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Author: William A. Stein
William Stein's Ph.D. thesis

Explicit approaches to modular abelian varieties

William A. Stein

Doctor of Philosophy in Mathematics

University of California at Berkeley

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       I investigate the Birch and Swinnerton-Dyer conjecture, which ties together the constellation of invariants attached to an abelian variety. I attempt to verify this conjecture for certain specific modular abelian varieties of dimension greater than one. The key idea is to use Barry Mazur's notion of visibility, coupled with explicit computations, to produce lower bounds on the Shafarevich-Tate group. I have not finished the proof of the conjecture in these examples; this would require computing explicit upper bounds on the order of this group.

       I next describe how to compute in spaces of modular forms of weight at least two. I give an integrated package for computing, in many cases, the following invariants of a modular abelian variety: the modular degree, the rational part of the special value of the L-function, the order of the component group at primes of multiplicative reduction, the period lattice, upper and lower bounds on the torsion subgroup, and the real measure. Taken together, these algorithms are frequently enough to compute the odd part of the conjectural order of the Shafarevich-Tate group of an analytic rank 0 optimal quotient of J0(N), with N square-free. I have not determined the exact structure of the component group, the order of the component group at primes whose square divides the level, or the exact order of the torsion subgroup in all cases. However, I do provide generalizations of some of the above algorithms to higher weight forms with nontrivial character.

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