BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Tristan Phillips (University of Arizona) DTSTART:20220222T210000Z DTEND:20220222T220000Z DTSTAMP:20260423T010005Z UID:UAANTS/37 DESCRIPTION:Title: Counting Elliptic Curves over Number Fields\nby Tristan Phillips (Univ ersity of Arizona) as part of University of Arizona Algebra and Number The ory Seminar\n\n\nAbstract\nIn this talk I will discuss some results on cou nting elliptic curves over number fields. In particular\, I will give asym ptotics for the number of isomorphism classes of elliptic curves over arbi trary number fields with certain prescribed level structures and prescribe d local conditions. This is done by counting the number of points of bound ed height on genus zero modular curves which are isomorphic to a weighted projective space. This includes the cases of X(N) for N\\in\\{1\,2\,3\,4\ ,5\\}\, X_1(N) for N\\in\\{1\,2\,\\dots\,10\,12\\}\, and X_0(N) for N\\in\ \{1\,2\,4\,6\,8\,9\,12\,16\,18\\}. Using these results for counting ellipt ic curves over number fields with a prescribed local condition\, one can s how that the average analytic rank of elliptic curves over any number fiel d K is bounded above by 3\\text{deg}(K)+1/2\, under the assumptions that a ll elliptic curves over K are modular and have L-functions which satisfy t he Generalized Riemann Hypothesis\n LOCATION:https://researchseminars.org/talk/UAANTS/37/ END:VEVENT END:VCALENDAR