BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Niven Achenjang (MIT) DTSTART:20240221T200000Z DTEND:20240221T210000Z DTSTAMP:20260422T172556Z UID:HarvardNT/102 DESCRIPTION:Title: The Average Size of 2-Selmer Groups of Elliptic Curves over Function F ields\nby Niven Achenjang (MIT) as part of Harvard number theory semin ar\n\nLecture held in Science Center Room 507.\n\nAbstract\nGiven an ellip tic curve $E$ over a global field $K$\, the abelian group $E(K)$ is finite ly generated\, and so much effort has been put into trying to understand t he behavior of $\\operatorname{rank}E(K)$\, as $E$ varies. Of note\, it is a folklore conjecture that\, when all elliptic curves $E/K$ are ordered b y a suitably defined height\, the average value of their ranks is exactly $1/2$. One fruitful avenue for understanding the distribution of $\\operat orname{rank}E(K)$ has been to first understand the distribution of the siz es of Selmer groups of elliptic curves. In this direction\, various author s (including Bhargava-Shankar\, Poonen-Rains\, and Bhargava-Kane-Lenstra-P oonen-Rains) have made conjectures which predict\, for example\, that the average size of the $n$-Selmer group of $E/K$ is equal to the sum of the d ivisors of $n$. In this talk\, I will report on some recent work verifying this average size prediction\, "up to small error term\," whenever $n=2$ and $K$ is any global *function* field. Results along these lines were pre viously known whenever $K$ was a number field or function field of charact eristic $\\ge 5$\, so the novelty of my work is that it applies even in "b ad" characteristic.\n LOCATION:https://researchseminars.org/talk/HarvardNT/102/ END:VEVENT END:VCALENDAR