c © 1997 Universitat de Barcelona Elliptic curves and special values of L-series

Let E/Q be an elliptic curve, and let L(E/Q, s) be its Hasse-Weil L-series. In this paper, working under certain simplifying assumptions, we sketch a proof of the following result: L(E/Q, 1) = 0 = ⇒ E(Q) finite. 0. Let E/Q be an elliptic curve, and let L(E/Q, s) be its Hasse-Weil L-series. Following the methods of our joint paper with Henri Darmon [2], and working under certain simplifying assumptions, we sketch a proof of the following result: L(E/Q, 1) = 0 = ⇒ E(Q) finite. Such a result was first proved for elliptic curves with complex multiplication by Coates and Wiles [3]. About ten years later, Kolyvagin [11], [12] found a proof for all modular elliptic curves. Kolyvagin’s method uses in a crucial way the family of Heegner points defined over the anticyclotomic extensions of an imaginary quadratic field, and the limit formula of Gross-Zagier [8]. Kato [10] has announced a new proof of Kolyvagin’s result, based on the construction of certain elements in the second K-group of modular function fields, defined over cyclotomic extensions of Q. In [2], Darmon and I present a proof of the above statement for modular elliptic curves having at least one prime of multiplicative reduction (we actually assume semistability, to simplify matters), which differs in essential ways from the othe