Visualizing elements in the Shafarevich-Tate group. (English) Zbl 0972.11049

Cremona, John E.; Mazur, Barry

Exp. Math. 9, No. 1, 13-28 (2000).

This paper is a nice contribution to producing data which allows one to “visualize” elements in the Shafarevich-Tate group of elliptic curves over number fields. The authors are specially interested in the case of elliptic curves defined over the rationals which are abelian subvarieties of the Jacobian variety \(J_0(N)\) of \(X_0(N)\), where \(N\) is the conductor of \(E\). If an elliptic curve \(E\) has a non-trivial Shafarevich-Tate group they ask the following question: “Are all curves of genus one representing elements of the Tate-Shafarevich group of \(E\) isomorphic (over the rationals) to curves contained in a (single) abelian surface \(A\) defined itself over the rationals, containing \(E\) as a subelliptic curve and contained in turn in the new part of \(J_0(N)\)?”
Although it might be expected that the answer to this question is yes, the authors found few examples where the answer is no, and it might remain so if the order of the Shafarevich-Tate group is large enough. However, for modular elliptic curves defined over the rationals of conductor at most 5500 and with no rational point of order 2, the authors found that in the majority of the cases the answer is yes. They present these data and wonder what is the conceptual reason behind such a behavior.
Given an elliptic curve \(E\) over a number field \(K\) two arithmetic objects are of main interest: its Mordell-Weil group \(E(K)\) and its Shafarevich-Tate group \(\text{ Ш}(E/K)\) which is defined as the set of isomorphism classes of pairs \((T,\imath)\), where \(T\) is a smooth projective curve defined over \(K\) of genus one having a \(K_v\)-rational point for each place \(v\) of \(K\) and \(\imath:E\to\text{Jac}(T)\) is a \(K\)-isomorphism, where \(\text{Jac}(T)\) denotes the Jacobian variety of \(T\).
In the literature more information on explicit examples are available about \(E(K)\) than about \(\text{ Ш}(E/K)\), since it is easier, in principle, to describe a rational point in terms of coordinates. If an element of \(\text{ Ш}(E/K)\) is annihilated by multiplication by a positive integer \(n\), then it can be obtained by push-out, starting from an appropriate \(1\)-cocycle on \(\text{Gal}(\overline{K}/K)\) with coefficients in the finite module \(E[n]\) of \(n\)-torsion points of \(E\). So this gives a “finitistic” way of representing the elements of \(\text{ Ш}(E/K)\). However, the authors’ wish and effort is to develop strategies which allow the “visualization” of the underlying curves more concretely.

Reviewer: Amílcar Pacheco (Rio de Janeiro)

MSC:

11G05	Elliptic curves over global fields
14G05	Rational points

Keywords:

Shafarevich-Tate group of elliptic curves

Software:

ecdata; mwrank

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