### Simple abelian varieties over finite fields with extreme point counts

Heilbronn Number Theory Seminar

16th November 2022, 11:00 am – 12:00 pm

Queens Building, 1.7

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.

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