Related papers: On the Formal Carlitz Module

The main objective of this article is to establish the $p$-adic Artin formalism for the algebraic $p$-adic $L$-functions attached to the adjoint representations of Coleman families of modular forms. In particular, we prove a factorization…

We introduce a class of rational functions $A:\,\mathbb C\mathbb P^1\rightarrow \mathbb C\mathbb P^1$ which can be considered as a natural extension of the class of Latt\`es maps and establish basic properties of functions from this class.

Let $k$ be a field of characteristic $p>0$ not necessarily perfect. Using Berthelot's theory of arithmetic $\mathcal{D}$-modules, we construct a $p$-adic formalism of Grothendieck's six operations for realizable $k$-schemes of finite type.

We survey our recent work on an extension of the theory of motivic integration, called arithmetic motivic integration. We developed this theory to understand how p-adic integrals of a very general type depend on p.

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating…

Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…

In this paper we find the genus field of finite abelian extensions of the global rational function field. We introduce the term conductor of constants for these extensions and determine it in terms of other invariants. We study the…

We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also…

Let $P$ be a monic prime of $\mathbb F_q[\theta]$, we define the $P$-adic $L$-series associated with Anderson $t$-modules and prove a $P$-adic class formula \`a la Taelman linking a $P$-adic regulator, the class module and a local factor at…

We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of…

We give an automata-theoretic description of the algebraic closure of the rational function field F_q(t) over a finite field, generalizing a result of Christol. The description takes place within the Hahn-Mal'cev-Neumann field of…

Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…

We give a sufficient condition for Kripke completeness of modal logics enriched with the transitive closure modality. More precisely, we show that if a logic admits what we call definable filtration (ADF), then such an expansion of the…

For a prime ideal $\mathfrak{P}$ of the ring of integers of a number field $K$, we give a general definition of $\mathfrak{P}$-adic continued fraction, which also includes classical definitions of continued fractions in the field of…

We develop a variant of Coleman and Perrin Riou's methods giving, for a de Rham $p$-adic Galois representation, a construction of $p$-adic $L$ functions from a compatible system of global elements. As a result, we construct analytic…

We investigate when the categories of all rational $A$-modules and of finite dimensional rational modules are closed under extensions inside the category of $C^*$-modules, where $C^*$ is the cofinite topological completion of $A$. We give a…

Let $K/k$ be a finite Galois extension of global function fields. Let $E$ be a Drinfeld module over $k$. We state and prove an equivariant refinement of Taelman's analogue of the analytic class number formula for $(E,K/k)$, and derive…

We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does…

Let $K$ be a field complete with respect to a nonarchimedean real-valued norm, and let $L/K$ be an algebraic extension. We show that there is a unique norm on $L$ extending the given norm on $K$, with an explicit description. As an…

We prove a kind of reflection principle for certain non-archimedean $L$-series in positive characteristic. We also prove the pseudo-cyclicity and pseudo-nullity of certain several variable generalizations of the class modules introduced by…