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SUMMARY:Debanjana Kundu (University of British Columbia)
DTSTART:20221003T180000Z
DTEND:20221003T190000Z
DTSTAMP:20260404T132231Z
UID:NTC/6
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/NTC/6
 /">Studying Hilbert's 10th problem via explicit elliptic curves</a>\nby De
 banjana Kundu (University of British Columbia) as part of Lethbridge numbe
 r theory and combinatorics seminar\n\nLecture held in University of Lethbr
 idge: M1040 (Markin Hall).\n\nAbstract\nIn 1900\, Hilbert posed the follow
 ing problem: "Given a Diophantine equation with integer coefficients: to d
 evise a process according to which it can be determined in a finite number
  of operations whether the equation is solvable in (rational) integers."\n
 \nBuilding on the work of several mathematicians\, in 1970\, Matiyasevich 
 proved that this problem has a negative answer\, i.e.\, such a general `pr
 ocess' (algorithm) does not exist.\n\nIn the late 1970's\, Denef--Lipshitz
  formulated an analogue of Hilbert's 10th problem for rings of integers of
  number fields. \n\nIn recent years\, techniques from arithmetic geometry 
 have been used extensively to attack this problem. One such instance is th
 e work of García-Fritz and Pasten (from 2019) which showed that the analo
 gue of Hilbert's 10th problem is unsolvable in the ring of integers of num
 ber fields of the form $\\mathbb{Q}(\\sqrt[3]{p}\,\\sqrt{-q})$ for positiv
 e proportions of primes $p$ and $q$. In joint work with Lei and Sprung\, w
 e improve their proportions and extend their results in several directions
 . We achieve this by using multiple elliptic curves\, and by replacing the
 ir Iwasawa theory arguments by a more direct method.\n
LOCATION:https://stable.researchseminars.org/talk/NTC/6/
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