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On the notion of visibility of torsors Amod Agashe
 

Summary: On the notion of visibility of torsors
Amod Agashe
Abstract
Let J be an abelian variety and A be an abelian subvariety of J,
both defined over Q. Let x be an element of H1
(Q, A). Then there are
at least two definitions of x being visible in J: one asks that the torsor
corresponding to x is isomorphic over Q to a subvariety of J, the other
asks that x is in the kernel of the natural map H1
(Q, A) H1
(Q, J).
In this article, we clarify the relation between the two definitions.
Mathematics Subject Classification Numbers: 11G35, 14G25.
1 Introduction and definitions
As in the abstract, let J be an abelian variety and A be an abelian subvariety
of J, both defined over Q. The concept of visibility of torsors of A (i.e.,
elements of H1(Q, A)) was introduced by Mazur [Maz98] in the context
where J is the Jacobian of a modular curve and A is an elliptic curve. He
was interested in visualizing elements of the Shafarevich-Tate group of A,
which is a subgroup of H1(Q, A), as subvarieties in an ambient space (i.e.,

  

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

 

Collections: Mathematics