770 research outputs found
tau-recurrent sequences and modular forms
In this paper we deal with Drinfeld modular forms, defined and taking values
in complete fields of positive characteristic. Our aim is to study a sequence
of families of Drinfeld modular forms depending on a parameter t that produces,
for certain values of t, several kinds of Eisenstein series considered by
Gekeler. We obtain formulas involving these functions. To obtain our results,
we introduce and discuss tau-linear recurrent sequences and deformations of
vectorial modular forms in this setting.Comment: improved the proof of Theorem 3 (now, it is shorter
Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms
We introduce and study certain deformations of Drinfeld quasi-modular forms
by using rigid analytic trivialisations of corresponding Anderson's t-motives.
We show that a sub-algebra of these deformations has a natural graduation by
the group Z^2 x Z/(q-1)Z and an homogeneous automorphism, and we deduce from
this and other properties multiplicity estimates
Hyperdifferential properties of Drinfeld quasi-modular forms
This article is divided in two parts. In the first part we endow a certain
ring of ``Drinfeld quasi-modular forms'' for \GL_2(\FF_q[T]) (where is a
power of a prime) with a system of "divided derivatives" (or hyperderivations).
This ring contains Drinfeld modular forms as defined by Gekeler in \cite{Ge},
and the hyperdifferential ring obtained should be considered as a close
analogue in positive characteristic of famous Ramanujan's differential system
relating to the first derivatives of the classical Eisenstein series of weights
2, 4 and 6. In the second part of this article we prove that, when ,
if is a non-zero hyperdifferential prime ideal, then it contains the
Poincar\'e series of \cite{Ge}. This last result is the analogue
of a crucial property proved by Nesterenko \cite{Nes} in characteristic zero in
order to establish a multiplicity estimate
Universal Gauss-Thakur sums and L-series
In this paper we study the behavior of the function omega of Anderson-Thakur
evaluated at the elements of the algebraic closure of the finite field with q
elements F_q. Indeed, this function has quite a remarkable relation to explicit
class field theory for the field K=F_q(T). We will see that these values,
together with the values of its divided derivatives, generate the maximal
abelian extension of K which is tamely ramified at infinity. We will also see
that omega is, in a way that we will explain in detail, an universal
Gauss-Thakur sum. We will then use these results to show the existence of
functional relations for a class of L-series introduced by the second author.
Our results will be finally applied to obtain a new class of congruences for
Bernoulli-Carlitz fractions, and an analytic conjecture is stated, implying an
interesting behavior of such fractions modulo prime ideals of A=F_q[T].Comment: Corrected several typos and an error in the proof of Proposition 21
Section 3. Improved the general presentation of the pape
On certain generating functions in positive characteristic
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.Comment: 18 pages. Refereed versio
Drinfeld -quasi-modular forms
The aim of this article is twofold: first, improve the multiplicity estimate
obtained by the second author for Drinfeld quasi-modular forms; and then, study
the structure of certain algebras of "almost--quasi-modular forms
Arithmetic of positive characteristic L-series values in Tate algebras
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.Comment: final versio
An introduction to Mahler's method for transcendence and algebraic independence
International audienceHere we propose a survey on Mahler's theory for transcendence and algebraic independence focusing on certain applications to the arithmetic of periods of Anderson t-motives
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