Introduction To Elliptic Curves And Modular Forms

We prove that the maximal abelian extension tamely ramified at infinity of the rational function field over $\mathbb {F}_q$ is generated by the values at the points in the algebraic closure of $\mathbb {F}_q$ of the higher derivatives of the so-called Anderson and Thakur function $\omega .$ We deduce a similar property for the special values of the higher derivatives of a new kind of $L$-series introduced by the second author.

Universal Gauss-Thakur sums and L-series ∗†

The second author has recently introduced a new class of L-series in the arithmetic theory of function fields over finite fields. We show that the value at one of these L-series encode arithmetic informations of certain Drinfeld modules defined over Tate algebras. This enables us to generalize Anderson's log-algebraicity Theorem and Taelman's Herbrand-Ribet Theorem.

Arithmetic of positive characteristic L-series values in Tate algebras

https://hal.archives-ouvertes.fr/hal-00955673/document

Functional identities for L-series values in positive characteristic

Arithmetic of positive characteristic -series values in Tate algebras

We focus on the generating series for the rational special values of Pellarin’s $L$-series in $1 \le s \le 2(q-1)$ indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the Carlitz exponential, the Anderson–Thakur function, and the Anderson generating functions for the Carlitz module. We draw several corollaries, including explicit formulae and recursive relations for Pellarin’s $L$-series in the same range of $s$, and divisibility results on the numerators of the Bernoulli–Carlitz numbers by monic irreducibles of degrees one and two.

Explicit formulae for $L$-values in positive characteristic

Necessary and sufficient conditions are given for a negative integer to be a trivial zero of a new type of $L$-series recently discovered by F. Pellarin, and it is shown that any such trivial zero is simple. We determine the exact degree of the special polynomials associated to Pellarin's $L$-series. The theory of Carlitz polynomial approximations is developed further for both additive and $\mathbb{F}_q$-linear functions. Using Carlitz' theory we give generating series for the power sums occurring as the coefficients of the special polynomials associated to Pellarin's series, and a connection is made between the Wagner representation for $\chi_t$ and the value of Pellarin's $L$-series at 1.

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On Pellarin's $L$-series

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithm introduced by M. Papanikolas and involve the Anderson-Thakur function and the Carlitz exponential function. They collect certain functional identities in families for a new class of L-functions introduced by the first author. This paper also deals with specializations at roots of unity of these generating functions, producing a link with Gauss-Thakur sums.

On certain generating functions in positive characteristic

In this paper, we study various twisted A-harmonic sums, named following the seminal log-algebraicity papers of G. Anderson. These objects are partial sums of new types of special zeta values introduced by the first author and linked to certain rank one Drinfeld modules over Tate algebras in positive characteristic by Angles, Tavares Ribeiro and the first author. We prove, by using techniques introduced by the second author, that various infinite families of such sums may be interpolated by polynomials, and we deduce, among several other results, properties of analogues of finite zeta values but inside the framework of the Carlitz module. In the theory of finite multi-zeta values in characteristic zero, finite zeta values are all zero. In the Carlitzian setting, there exist non-vanishing finite zeta values, and we study some of their properties in the present paper.

Rudolph Perkins

Papers

Explicit formulae for \(L\)-values in positive characteristic

On Pellarin's $L$-series

On certain generating functions in positive characteristic

On twisted $A$-harmonic sums and Carlitz finite zeta values

On certain generating functions in positive characteristic