BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Ashvin Swaminathan (Harvard University)
DTSTART:20260406T200000Z
DTEND:20260406T210000Z
DTSTAMP:20260423T022229Z
UID:NTBU/47
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/NTBU/47/">Tw
 ist hypercubes and the distribution of $2$-Selmer ranks of elliptic curves
 </a>\nby Ashvin Swaminathan (Harvard University) as part of Boston Univers
 ity Number Theory Seminar\n\nLecture held in CDS Room 548 in Boston Univer
 sity.\n\nAbstract\nThe Poonen–Rains heuristics conjecture an explicit di
 stribution for the $2$-Selmer ranks of elliptic curves over $\\mathbb{Q}$.
  Their conjecture predicts in particular that every nonnegative integer sh
 ould occur as the $2$-Selmer rank of a positive proportion of curves. This
  qualitative prediction has remained entirely open: prior to this work\, n
 ot a single value of $r$ was known to occur with positive proportion. We p
 rove this prediction for every $r$.\n\nOur method organizes quadratic twis
 ts of elliptic curves into hypercubes whose $2$-Selmer ranks are tightly c
 onstrained by Poitou–Tate duality. We classify all valid rank patterns b
 y simple graphs and use this classification to obtain the first two-sided 
 bounds on rank densities in congruence families. In a complementary direct
 ion\, we show that the $2$-Selmer rank evolves as a birth–death chain ac
 ross the hypercube\, and prove that this chain converges to the Poonen–R
 ains distribution. Analogous results hold for Jacobians of hyperelliptic c
 urves of any genus.\n\nThis is joint work with Manjul Bhargava\, Wei Ho\, 
 Ari Shnidman\, and Alexander Smith.\n
LOCATION:https://researchseminars.org/talk/NTBU/47/
END:VEVENT
END:VCALENDAR