0, and $S$ a separated $k$-scheme of finite type. When $S$ is smooth…

This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism…

For a prime number p and a number field k, we first study certain etale cohomology groups with coefficients associated to a p-adic Artin representation of its Galois group, where we twist the coefficients using a modified Tate twist with a…

We prove that algebraic de Rham cohomology as a functor defined on smooth $\mathbb{F}_p$-algebras is formally \'etale in a precise sense. This result shows that given de Rham cohomology, one automatically obtains the theory of crystalline…

We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and…

This is the first in a sequence of four papers, where we prove the arithmetic Siegel--Weil formula in co-rank $1$ for Kudla--Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even…

Let $p$ be a prime number and $\mathbb{C}_p$ the completion of algebraic closure of $\mathbb{Q}_p$. Let $K$ be an algebraic number field. We fix an embedding $\iota_p:\overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ and denote $K_p$ the…

A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees.…

Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ let $\m=(x_1,..., x_n)$ be the maximal ideal generated by the variables, let $^*E$ be the naturally graded injective hull of $R/\m$ and let $^*E(n)$ be…

In this paper we use automorph class theory formalism to construct a lifting of similitudes of quadratic Z-modules of arbitrary ternary nondegenerate quadratic forms to morphisms between certain subrings of associated Clifford algebras. The…

In this paper, we study the Koshliakov zeta function $\eta_p(s)$, whose theory appears to be more involved than that of its counterpart $\zeta_p(s)$, owing to the fact that its defining series is not of Dirichlet type. We derive formulas…

Classifying endotrivial kG-modules, i.e., elements of the Picard group of the stable module category for an arbitrary finite group G, has been a long-running quest. By deep work of Dade, Alperin, Carlson, Thevenaz, and others, it has been…

In this note, for each odd prime $p$, we show that the orders of the $KU$-local homotopy groups of the mod $p$ Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number…

Let $p:X \rightarrow S$ be a flat, proper and regular scheme over a strictly henselian discrete valuation ring. We prove that the singularity category of the special fiber with its natural two-periodic structure allows to recover the…

We investigate the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for the algebraic $K$-theory of twisted group rings of a group G with coefficients in a regular ring R or, more…

Recently, the author defined multiple Dedekind zeta values [5] associated to a number K field and a cone C. These objects are number theoretic analogues of multiple zeta values. In this paper we prove that every multiple Dedekind zeta value…

In this article we study in detail the category of noncommutative motives of separable algebras Sep(k) over a base field k. We start by constructing four different models of the full subcategory of commutative separable algebras CSep(k).…

We give a new formula for the special value at s=2 of the Hasse-Weil zeta function for smooth projective fourfolds under some assumptions (the Tate and Beilinson conjecture, finiteness of some cohomology groups, etc.). Our formula may be…

We prove homology stability for elementary and special linear groups over rings with many units improving known stability ranges. Our result implies stability for unstable Quillen K-groups and proves a conjecture of Bass. For commutative…

Let $R$ be the ring of integers in a finite extension $K$ of $\mathbb{Q}_p$, let $k$ be its residue field and let $\chi:\pi_1(X)\to R^{\times}=GL_{1}(R)$ be a "geometric" rank one representation of the arithmetic fundamental group of a…