BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Harris Daniels (Amherst College) DTSTART:20251007T203000Z DTEND:20251007T213000Z DTSTAMP:20260423T130123Z UID:MITNT/121 DESCRIPTION:Title: Near coincidences and nilpotent division fields of elliptic curves\nby Harris Daniels (Amherst College) as part of MIT number theory seminar\n\n Lecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstrac t\nA natural question about the division fields of a fixed elliptic curve $E/\\mathbb{Q}$ is whether there is a coincidence between the division fie lds. I.e. Are there distinct integers $m \\neq n$ such that the $m$-divis ion field equals the $n$-division field. In 2023\, Daniels and Lozano-Robl edo gave partial answers to this question\, using (among other tools) the fact that the $n$-th roots of unity often fail to lie in the $m$-division field\, thereby preventing such coincidences.\n\nMotivated by this\, we co nsider a broader notion of \\emph{near coincidences}: when does there exis t $E/\\mathbb{Q}$ and distinct $m\,n$ such that\n\\[\n\\mathbb{Q}(E[n]) = \\mathbb{Q}(E[m]\, \\zeta_n)?\n\\]\nIn the first part of this talk\, we an swer this question completely in the case where $m$ and $n$ are powers of the same prime.\n\nIn the second part\, we turn to a seemingly unrelated b ut natural problem: classifying all elliptic curves $E/\\mathbb{Q}$ and po sitive integers $n$ such that\n\\[\n\\operatorname{Gal}(\\mathbb{Q}(E[n])/ \\mathbb{Q})\n\\]\nis a nilpotent group. This question generalizes the cla ssification of abelian division fields obtained by Gonz\\'alez-Jim\\'enez and Lozano-Robledo (2016). We present a conditionally complete classificat ion of nilpotent division fields\, under either a standard conjecture abou t rational points on modular curves attached to normalizers of non-split C artan subgroups or a full classification of the Mersenne primes. This is j oint work with Jeremy Rouse.\n LOCATION:https://researchseminars.org/talk/MITNT/121/ END:VEVENT END:VCALENDAR