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CATEGORIES:Number Theory Seminar
SUMMARY:Cohen-Lenstra heuristic revisited - Alex Bartel (W
arwick)
DTSTART;TZID=Europe/London:20140603T161500
DTEND;TZID=Europe/London:20140603T171500
UID:TALK52710AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/52710
DESCRIPTION:The original Cohen-Lenstra heuristic predicts freq
uencies with which a given abelian group appears a
s the class group of a quadratic field. Postulated
just over 30 years ago\, the heuristic helped exp
lain in a very compelling and intuitive way variou
s phenomena in the study of class groups that had
been observed over the preceding decades and centu
ries. Roughly speaking\, the probability of an abe
lian group A occurring as the class group of an im
aginary quadratic field is inverse proportional to
#Aut(A) - a random object tends to have few symme
tries. Already for real quadratic fields\, the wei
ghts are postulated to be different\, and the heur
istic explanation is less intuitive. Later\, the h
euristic was extended by Cohen and Martinet to mor
e general number fields\, this time without even a
n attempt at an intuitive explanation of why these
should be the right weights. I will explain that
in fact\, the intuitive heuristic that the rarity
of an algebraic object is proportional to the numb
er of symmetries does explain distributions of cla
ss groups of arbitrary number fields and recovers
all the above heuristics\, but the object one has
to look at is not the class group. This is still w
ork in progress\, jointly with Hendrik Lenstra.
LOCATION:MR13
CONTACT:James Newton
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