BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Michael Allen (Wesleyan University) DTSTART:20251021T200000Z DTEND:20251021T210000Z DTSTAMP:20260824T073605Z UID:FCNTS/9 DESCRIPTION:Title: Ex plicit Modularity of Hypergeometric Motives\nby Michael Allen (Wesleya n University) as part of Pioneer Valley Number Theory Seminar\n\nLecture h eld in Seeley Mudd 207 @Amherst College.\n\nAbstract\nThe Modularity Theor em states that given an elliptic curve one can find an associated modular form. One of the more striking aspects of the Modularity Theorem is the v ariety of seemingly unrelated ways in which the relationship between the e lliptic curve and the modular form can be stated. For this talk\, the pri mary formulations of modularity we will be interested in are the equality of elliptic and modular $L$-functions\, equality between the number of poi nts on the elliptic curve mod $p$ with the Fourier coefficients of the mod ular form\, and finally an isomorphism between elliptic and modular Galois representations. Each of these connections can be made explicit by expre ssing both sides in terms of hypergeometric functions (over $\\mathbb{C}$) \, hypergeometric character sums (over $\\mathbb{F}_p$)\, and hypergeometr ic Galois representations (over $\\mathbb{Q}_\\ell)$. More generally\, ea ch of these connections correspond to De Rham\, crystalline\, and étale r ealizations of hypergeometric motives. We discuss recent and upcoming wor k with Grove\, Long\, and Tu using these hypergeometric perspectives towar ds understanding generalizations of the Modularity Theorem for these hyper geometric motives.\n LOCATION:https://researchseminars.org/talk/FCNTS/9/ END:VEVENT END:VCALENDAR