International Congress of Mathematicians

Elliptic Curves and Class Field Theory 1915. The arithmetic theoryFor every algebraic extension F of K, let Sel p (E/ F ) denote the p-power Selmergroup of F over F, the subgroup of ^(Gp, E[p°°]) that sits in an exact sequence0 —• E(F) ® Qp/Zp —• Selp(F /F ) —• HI(F /F )[p°°] —• 0where IH(E/ F )is the Shafarevich-Tate group of F over F. Also writeS p (E/p)= Hom(Selp(F /F ), Qp/Z p ) ® Q pfor the tensor product of Q p with the Pontrjagin dual of the Selmer group.The following theorem is proved using techniques which go back to [7]; see [2]and [12] Lemme 5, §2.2.Theorem 12 (Control Theorem). Suppose K C F C K œ .(i) The natural restriction map ^(FjEfy 00 ]) —t F 1 (K 0O ,F[p°°]) induces an isomorphismSp(E/j^ao ) ®A Ap ^y S p (E/p).(ii) There is a canonical isomorphism U(F) ^y Ylom Af ,(S p (E/p),Ap).Conjecture 13 (Two-variable main conjecture [8, 12]). The two-variable p-adic L-function L generates the ideal chax\(S p (E/ Kao )) of A.Restricting the two-variable main conjecture to the cyclotomic and anticyclotomiclines leads to the following "one-variable" conjectures originally formulatedin [9] and [12], respectively. Let F' denote the image of L in F cyc i ® Zp A ant ; as inConjecture 9, and S P (E/K^itors the A ant i-torsion submodule of S P (E/ K^; ti j)-Conjecture 14 (Cyclotomic and anticyclotomic main conjectures).(i) F cyc i generates the ideal char A