Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
Theorie des nombres/Number Theory On invisible elements of the Tate-Shafarevich group
 

Summary: Th´eorie des nombres/Number Theory
On invisible elements of the Tate-Shafarevich group
Amod AGASH´E
Department of Mathematics, 940 Evans Hall, University of California, Berkeley, CA 94720, U.S.A.
Email: amod@math.berkeley.edu
Abstract. Mazur [8] 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
group H1
(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 [8] 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

  

Source: Agashe, Amod - Department of Mathematics, Florida State University

 

Collections: Mathematics