Ranks of elliptic curves and Selmer groups

James E Weigandt, Purdue University

Abstract

This dissertation concerns the computation of m-Selmer groups of elliptic curves via the number field method of descent. We lay out a strategy for explicitly representing cohomology classes ξ∈Sel (m)(E/K) by those m-th power classes of an étale K-algebra belonging to a naturally defined group and subject to finitely many additional local conditions. Under suitable hypotheses, we relate this group to the m-torsion in the class group and m-th power classes of the unit group of an appropriate subring of the étale K-algebra. We work out the strategy in detail when m =2$, getting the cubic étale algebra used in the classical number field method of 2-descent. When K = Q and this algebra splits as three copies of Q, we recast the theory of complete 2-descent in terms of the Frey-Hellegouarch correspondence between solutions to A + B + C = 0 and elliptic curves with E[2] ⊆ E(Q). We classify those cases with at most three local conditions to compute, and describe Sel(2)(E/ Q) explicitly for such curves.

Degree

Ph.D.

Advisors

Goins, Purdue University.

Subject Area

Mathematics

COinS