p-adic L-functions and Selmer varieties associated to elliptic curves with complex multiplication

The study of non-abelian fundamental groups renders it plausible that the principle of Birch and Swinnerton-Dyer, whereby non-vanishing of L-values, in some appropriate sense, accounts for the finiteness of integral points, can eventually be extended to hyperbolic curves. Here we will discuss the very simple case of a genus 1 hyperbolic curve X/Q obtained by removing the origin from an elliptic curve E defined over Q with complex multiplication by an imaginary quadratic field K. Denote by E a Weierstrass minimal model of E and by X the integral model of X obtained as the complement in E of the closure of the origin. Let S be a set of primes including the infinite place and those of bad-reduction for E . We wish to examine the theorem of Siegel, asserting the finiteness of X (ZS), the S-integral points of X , from the point of view of fundamental groups and Selmer varieties. In particular, we show how the finiteness of points can be proved using ‘the method of Coates and Wiles’ which, in essence, makes use of the non-vanishing of p-adic L-functions arising from the situation. That is to say, in studying the set E(ZS)(= E(Z) = E(Q)), Coates and Wiles showed the special case of the conjecture of Birch and Swinnerton-Dyer by deriving the finiteness of E(ZS) from the non-vanishing of L(E/Q, s) at s = 1. Of course the L-function can vanish at 1 in general, in which case E(ZS) is supposed to be infinite. But we know that X (ZS) is always finite. From the perspective of this paper, this is a consequence of the fact that appropriate p-adic L-functions have only finitely many zeros. More precisely, if we choose a prime p that splits as p = ππ̄ in K and let L and L̄ denote the p-adic L-functions associated to the π and π̄-power torsion points of E/K [7], we know that they have only finitely many zeros. And then, the non-vanishing of L-functions forces the vanishing of infinitely many Qp−Selmer groups for a family of Galois representations naturally associated to X . The motivic tool used to put this information together in the present approach is a natural quotient