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SUMMARY:Michelle Manes (UH/NSF)
DTSTART;VALUE=DATE-TIME:20200926T160500Z
DTEND;VALUE=DATE-TIME:20200926T165500Z
DTSTAMP;VALUE=DATE-TIME:20201101T002541Z
UID:FRNTDFall2020/1
DESCRIPTION:Title: Complex multiplication in arithmetic dynamics\nby Miche
lle Manes (UH/NSF) as part of Front Range Number Theory Day\n\n\nAbstract\
nArithmetic dynamics is the study of number theoretic properties of iterat
ed functions.\nThe field draws inspiration from dynamical analogues of the
orems and conjectures in classical\narithmetic geometry. In this talk\, I
will describe some of these analogues with a focus on\nattempts to develop
a “dynamical” theory of complex multiplication.\n
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SUMMARY:Andrew Sutherland (MIT)
DTSTART;VALUE=DATE-TIME:20200926T191500Z
DTEND;VALUE=DATE-TIME:20200926T200500Z
DTSTAMP;VALUE=DATE-TIME:20201101T002541Z
UID:FRNTDFall2020/2
DESCRIPTION:Title: Sato-Tate groups of abelian threefolds\nby Andrew Suthe
rland (MIT) as part of Front Range Number Theory Day\n\n\nAbstract\nLet $A
$ be an abelian variety of dimension $g$ defined over a\nnumber field $K$.
As defined by Serre\, the Sato-Tate group $\\mathrm{ST}(A)$ is a\ncompact
subgroup of the unitary symplectic group $\\mathrm{USp}(2g)$ equipped wit
h\na map that sends each Frobenius element of the absolute Galois group\no
f $K$ at primes $\\mathfrak p$ of good reduction for $A$ to a conjugacy cl
ass of $\\mathrm{ST}(A)$\nwhose characteristic polynomial is determined by
the zeta function of\nthe reduction of $A$ at $\\mathfrak p$. Under a set
of axioms proposed by Serre that\nare known to hold for $g \\le 3$\, up t
o conjugacy in $\\mathrm{Usp}(2g)$ there is a\nfinite list of possible Sat
o-Tate groups that can arise for abelian\nvarieties of dimension $g$ over
number fields.\n\nFor $g = 1$ there are $3$ possibilities for $\\mathrm{ST
}(A)$\, for $g = 2$ there are\n$52$\, and last year it was shown that for
$g = 3$ there are $410$. In this\ntalk I will give a brief overview of thi
s classification and then\ndiscuss ongoing efforts to produce explicit exa
mples that realize\nthese $410$ possibilities.\n\nThis is joint work with
Kiran Kedlaya and Francesc Fité\n
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BEGIN:VEVENT
SUMMARY:Christelle Vincent (UVM)
DTSTART;VALUE=DATE-TIME:20200926T202000Z
DTEND;VALUE=DATE-TIME:20200926T211000Z
DTSTAMP;VALUE=DATE-TIME:20201101T002541Z
UID:FRNTDFall2020/3
DESCRIPTION:Title: Computing hyperelliptic invariants from period matrices
\nby Christelle Vincent (UVM) as part of Front Range Number Theory Day\n\n
\nAbstract\nIn this talk we present an obstacle to computing invariants of
curves whose Jacobian\nhas CM (complex multiplication)\, when the genus o
f the curve is greater than 1. The problem is\nessentially that while the
Jacobian has everywhere potential good reduction\, the curve does not.\nWe
show the connection between this obstacle and a certain embedding problem
which we\ndefine in the talk\, and present our progress on analyzing the
embedding problem. This is joint\nwork with Ionica\, Kilicer\, Lauter\, Lo
renzo Garcia and Manzateanu.\n
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