BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Christopher Keyes (Emory) DTSTART:20221201T220000Z DTEND:20221201T230000Z DTSTAMP:20260423T022806Z UID:UCSD_NTS/78 DESCRIPTION:Title: Local solubility in families of superelliptic curves\nby Christopher Keyes (Emory) as part of UCSD number theory seminar\n\nLecture held in AP M 6402 and online.\n\nAbstract\nIf we choose at random an integral binary form $f(x\, z)$ of fixed degree $d$\, what is the probability that the sup erelliptic curve with equation $C \\colon: y^m = f(x\, z)$ has a $p$-adic point\, or better\, points everywhere locally? In joint work with Lea Ben eish\, we show that the proportion of forms $f(x\, z)$ for which $C$ is ev erywhere locally soluble is positive\, given by a product of local densiti es. By studying these local densities\, we produce bounds which are suitab le enough to pass to the large $d$ limit. In the specific case of curves o f the form $y^3 = f(x\, z)$ for a binary form of degree 6\, we determine t he probability of everywhere local solubility to be 96.94%\, with the exac t value given by an explicit infinite product of rational function express ions.\n\npre-talk\n LOCATION:https://researchseminars.org/talk/UCSD_NTS/78/ END:VEVENT END:VCALENDAR