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SERMON (South East Regional Meeting On Numbers)
April 17-18, 1999
University of South Carolina
Columbia, SC, USA

Organizers
Michael Filaseta, Kevin Ford, Richard Hudson, Robert Murphy, Kostya Oskolkov, Ognian Trifonov

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Average size of 2-Selmer groups of a family of elliptic curves
by
Gang Yu
Department of Mathematics, The University of Georgia, Athens, GA 30605

Suppose X is a large positive real number. We show an asymptotic formula for the average size of 2-Selmer groups of elliptic curves given by the equation
y2=x(x+p)(x+q),
where -X <= p =/= q <= X with both |p | and |q | being prime. For curves given by the equation
y2=x(x+a)(x+b)
with a a fixed non-zero integer, we show that, when varying b in [-X, X], the average size of the 2-Selmer groups is unbounded if |a | is a perfect square, and is bounded otherwise. The results provide an approach to a conjecture of A. Brumer.

Date received: April 14, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacu-10.