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CATEGORIES:Number Theory Seminar
SUMMARY:Commensurability of automorphism groups\, and numb
er theoretic applications - Alex Bartel (Warwick U
niversity)
DTSTART;TZID=Europe/London:20151124T141500
DTEND;TZID=Europe/London:20151124T151500
UID:TALK61452AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/61452
DESCRIPTION:There is a general philosophy that if a family of
algebraic objects behaves randomly\, then the prob
ability that an object from this family is isomorp
hic to a given object A is inverse proportional to
#Aut(A). This has first been observed by Cohen an
d Lenstra in the case of class groups of imaginary
quadratic number fields. That so-called Cohen-Len
stra heuristic was later extended to other familie
s of number fields\, at which point much less natu
rally looking probability weights started occurrin
g. It turns out that if instead of class groups\,
one talks about Arakelov class groups\, then the o
riginal heuristic holds in great generality\, prov
ided one can make sense of "inverse proportional t
o #Aut(A)" in cases where the automorphism group i
s infinite. In this talk I will present a theory o
f commensurability of modules over certain rings\,
and of their endomorphism rings and automorphism
groups\, and will use it to formulate a heuristic
for Arakelov class groups of number fields\, with
a surprising twist at the end. This is joint work
with Hendrik Lenstra.
LOCATION:MR13
CONTACT:Jack Thorne
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