On the Tate-shafarevich Group of a Number Field
نویسنده
چکیده
For an elliptic curve E defined over a fieldK, the Tate-Shafarevich group X(E/K) encodes important arithmetic and geometric information. An important conjecture of Tate and Shafarevich states X(E/K) is always finite. Supporting this conjecture is a cohomological analogy between Mordell-Weil groups of elliptic curves and unit groups of number fields. In this note, we follow the analogy to construct, for each number field K, a “Tate-Shafarevich group” X(K) and prove that X(K) is canonically isomorphic to the ideal class group Cl(K) (which is finite by a classical result of Dedekind). Prerequisites are some knowledge of algebraic number theory and elliptic curve theory.
منابع مشابه
787 The p - part of Tate - Shafarevich groups of elliptic curves can be arbitrarily large par REMKE KLOOSTERMAN
In this paper we show that for every prime p ~ 5 the dimension of the p-torsion in the Tate-Shafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K : Q] bounded by a constant depending only on p. From this we deduce that the dimension of the ptorsion in the Tate-Shafarevich group of A/Q can be arbitrarily large, where A is an abelia...
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