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SUMMARY:Charlotte Chan (Michigan)
DTSTART:20220205T173000Z
DTEND:20220205T183000Z
DTSTAMP:20260404T094547Z
UID:SCNTD2022/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/SCNTD
 2022/1/">Geometric L-packets of toral supercuspidal representations</a>\nb
 y Charlotte Chan (Michigan) as part of Southern California Number Theory D
 ay\n\nLecture held in APM 6402 and online.\n\nAbstract\nIn 2001\, Yu gave 
 an algebraic construction of supercuspidal representations of p-adic group
 s. There has since been a lot of progress towards explicitly constructing 
 the local Langlands correspondence for supercuspidal representations: Kazh
 dan-Varshavsky and DeBacker-Reeder (depth zero)\, Reeder and DeBacker-Spic
 e (toral)\, and Kaletha (regular). In this talk\, we present recent and on
 going work investigating a geometric counterpart to this story. This is ba
 sed on joint work with Masao Oi.\n
LOCATION:https://stable.researchseminars.org/talk/SCNTD2022/1/
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SUMMARY:Evan O'Dorney (Notre Dame)
DTSTART:20220205T190000Z
DTEND:20220205T200000Z
DTSTAMP:20260404T094547Z
UID:SCNTD2022/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/SCNTD
 2022/2/">Reflection theorems for counting quadratic and cubic polynomials<
 /a>\nby Evan O'Dorney (Notre Dame) as part of Southern California Number T
 heory Day\n\nLecture held in APM 6402 and online.\n\nAbstract\nScholz's ce
 lebrated 1932 reflection principle\, relating the 3-torsion in the class g
 roups of $\\mathbf{Q}(\\sqrt{D})$ and $\\mathbf{Q}(\\sqrt{-3D})$\, can be 
 viewed as an equality among the numbers of cubic fields of different discr
 iminants. In 1997\, Y. Ohno discovered (quite by accident) a beautiful ref
 lection identity relating the number of binary cubic forms\, equivalently 
 cubic rings\, of discriminants D and -27D\, where D is not necessarily squ
 arefree. This was proved in 1998 by Nakagawa\, establishing an "extra func
 tional equation" for the Shintani zeta functions counting binary cubic for
 ms. In my talk\, I will present a new and more illuminating method for pro
 ving identities of this type\, based on Poisson summation on adelic cohomo
 logy (in the style of Tate's thesis). Also\, I will present a correspondin
 g reflection theorem for quadratic polynomials of a quite unexpected shape
 . The corresponding Shintani zeta function is in two variables\, counting 
 by both discriminant and leading coefficient\, and finding its analytic pr
 operties is a work in progress.\n
LOCATION:https://stable.researchseminars.org/talk/SCNTD2022/2/
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SUMMARY:Congling Qiu (Yale)
DTSTART:20220205T220000Z
DTEND:20220205T230000Z
DTSTAMP:20260404T094547Z
UID:SCNTD2022/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/SCNTD
 2022/3/">Injectivity of the Abel-Jacobi map and Gross-Kudla-Schoen cycles<
 /a>\nby Congling Qiu (Yale) as part of Southern California Number Theory D
 ay\n\nLecture held in APM 6402 and online.\n\nAbstract\nOn the triple prod
 uct of a quaternionic Shimura curve over a totally real field\, the inject
 ivity of the Abel-Jacobi map implies an automorphic decomposition of the C
 how groups. Then Prasad's theorem on trilinear forms implies the vanishing
  of the isotypic component of the Gross-Kudla-Schoen modified diagonal cyc
 le with a certain local root number. We define such a decomposition uncond
 itionally and prove the vanishing. This is a special case of some general 
 results.\n
LOCATION:https://stable.researchseminars.org/talk/SCNTD2022/3/
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SUMMARY:Alex Smith (Stanford)
DTSTART:20220205T233000Z
DTEND:20220206T003000Z
DTSTAMP:20260404T094547Z
UID:SCNTD2022/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/SCNTD
 2022/4/">Simple abelian varieties over finite fields with extreme point co
 unts</a>\nby Alex Smith (Stanford) as part of Southern California Number T
 heory Day\n\nLecture held in APM 6402 and online.\n\nAbstract\nGiven a com
 pactly supported probability measure on the reals\, we will give a necessa
 ry and sufficient condition for there to be a sequence of totally real alg
 ebraic integers whose distribution of conjugates approaches the measure. W
 e use this result to prove that there are infinitely many totally positive
  algebraic integers X satisfying tr(X)/deg(X) < 1.899\; previously\, there
  were only known to be infinitely many such integers satisfying tr(X)/deg(
 X) < 2. We also will explain how our method can be used in the search for 
 simple abelian varieties with extreme point counts.\n
LOCATION:https://stable.researchseminars.org/talk/SCNTD2022/4/
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