Explicit approaches to modular abelian varieties

William Arthur Stein

Doctor of Philosophy in Mathematics

University of California at Berkeley

Professor Hendrik Lenstra, Chair

Many spectacular recent successes in number theory, such as the resolution of Fermat's Last Theorem due to A. Wiles, have involved proving modularity conjectures. These state that certain purely number theoretic objects, such as elliptic curves, arise from entirely different analytic objects like modular forms. These conjectural links indicate why my thesis topic, which is explicit computations with modular objects, is of central importance.

Elliptic curves are certain types of plane cubic curves whose points form an abelian group. In the 1960s B. Birch and others embarked upon a program to enumerate extensive information about the first few hundred elliptic curves; in the course of their work, they uncovered many remarkable properties and conjectures flourished. Later, J. Cremona, J.-F. Mestre, and N. Elkies greatly extended Birch's computations. Likewise, the first part of my thesis lays the groundwork necessary to carry out similar investigations in directions not considered by Cremona. I give algorithms enumerating higher degree and weight modular forms, and for computing information about abelian varieties, which are the higher dimensional analogues of elliptic curves. These include algorithms for computing congruences, the modular degree, rational parts of special values of L-functions, component groups at primes of multiplicative reduction, the period lattice, and the real volume.

In part two of my thesis, I use the well-developed theory of modular forms, combined with the algorithms of part one, to investigate two outstanding conjectures. In the 1960s, based on computations and analogy, Birch, P. Swinnerton-Dyer, and J. Tate made conjectures that tie together the constellation of arithmetic invariants associated with an abelian variety. Because these conjectures have remained mostly open, any successful attempt to verify them for specific abelian varieties is likely to lead to new insights. I have attempted such verifications in many cases in which the dimension of the abelian variety involved is large. My long-term hope is that my approach, which depends on B. Mazur's notion of visibility of Shafarevich-Tate groups and K. Ribet's theory of congruences between modular forms, will lead me to find a proof of their conjectures in infinitely many cases.

I also consider a modularity question. In 1924 E. Artin conjectured that the L-series associated to a complex representation of the Galois group of the rational numbers has an extension to the whole complex plane. The only known way to prove this is to show that the representation comes from an appropriate modular form. A special case of this conjecture, which was proved by R. P. Langlands, played a key role in Wiles's proof of Fermat Last Theorem. Jointly with K. Buzzard, I obtained previously unknown examples in which this conjecture holds; the proof relies on a theorem of Buzzard and Taylor along with modular forms computations that use mod 5 variants of the above algorithms.