Upper bound for the height of S-integral points on elliptic curves

We establish new upper bounds for the height of the S-integral points of an elliptic curve. This bound is explicitly given in terms of the set S of places of the number field K involved, but also in terms of the degree of K, as well as the rank, the regulator and the height of a basis of the Mordell-Weil group of the curve. The proof uses the elliptic analogue of Baker's method, based on lower bounds for linear forms in elliptic logarithms.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1203.386

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 $q$ 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 $q\not=2,3$, if ${\cal P}$ is a non-zero hyperdifferential prime ideal, then it contains the Poincar\'e series $h=P_{q+1,1}$ 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

Drinfeld AA-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-$A$-quasi-modular forms

Hankel-type determinants and Drinfeld quasi-modular forms

International audienceIn this paper we introduce a class of determinants ''of Hankel type". We use them to compute certain remarkable families of Drinfeld quasi-modular forms

Arithmetic of characteristic p special L-values (with an appendix by V. Bosser)

Recently the second author has associated a finite \F_q[T]-module $H$ to the Carlitz module over a finite extension of \F_q(T). This module is an analogue of the ideal class group of a number field. In this paper we study the Galois module structure of this module $H$ for `cyclotomic' extensions of \F_q(T). We obtain function field analogues of some classical results on cyclotomic number fields, such as the $p$-adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module $H$ to Anderson's module of circular units, and give a negative answer to Anderson's Kummer-Vandiver-type conjecture. These results are based on a kind of equivariant class number formula which refines the second author's class number formula for the Carlitz module.Comment: (v2: several corrections in section 9; v3: minor corrections, improved exposition; v4: minor corrections; v5 minor corrections

Drinfeld singular moduli, hyperbolas, units

Let $q\geq2$ be a prime power and consider Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$. We prove that there are no points with coordinates being Drinfeld singular moduli, on a family of hyperbolas $XY=\gamma$, where $\gamma$ is a polynomial of small degree. This is an effective Andr\'e-Oort theorem for these curves. We also prove that there are at most finitely many Drinfeld singular moduli that are algebraic units, for every fixed $q\geq2$, and we give an effective bound on the discriminant of such singular moduli. We give in an appendix an inseparability criterion for values of some classical modular forms, generalising an argument used in the proof of our first result

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach

Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2, three references added ([Mas75], [MB90] and [Yu94]) and one reference updated [BS12]. Accepted in Bull. Brazil. Mat. So

A-expansions of Drinfeld modular forms

We introduce the notion of Drinfeld modular forms with $A$-expansions, where instead of the usual Fourier expansion in $t^n$ ($t$ being the uniformizer at `infinity'), parametrized by $n \in \mathbb{N}$, we look at expansions in $t_a$, parametrized by a \in A = \Fq [T]. We construct an infinite family of eigenforms with $A$-expansions. Drinfeld modular forms with $A$-expansions have many desirable properties that allow us to explicitly compute the Hecke action. The applications of our results include: (i) various congruences between Drinfeld eigenforms; (ii) the computation of the eigensystems of Drinfeld modular forms with $A$-expansions; (iii) examples of failure of multiplicity one result, as well as a restrictive multiplicity one result for Drinfeld modular forms with $A$-expansions; (iv) examples of eigenforms that can be represented as `non-trivial' products of eigenforms; (v) an extension of a result of B\"{o}ckle and Pink concerning the Hecke properties of the space of cuspidal modulo double-cuspidal forms for $\Gamma_1(T)$ to the groups \text{GL}_2 (\Fq [T]) and $\Gamma_0(T)$.Comment: This version does not use normalized Goss polynomials, which fixes various inaccuracies in the previous versio