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), which 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 E is the dual of an optimal elliptic curve quotient of J_{0}(N), where N is the conductor of E. We call such an elliptic curve an optimal sub-elliptic curve of J_{0}(N). 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.

There is also a method of proving the existence of visible elements of � (E/Q) for optimal sub-elliptic curves E of J_{0}(N). Suppose F is another elliptic curve contained in J_{0}(N), such that E[p] = F [p] as subgroups of J_{0}(N) for some prime p. Then, subject to some additional hypotheses, one can show that there is a natural injection of F (Q)/pF (Q) into � (E/Q) and the elements in the image of this map are visible in J_{0}(N). This can be concluded from the discussion on p. 19 of the article; it is also explained in a preprint of the author and W. Stein. Note that this strategy does not assume the BSD conjecture.

The article reports the results of investigations (conducted by J. Cremona) following the lines of the last two paragraphs above. The data is quite striking: 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.

The article also discusses some other methods to represent elements of � (E/K), where K is a number field and E is an elliptic curve over K, and includes a proof due to J. de Jong showing that there is some abelian variety J over K containing E such that all elements of � (E/K) are 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 any number field and E is an elliptic curve over K, then there is an abelian surface J over K such that every element of order 3 of � (E/K) 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 author generalized the notion of visibility to abelian varieties and extended the strategy mentioned in the second paragraph of this review to certain quotients of J_{0}(N). In a preprint, the author and W. Stein have extended most of the results and calculations of the article being reviewed to abelian variety quotients of J_{0}(N) associated to modular forms by Shimura (the elliptic curves considered in the second and third paragraph of this review are a special case of this construction).