BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Debanjana Kundu (University of British Columbia) DTSTART:20221003T180000Z DTEND:20221003T190000Z DTSTAMP:20260423T035536Z UID:NTC/6 DESCRIPTION:Title: Stud ying Hilbert's 10th problem via explicit elliptic curves\nby Debanjana Kundu (University of British Columbia) as part of Lethbridge number theor y and combinatorics seminar\n\nLecture held in University of Lethbridge: M 1040 (Markin Hall).\n\nAbstract\nIn 1900\, Hilbert posed the following pro blem: "Given a Diophantine equation with integer coefficients: to devise a process according to which it can be determined in a finite number of ope rations whether the equation is solvable in (rational) integers."\n\nBuild ing on the work of several mathematicians\, in 1970\, Matiyasevich proved that this problem has a negative answer\, i.e.\, such a general `process' (algorithm) does not exist.\n\nIn the late 1970's\, Denef--Lipshitz formul ated an analogue of Hilbert's 10th problem for rings of integers of number fields. \n\nIn recent years\, techniques from arithmetic geometry have be en used extensively to attack this problem. One such instance is the work of García-Fritz and Pasten (from 2019) which showed that the analogue of Hilbert's 10th problem is unsolvable in the ring of integers of number fie lds of the form $\\mathbb{Q}(\\sqrt[3]{p}\,\\sqrt{-q})$ for positive propo rtions of primes $p$ and $q$. In joint work with Lei and Sprung\, we impro ve their proportions and extend their results in several directions. We ac hieve this by using multiple elliptic curves\, and by replacing their Iwas awa theory arguments by a more direct method.\n LOCATION:https://researchseminars.org/talk/NTC/6/ END:VEVENT END:VCALENDAR