Totally positive integers of small trace and extreme orders of abelian varieties over finite fields

Abstract: Outside of finitely many exceptions, we show that the average real valuation of a totally positive algebraic integer is at least $1.80$, improving the prior best of $1.7919$. As a consequence, for a sufficiently large square prime power $q$, we show that all but finitely many simple abelian varieties $A/\mathbb{F}_q$ satisfy \[(q - 2q^{1/2} + 2.8)^{\dim A} \le \#A(\mathbb{F}_q) \le (q + 2q^{1/2} - 0.8)^{\dim A},\] and we explain how are approach can be adapted to other $q$. We will also give some evidence that there are infinitely many totally positive algebraic integers whose average valuation is less than $1.82$ and explain the implications of such a result for abelian varieties over finite fields.

Our starting point is the fact that the discriminant of a rational integer polynomial must be a rational integer. We are able to take advantage of this fact in our computational approach by using logarithmic potential theory.

number theory

Audience: researchers in the topic

Around Frobenius distributions and related topics II