BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Carlo Pagano (Concordia University)
DTSTART:20250429T190000Z
DTEND:20250429T200000Z
DTSTAMP:20260423T130534Z
UID:MITNT/115
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITNT/115/">
 Additive combinatorics and descent</a>\nby Carlo Pagano (Concordia Univers
 ity) as part of MIT number theory seminar\n\nLecture held in Room 2-255 in
  the Simons Building (building 2).\n\nAbstract\nWe shall discuss the probl
 em of constructing elliptic curves over number fields with positive but "c
 ontrolled" rank. An example of this problem is: given a quadratic extensio
 n $L/K$ of number fields\, construct an elliptic curve $E/K$ such that $0<
 \\text{rk}(E(K))=\\text{rk}(E(L))$. Another example is: for a number field
  $K$\, find an elliptic curve $E/K$ such that $\\text{rk}(E(K))=1$. \nWith
  Peter Koymans we introduced a method to tackle this type of problems\, co
 mbining additive combinatorics with 2-descent. I will explain our past wor
 k on the former problem\, where we showed that Hilbert 10th problem has ne
 gative answer on ring of integers of general number fields. Next\, I will 
 explain our joint work in progress\, where we settle the latter question\,
  showing the following stronger result: if $E/K$ has full rational $2$-tor
 sion and no cyclic degree $4$ isogeny defined over $K$\, and it has at lea
 st one quadratic twist with odd root number\, then it has infinitely many 
 quadratic twists $d$ in $K^{\\ast}/K^{\\ast 2}$ such that $\\text{rk}(E^d(
 K))=1$.\n
LOCATION:https://researchseminars.org/talk/MITNT/115/
END:VEVENT
END:VCALENDAR