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SUMMARY:Michael Allen (Wesleyan University)
DTSTART:20251021T200000Z
DTEND:20251021T210000Z
DTSTAMP:20260422T173622Z
UID:FCNTS/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/FCNTS/9/">Ex
 plicit Modularity of Hypergeometric Motives</a>\nby Michael Allen (Wesleya
 n University) as part of Five College Number Theory Seminar\n\nLecture hel
 d in Seeley Mudd 207 @Amherst College.\n\nAbstract\nThe Modularity Theorem
  states that given an elliptic curve one can find an associated modular fo
 rm.  One of the more striking aspects of the Modularity Theorem is the var
 iety of seemingly unrelated ways in which the relationship between the ell
 iptic curve and the modular form can be stated.  For this talk\, the prima
 ry formulations of modularity we will be interested in are the equality of
  elliptic and modular $L$-functions\, equality between the number of point
 s on the elliptic curve mod $p$ with the Fourier coefficients of the modul
 ar form\, and finally an isomorphism between elliptic and modular Galois r
 epresentations.  Each of these connections can be made explicit by express
 ing both sides in terms of hypergeometric functions (over $\\mathbb{C}$)\,
  hypergeometric character sums (over $\\mathbb{F}_p$)\, and hypergeometric
  Galois representations (over $\\mathbb{Q}_\\ell)$.  More generally\, each
  of these connections correspond to De Rham\, crystalline\, and étale rea
 lizations of hypergeometric motives.  We discuss recent and upcoming work 
 with Grove\, Long\, and Tu using these hypergeometric perspectives towards
  understanding generalizations of the Modularity Theorem for these hyperge
 ometric motives.\n
LOCATION:https://researchseminars.org/talk/FCNTS/9/
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