Some remarks on the L-conjecture

Abstract

In this paper, we show several connections between the L-conjecture, proposed by Burgisser [3], and the boundedness theorem for the torsion of elliptic curves. Assuming the L-conjecture, a sharper bound is obtained for the number of torsions over extensions of k on an elliptic curve over a number field k, which improves Masser's result [6]. It is also shown that the Torsion Theorem for elliptic curves [10] follows directly from the WL-conjecture, which is a much weaker version of the L-conjecture. Since the WL-conjecture differs from the trivial lower bound only at the coefficient, this result provides an interesting example where increasing the coefficient in a trivial lower bound of straight-line complexity is difficult and important. © Springer-Verlag Berlin Heidelberg 2002.