Simple abelian varieties over finite fields with extreme point counts

Alexander D. Smith (UCLA/Stanford)

Given a compactly supported probability measure on the reals, we will give a necessary and sufficient condition for there to be a sequence of totally real algebraic integers whose distribution of conjugates approaches the measure. We use this result to prove that there are infinitely many totally positive algebraic integers X satisfying tr(X)/deg(X) < 1.899; previously, there were only known to be infinitely many such integers satisfying tr(X)/deg(X) < 2. We also will explain how our method can be used in the search for simple abelian varieties with extreme point counts.

Arithmetic progressions in sumsets of geometric progressions

Michael Bennett (University of British Columbia)

If A and B are two geometric progressions, we characterize all 3-term arithmetic progressions in the sumset A+B. Somewhat surprisingly, while mostly elementary, this appears to require quite deep machinery from Diophantine Approximation.