Constructing elements in Shafarevich–Tate groups of modular motives (2003)
| Venue: | in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303 |
| Citations: | 11 - 1 self |
BibTeX
@INPROCEEDINGS{Dummigan03constructingelements,
author = {Neil Dummigan and William Stein and Mark Watkins},
title = {Constructing elements in Shafarevich–Tate groups of modular motives},
booktitle = {in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303},
year = {2003},
pages = {91--118},
publisher = {University Press}
}
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Abstract
We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich–Tate groups of modular motives of low level and weight ≤ 12. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives rather than abelian varieties. 1
