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SUMMARY:Charlotte Chan (Michigan)
DTSTART;VALUE=DATE-TIME:20220205T173000Z
DTEND;VALUE=DATE-TIME:20220205T183000Z
DTSTAMP;VALUE=DATE-TIME:20240423T103247Z
UID:SCNTD2022/1
DESCRIPTION:Title: Geometric L-packets of toral supercuspidal representations\nby Charl
otte Chan (Michigan) as part of Southern California Number Theory Day\n\nL
ecture held in APM 6402 and online.\n\nAbstract\nIn 2001\, Yu gave an alge
braic construction of supercuspidal representations of p-adic groups. Ther
e has since been a lot of progress towards explicitly constructing the loc
al Langlands correspondence for supercuspidal representations: Kazhdan-Var
shavsky and DeBacker-Reeder (depth zero)\, Reeder and DeBacker-Spice (tora
l)\, and Kaletha (regular). In this talk\, we present recent and ongoing w
ork investigating a geometric counterpart to this story. This is based on
joint work with Masao Oi.\n
LOCATION:https://researchseminars.org/talk/SCNTD2022/1/
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SUMMARY:Evan O'Dorney (Notre Dame)
DTSTART;VALUE=DATE-TIME:20220205T190000Z
DTEND;VALUE=DATE-TIME:20220205T200000Z
DTSTAMP;VALUE=DATE-TIME:20240423T103247Z
UID:SCNTD2022/2
DESCRIPTION:Title: Reflection theorems for counting quadratic and cubic polynomials\nby
Evan O'Dorney (Notre Dame) as part of Southern California Number Theory D
ay\n\nLecture held in APM 6402 and online.\n\nAbstract\nScholz's celebrate
d 1932 reflection principle\, relating the 3-torsion in the class groups o
f $\\mathbf{Q}(\\sqrt{D})$ and $\\mathbf{Q}(\\sqrt{-3D})$\, can be viewed
as an equality among the numbers of cubic fields of different discriminant
s. In 1997\, Y. Ohno discovered (quite by accident) a beautiful reflection
identity relating the number of binary cubic forms\, equivalently cubic r
ings\, of discriminants D and -27D\, where D is not necessarily squarefree
. This was proved in 1998 by Nakagawa\, establishing an "extra functional
equation" for the Shintani zeta functions counting binary cubic forms. In
my talk\, I will present a new and more illuminating method for proving id
entities of this type\, based on Poisson summation on adelic cohomology (i
n the style of Tate's thesis). Also\, I will present a corresponding refle
ction theorem for quadratic polynomials of a quite unexpected shape. The c
orresponding Shintani zeta function is in two variables\, counting by both
discriminant and leading coefficient\, and finding its analytic propertie
s is a work in progress.\n
LOCATION:https://researchseminars.org/talk/SCNTD2022/2/
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SUMMARY:Congling Qiu (Yale)
DTSTART;VALUE=DATE-TIME:20220205T220000Z
DTEND;VALUE=DATE-TIME:20220205T230000Z
DTSTAMP;VALUE=DATE-TIME:20240423T103247Z
UID:SCNTD2022/3
DESCRIPTION:Title: Injectivity of the Abel-Jacobi map and Gross-Kudla-Schoen cycles\nby
Congling Qiu (Yale) as part of Southern California Number Theory Day\n\nL
ecture held in APM 6402 and online.\n\nAbstract\nOn the triple product of
a quaternionic Shimura curve over a totally real field\, the injectivity o
f the Abel-Jacobi map implies an automorphic decomposition of the Chow gro
ups. Then Prasad's theorem on trilinear forms implies the vanishing of the
isotypic component of the Gross-Kudla-Schoen modified diagonal cycle with
a certain local root number. We define such a decomposition unconditional
ly and prove the vanishing. This is a special case of some general results
.\n
LOCATION:https://researchseminars.org/talk/SCNTD2022/3/
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SUMMARY:Alex Smith (Stanford)
DTSTART;VALUE=DATE-TIME:20220205T233000Z
DTEND;VALUE=DATE-TIME:20220206T003000Z
DTSTAMP;VALUE=DATE-TIME:20240423T103247Z
UID:SCNTD2022/4
DESCRIPTION:Title: Simple abelian varieties over finite fields with extreme point counts\nby Alex Smith (Stanford) as part of Southern California Number Theory D
ay\n\nLecture held in APM 6402 and online.\n\nAbstract\nGiven a compactly
supported probability measure on the reals\, we will give a necessary and
sufficient condition for there to be a sequence of totally real algebraic
integers whose distribution of conjugates approaches the measure. We use t
his result to prove that there are infinitely many totally positive algebr
aic integers X satisfying tr(X)/deg(X) < 1.899\; previously\, there were o
nly 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://researchseminars.org/talk/SCNTD2022/4/
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