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SUMMARY:Takuya Yamauchi (Tohoku)
DTSTART:20201120T153000Z
DTEND:20201120T170000Z
DTSTAMP:20260423T024515Z
UID:CAFAS/11
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAFAS/11/">A
 utomorphy of mod 2 Galois representations associated to the quintic Dwork 
 family and reciprocity of some quintic trinomials</a>\nby Takuya Yamauchi 
 (Tohoku) as part of Columbia Automorphic Forms and Arithmetic Seminar\n\n\
 nAbstract\nIn this talk\, I will explain my recent work with Tsuzuki Nobuo
  on computing\nmod $2$ Galois representations $\\overline{\\rho}_{\\psi\,2
 }:G_K:={\\rm Gal}(\\overline{K}/K)\\longrightarrow {\\rm GSp}_4(\\F_2)$\na
 ssociated to the mirror motives of rank 4 with pure weight 3 coming from t
 he\nDwork quintic family\n$$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5\\psi X_0X_1X_2
 X_3X_4=0\,\\ \\psi\\in K$$\ndefined over a number field $K$ under the irre
 ducibility condition of the quintic trinomial\n$f_\\psi(x)=4x^5-5\\psi x^4
 +1$.\nIn the course of the computation\, we observe that the image of such
  a mod $2$ representation is governed by reciprocity of\n$f_\\psi(x)$ whos
 e decomposition field is generically of type\n5-th symmetric group $S_5$.\
 nWhen K=F is totally real field\, we apply the modularity of\n2-dimensiona
 l\, totally odd Artin representations of ${\\rm Gal}(\\overline{F}/F)$ due
  to Shu Sasaki\nto obtain automorphy of $\\overline{\\rho}_{\\psi\,2}$ aft
 er a suitable (at most) quadratic base extension.\n
LOCATION:https://researchseminars.org/talk/CAFAS/11/
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