Summary: Th’eorie des nombres/Number Theory
On invisible elements of the TateShafarevich group
Amod AGASH ’
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 TateShafarevich 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
SwinnertonDyer conjecture, that there are elements of the TateShafarevich group of
certain subabelian 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
group H 1 (Q, A) is isomorphic to the group of principal homogeneous spaces, or torsors, of A. An
Atorsor V is said to be visible in J if it is isomorphic over Q to a sub variety of J . An element of
the TateShafarevich 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 TateShafarevich groups of optimal
modular elliptic curves. Adam Logan, based on Cremona's tables, studied instances of nontrivial
TateShafarevich groups for modular elliptic curves of squarefree conductor < 3000. The order