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SUMMARY:Michelle Manes (UH/NSF)
DTSTART;VALUE=DATE-TIME:20200926T160500Z
DTEND;VALUE=DATE-TIME:20200926T165500Z
DTSTAMP;VALUE=DATE-TIME:20220528T190539Z
UID:FRNTDFall2020/1
DESCRIPTION:Title: Complex multiplication in arithmetic dynamics\nby Michelle Manes
(UH/NSF) as part of Front Range Number Theory Day\n\n\nAbstract\nArithmet
ic dynamics is the study of number theoretic properties of iterated functi
ons.\nThe field draws inspiration from dynamical analogues of theorems and
conjectures in classical\narithmetic geometry. In this talk\, I will desc
ribe some of these analogues with a focus on\nattempts to develop a “dyn
amical” theory of complex multiplication.\n
LOCATION:https://researchseminars.org/talk/FRNTDFall2020/1/
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SUMMARY:Andrew Sutherland (MIT)
DTSTART;VALUE=DATE-TIME:20200926T191500Z
DTEND;VALUE=DATE-TIME:20200926T200500Z
DTSTAMP;VALUE=DATE-TIME:20220528T190539Z
UID:FRNTDFall2020/2
DESCRIPTION:Title: Sato-Tate groups of abelian threefolds\nby Andrew Sutherland (MI
T) as part of Front Range Number Theory Day\n\n\nAbstract\nLet $A$ be an a
belian variety of dimension $g$ defined over a\nnumber field $K$. As defin
ed by Serre\, the Sato-Tate group $\\mathrm{ST}(A)$ is a\ncompact subgroup
of the unitary symplectic group $\\mathrm{USp}(2g)$ equipped with\na map
that sends each Frobenius element of the absolute Galois group\nof $K$ at
primes $\\mathfrak p$ of good reduction for $A$ to a conjugacy class 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 axiom
s proposed by Serre that\nare known to hold for $g \\le 3$\, up to conjuga
cy in $\\mathrm{Usp}(2g)$ there is a\nfinite list of possible Sato-Tate gr
oups that can arise for abelian\nvarieties of dimension $g$ over number fi
elds.\n\nFor $g = 1$ there are $3$ possibilities for $\\mathrm{ST}(A)$\, f
or $g = 2$ there are\n$52$\, and last year it was shown that for $g = 3$ t
here are $410$. In this\ntalk I will give a brief overview of this classif
ication and then\ndiscuss ongoing efforts to produce explicit examples tha
t realize\nthese $410$ possibilities.\n\nThis is joint work with Kiran Ked
laya and Francesc Fité\n
LOCATION:https://researchseminars.org/talk/FRNTDFall2020/2/
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SUMMARY:Christelle Vincent (UVM)
DTSTART;VALUE=DATE-TIME:20200926T202000Z
DTEND;VALUE=DATE-TIME:20200926T211000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190539Z
UID:FRNTDFall2020/3
DESCRIPTION:Title: Computing hyperelliptic invariants from period matrices\nby Chri
stelle Vincent (UVM) as part of Front Range Number Theory Day\n\n\nAbstrac
t\nIn this talk we present an obstacle to computing invariants of curves w
hose Jacobian\nhas CM (complex multiplication)\, when the genus of the cur
ve 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\, Lorenzo Gar
cia and Manzateanu.\n
LOCATION:https://researchseminars.org/talk/FRNTDFall2020/3/
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