Amod Agashe 024547 2479 Virginia St., Apt. A Berkeley, CA 94709 amod@math.berkeley.edu remona, John E., Mazur, Barry Visualizing elements in the Shafarevich-Tate group 1 758797 11G05 14G05

Suppose K is a number field and E is an elliptic curve over K. If A is an abelian variety over K, let � (A/K) denote the Shafarevich-Tate group of A, which consists of isomorphism classes of principal homogeneous spaces for A that are locally trivial everywhere. The article under review explores the notion of visibility of elements of � (E/K); this notion was introduced by B. Mazur in his lectures at the 1998 Arizona Winter School. He posed the question of how one can “visualize” the curves of genus 1 that represent elements of � (E/K). Suppose we are given an embedding over K of E into an abelian variety J. An element σ of � (E/K) is said to be visible in J if it is in the kernel of the natural homomorphism � (E/K) → � (J/K). Under certain conditions (see Remark 2 of §3 of the article), this definition is equivalent to the assertion that the curve of genus 1 that represents σ is isomorphic over K to a curve contained in the variety J, hence the terminology “visible”.

The primary case of interest in the article being reviewed is when K = Q and the elliptic curve E is an abelian subvariety of J_{0}(N), where N is the conductor of E. It is easy to see (p. 19 of the article) that the elements of � (E/Q) that are visible in J_{0}(N) are annihilated by multiplication by the modular degree of E. Thus by comparing the order of � (E/Q) (as predicted by the Birch and Swinnerton-Dyer (BSD) conjecture) with the modular degree of E, both of which can be computed (see [J. E. Cremona, Algorithms for modular elliptic curves, Second edition, Cambridge Univ. Press, Cambridge, 1997; MR 99e:11068]), one can try to detect whether there are elements of � (E/Q) that are not visible in J_{0}(N). Note that this strategy assumes the BSD conjecture.

The authors also mention two methods that can be used to prove the existence of non-trivial visible elements of � (E/Q) for elliptic curves E as in the last paragraph. Suppose F is another elliptic curve of conductor N that is an abelian subvariety of J_{0}(N) such that E[p] = F [p] as subgroups of J_{0}(N) for some prime p. In this situation, the first strategy to show visibility is outlined in the discussion on p. 19 of the article; the authors assume the BSD conjecture to use this strategy ^{1}. Another strategy, mentioned in Remark 1 on p. 16 and 1 used on p. 22, uses the fact that if F has non-zero Mordell-Weil rank, then, under certain additional hypotheses, one can prove the existence of non-trivial visible elements of � (E/Q) ^{2}. Note that this strategy can be used without 2 assuming any conjectures.

The article reports the results of investigations conducted by J. Cremona following the lines of the last two paragraphs above. The data is quite interesting: amongst all optimal sub-elliptic curves E of J_{0}(N) with N ≤ 5500 and having no rational point of order 2, the elements of � (E/Q) are visible in J_{0}(N) in the vast majority of cases ^{3}. Since the article is computational in nature, some 3 of the details of the proofs of visibility have been omitted; the reader may refer to a forthcoming article of the reviewer and W. Stein, among other places, for some more details.

1This strategy does not assume F has non-zero MW rank, but focusses on the � side to prove visibility; however, it follows a posteriori that F has non-zero MW rank.

2This is as in your proof in our paper. Maybe the two strategies will be equivalent in the end, but since BSD is not proved, I am not sure at this point.

The authors also discuss some other methods to represent elements of � (E/K), where K is a number field and E is an elliptic curve over K, and include a proof due to J. de Jong of the fact that if σ is an element of � (E/K), then there exists an abelian variety J over K containing E as an abelian subvariety such that σ is visible in J. The article closes with some open questions.

Since the notion of visibility is quite new, we briefly mention some related work. In a preprint, B. Mazur shows that if K is a number field, E is an elliptic curve over K and σ is an element of � (E/K) of order 3 , then there is an abelian surface J over K such that σ is visible in J. Also, the original lectures of B. Mazur at the 1998 Arizona Winter School are available at

[http://swc.math.arizona.edu/~swcenter/aws98/Texts.html], and contain

some additional discussion. In [A. Agashé, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 369–374; MR 2000e:11083], the notion of visibility is generalized to abelian varieties and the strategy mentioned in the second paragraph of this review is extended to certain quotients of J_{0}(N). In a preprint, the reviewer and W. Stein have extended most of the results and calculations of the article being reviewed to abelian varieties associated to newforms; the elliptic curves considered in the second and third paragraph of this review are a special case of this construction.

3Will, you had taken objection to this, but I found that in the second table invisibility is really rare.