Visibility of Ideal Classes

Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich-Tate groups of elliptic curves. The analogue for an abelian number field K is capitulation of ideal classes of K in the minimal cyclotomic field containing K. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of nontrivial Shafarevich-Tate elements might be much more common for elliptic curves of positive rank than for curves of rank 0. Let E be an elliptic curve over Q of conductor N . Then there is a modular parametrizationX0(N ) → E and a corresponding map E → J0(N ) ofE into the jacobian of X0(N ). This induces a map of Shafarevich-Tate groups III(E) → III(J0(N )). Cremona and Mazur [6] study the question of when an element s of III(E) is in the kernel of this map. When this happens, there is a curve defined over Q and contained in J0(N ) that represents s. In other words, s is “visible” in J0(N ). Further work on this topic has been done by Agashe, Stein, and others [1], [2]. Let K/Q be an abelian extension of conductor n, so K ⊆ Q(ζn), where n is the conductor of K. This is the analogue of the modular parametrization above. The ideal class group is the analogue of the Shafarevich-Tate group (this will be made more precise in Section 1), so the analogue of the above question is to ask when ideal classes of K become principal in Q(ζn). Let L/K be an extension of number fields. An ideal class of K that becomes principal in L is said to capitulate. Many authors have discussed capitulation in various contexts. The hope of the present paper is to use results about capitulation for Q(ζn)/K to gain some insight into the situation for ShafarevichTate groups. For example, for imaginary quadratic fields K, the capitulation in Q(ζn)/K is mostly accounted for by the ambiguous classes, namely those produced by ∗Dipartimento di Matematica, 2a Università di Roma “Tor Vergata”, I-00133 Roma, Italy, [email protected] †Department of Mathematics, University of Maryland, College Park, MD 20742, [email protected]