15 research outputs found
Explicit towers of Drinfeld modular curves
We give explicit equations for the simplest towers of Drinfeld modular curves
over any finite field, and observe that they coincide with the asymptotically
optimal towers of curves constructed by Garcia and Stichtenoth.Comment: 10 pages. For mini-symposium on "curves over finite fields and codes"
at the 3rd European Congress in Barcelona 7/2000 Revised to correct minor
typographical and grammatical error
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