Visibility of Shafarevich–Tate Groups of Abelian Varieties

Authors: Agashe A.¹; Stein W.²

Source: Journal of Number Theory, Volume 97, Number 1, November 2002 , pp. 171-185(15)

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Abstract:

We investigate Mazur's notion of visibility of elements of Shafarevich–Tate groups of abelian varieties. We give a proof that every cohomology class is visible in a suitable abelian variety, discuss the visibility dimension, and describe a construction of visible elements of certain Shafarevich–Tate groups. This construction can be used to give some of the first evidence for the Birch and Swinnerton–Dyer conjecture for abelian varieties of large dimension. We then give examples of visible and invisible Shafarevich–Tate groups. © 2002 Elsevier Science (USA).

Keywords: visibility; Shafarevich–; Tate group; Birch and Swinnerton–; Dyer conjecture; modular abelian variety.

Language: English

Document Type: Research article

DOI: http://dx.doi.org/10.1006/jnth.2002.2810

Affiliations: 1: Department of Mathematics, University of Texas, Austin, Texas, 78712 2: Department of Mathematics, Harvard University, Cambridge, Massachusetts

Publication date: 2002-11-01