19 research outputs found
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
Hyperbolicity of varieties of log general type
These notes provide an overview of various notions of hyperbolicity for
varieties of log general type from the viewpoint of both arithmetic and
birational geometry. The main results are based on our paper entitled
"Hyperbolicity and uniformity of varieties of log general type." They are
expanded notes from a minicourse the authors gave as part of the Geometry and
arithmetic of orbifolds workshop at UQ\'AM.Comment: Addressed some inaccuracies and typos pointed out by the referees and
some readers. Slight change of title. To appear in CRM short courses
(Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces:
Hyperbolicity in Montr\'eal
On modular symbols and the cohomology of Hecke triangle surfaces
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the standard algorithms for the computation of holomorphic modular forms. Precise and explicit connections are established to the cohomology of Hecke triangle surfaces and group cohomology. In all the note a general commutative ring is used as coefficient ring in view of applications to the computation of modular forms over rings different from the complex numbers