Related papers: Special L-values of t-motives: a conjecture

The purpose of this article is to generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic $\mathbb{Z}_{p}$-extension. Let $g$ be a cuspidal Hilbert modular form of parallel weight…

We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out…

We prove that a certain conjecture holds true and the conjecture states a relationship between the zeta function of a finite category and the Euler characteristic of a finite category.

In analogy with values of the classical Euler Gamma-function at rational numbers and the Riemann zeta-function at positive integers, we consider Thakur's geometric Gamma-function evaluated at rational arguments and Carlitz zeta-values at…

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…

Using the method of multiple Dirichlet series, we develop L-functions ratios conjecture with one shift in both the numerator and denominator in certain ranges for quadratic families of Dirichlet and Hecke L-functions of primerelated moduli…

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of…

In this article we introduce and study a motivic category in the arithmetic of function fields, namely the category of motives over an algebraic closure $L$ of a finite field with coefficients in a global function field over this finite…

Let $f$ be a newform of weight $k\geq 2$, level $N$ with coefficients in a number field $K$, and $A$ the adjoint motive of the motive $M$ associated to $f$. We carefully discuss the construction of the realisations of $M$ and $A$, as well…

In this article, we establish an additive decomposition of the discrete zeta function (for $s \in \mathbb{N}^*$, $s > 1$), more precisely of the function $4(\zeta(s)-1)$, as a series whose general term is of the form $1/x_n(s) + 1/y_n(s) +…

In this paper, we explain several conjectures about how a product of two Carlitz-Goss zeta values can be expressed as a F_p-linear combination of Thakur's multizeta values, generalizing the q=2 case dealt by D. Thakur in Relations between…

Let $\mathcal{O}_K$ be the ring of integers in an algebraic number field $K$ and let $S:=\operatorname{Spec}(\mathcal{O}_K)$. Let $T_0,\ldots,T_n$ be regular schemes of finite type over $S$ and let $X$ be a scheme of finite type over $T_n$…

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of $L$-functions attached to these objects. We computationally investigate this principle for…

This article investigates congruences of $\mathfrak{p}$-adic representations arising from effective $A$-motives defined over a global function field $K$. We give a criterion for two congruent $\mathfrak{p}$-adic representations coming from…

Let $M$ be a T-motive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic…

In this note we describe weight functions that exhibit a transitional behavior between weak and strong correlation with the Liouville function. We also describe a binary problem which may be considered as an interpolation between Chowla's…

In the present paper, we study linear equations on tensor powers of the Carlitz module using the theory of Anderson dual $t$-motives and a detailed analysis of a specific Frobenius difference equation. As an application, we derive some…

We present a remarkably simple and surprisingly natural interpretation of the values of zeta functions at negative integers and zero. Namely we are able to relate these values to areas related to partial sums of powers. We apply these…

The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function,…

Lichtenbaum conjectured the existence of a Weil-\'etale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler-Poincar\'e…