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SUMMARY:Cohen-Lenstra heuristic revisited - Alex Bartel (Warwick)
DTSTART:20140603T151500Z
DTEND:20140603T161500Z
UID:TALK52710@talks.cam.ac.uk
CONTACT:James Newton
DESCRIPTION:The original Cohen-Lenstra heuristic predicts frequencies with
  which a given abelian group appears as the class group of a quadratic fie
 ld. Postulated just over 30 years ago\, the heuristic helped explain in a 
 very compelling and intuitive way various phenomena in the study of class 
 groups that had been observed over the preceding decades and centuries. Ro
 ughly speaking\, the probability of an abelian group A occurring as the cl
 ass group of an imaginary quadratic field is inverse proportional to #Aut(
 A) - a random object tends to have few symmetries. Already for real quadra
 tic fields\, the weights are postulated to be different\, and the heuristi
 c explanation is less intuitive. Later\, the heuristic was extended by Coh
 en and Martinet to more general number fields\, this time without even an 
 attempt at an intuitive explanation of why these should be the right weigh
 ts. I will explain that in fact\, the intuitive heuristic that the rarity 
 of an algebraic object is proportional to the number of symmetries does ex
 plain distributions of class groups of arbitrary number fields and recover
 s all the above heuristics\, but the object one has to look at is not the 
 class group. This is still work in progress\, jointly with Hendrik Lenstra
 .
LOCATION:MR13
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