BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alexander Smith (Massachusetts Institute of Technology) DTSTART:20210629T151000Z DTEND:20210629T160000Z DTSTAMP:20260419T121347Z UID:AFDRT/8 DESCRIPTION:Title: To tally positive integers of small trace and extreme orders of abelian varie ties over finite fields\nby Alexander Smith (Massachusetts Institute o f Technology) as part of Around Frobenius distributions and related topics II\n\n\nAbstract\nOutside 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 s ufficiently large square prime power $q$\, we show that all but finitely m any simple abelian varieties $A/\\mathbb{F}_q$ satisfy\n\\[(q - 2q^{1/2} + 2.8)^{\\dim A} \\le \\#A(\\mathbb{F}_q) \\le (q + 2q^{1/2} - 0.8)^{\\dim A}\,\\]\nand we explain how are approach can be adapted to other $q$. We will also give some evidence that there are infinitely many totally positi ve algebraic integers whose average valuation is less than $1.82$ and expl ain the implications of such a result for abelian varieties over finite fi elds.\n\nOur starting point is the fact that the discriminant of a rationa l integer polynomial must be a rational integer. We are able to take advan tage of this fact in our computational approach by using logarithmic poten tial theory.\n LOCATION:https://researchseminars.org/talk/AFDRT/8/ END:VEVENT END:VCALENDAR