Visualizing Elements in the Shafarevich - Tate Group

As is well known, E(K) and X(E/K) are somehow linked in the sense that it is often easier to come by information about the Selmer group of E over K which is built out of both E(K) and X(E/K) than it is to get information about either of these groups separately. It occurred to us that, although these two groups (Mordell-Weil and X) are partners, so to speak, in the arithmetic analysis of the elliptic curve E, there seems to be a slight discrepancy in their treatment in the existent mathematical literature, for this literature does a much more thorough job of helping one (at least in specific instances) to compute rational points, i.e., to exhibit elements of Mordell-Weil, than it does in helping one to find (in an explicit way) the curves of genus one which represent elements of X (especially if one is interested in elements of X of order > 2). This is perhaps understandable in that it is usually quite clear how to present a rational point (e.g., if E is given in Weierstrass form, giving just its x-coordinate determines the rational point up to sign) but it is less clear what manner one should choose to exhibit the curves of genus 1 representing the elements of X. Of course (for a fixed integer n) an element in X annihilated by multiplication by n can always be obtained by push-out, starting with an appropriate 1-cocycle on the Galois group GK = Gal(K/K) with coefficients in the finite Galois module E[n] ⊂ E, the kernel of multiplication by n in E, (the 1-cocycle being unramified outside the primes dividing n and the places of bad reduction for E) and so therefore, there is indeed, a “finitistic” way of representing these elements of X. Our aim here is rather to develop strategies that might enable us to “visualize” the underlying curves more concretely. There are, for example, two standard ways of representing elements of X, both of which we will briefly review below, and we will also suggest a third (where the curves of genus 1 in question are sought as subcurves of abelian varieties). It is this third mode of visualizing elements of the Shafarevich-Tate group together with data regarding it (See Tables 1 and 2 below) that is the principal theme of our article.