William A. Stein We study visibility of Shafarevich-Tate groups of modular abelian varieties in Jacobians of modular curves of higher level. We prove a theorem about the existence of visible elements at a specific higher level under hypotheses that can be verified explicitely. We also provide a table of examples of visible subgroups at higher level and state conjectures inspired by our data.

Dimitar P. Jetchev
Department of Mathematics
University of California
Berkeley, CA 94720-3840
`[email protected]`
William A. Stein
Department of Mathematics
University of Washington
Seattle, WA 98195-4350
`[email protected]`

15pt

- Introduction

- Notation
- Visible Subgroups of Shafarevich-Tate Groups
- Equivariant Visibility

- Strong Visibility at Higher Level
- Strongly visible subgroups
- Some auxiliary lemmas
- Proof of Theorem 5.1.3
- A Variant of Theorem 5.1.3 with Simpler Hypothesis

- Computational Examples

- Conjecture, evidence and more computational data
- The conjecture
- Theoretical Evidence for the Conjectures
- Visibility of Kolyvagin cohomology classes
- Table of Strong Visibility at Higher Level

- Bibliography
- Bibliography
- About this document ...

William Stein 2006-06-21