Summary: Th´eorie des nombres/Number Theory
On invisible elements of the Tate-Shafarevich group
Department of Mathematics, 940 Evans Hall, University of California, Berkeley, CA 94720, U.S.A.
Abstract. Mazur  has introduced the concept of visible elements in the Tate-Shafarevich group
of optimal modular elliptic curves. We generalized the notion to arbitrary abelian sub-
varieties of abelian varieties and found, based on calculations that assume the Birch-
Swinnerton-Dyer conjecture, that there are elements of the Tate-Shafarevich group of
certain sub-abelian varieties of J0(p) and J1(p) that are not visible.
1. Introduction and definitions
Let J be an abelian variety and A be any abelian subvariety of J, both defined over Q. The
(Q, A) is isomorphic to the group of principal homogeneous spaces, or torsors, of A. An
A-torsor V is said to be visible in J if it is isomorphic over Q to a sub variety of J. An element of
the Tate-Shafarevich of A group is said to be visible (in J) if the corresponding torsor is visible.
We say that an element is invisible if it is not visible.
Mazur  introduced the concept of visible elements in the Tate-Shafarevich groups of optimal
modular elliptic curves. Adam Logan, based on Cremona's tables, studied instances of non-trivial
Tate-Shafarevich groups for modular elliptic curves of square-free conductor < 3000. The order