Name
Title | ||
|---|---|---|
Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero |
Abstract | ||
|---|---|---|
This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties A(f) that are optimal quotients of J(0)(N) attached to newforms. We prove theorems about the ratio L(A(f), 1)/OmegaA(f), develop tools for computing with A(f), and gather data about certain arithmetic invariants of the nearly 20, 000 abelian varieties Af of level less than or equal to 2333. Over half of these A(f) have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of III(A(f)). We find that there are at least 168 such A(f) for which the Birch and Swinnerton-Dyer conjecture implies that III( A(f)) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of III(A(f)) really divides # III(A(f)) by constructing nontrivial elements of III(A(f)) using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves. |
Year | DOI | Venue |
|---|---|---|
2005 | 10.1090/S0025-5718-04-01644-8 | MATHEMATICS OF COMPUTATION |
Keywords | Field | DocType |
Birch and Swinnerton-Dyer conjecture,modular abelian variety,visibility,Shafarevich-Tate groups | Prime (order theory),Abelian group,Combinatorics,Upper and lower bounds,Mathematical analysis,Quotient,Pure mathematics,Invariant (mathematics),Birch and Swinnerton-Dyer conjecture,Conjecture,Elliptic curve,Mathematics | Journal |
Volume | Issue | ISSN |
74 | 249 | 0025-5718 |
Citations | PageRank | References |
2 | 0.66 | 5 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Amod Agashe | 1 | 4 | 2.81 |
| William Stein | 2 | 2 | 1.68 |
| B. Mazur | 3 | 11 | 2.59 |
| John Cremona | 4 | 14 | 4.46 |