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SUMMARY:Commensurability of automorphism groups\, and number theoretic app
 lications - Alex Bartel (Warwick University)
DTSTART:20151124T141500Z
DTEND:20151124T151500Z
UID:TALK61452@talks.cam.ac.uk
CONTACT:Jack Thorne
DESCRIPTION:There is a general philosophy that if a family of algebraic ob
 jects behaves randomly\, then the probability that an object from this fam
 ily is isomorphic to a given object A is inverse proportional to #Aut(A). 
 This has first been observed by Cohen and Lenstra in the case of class gro
 ups of imaginary quadratic number fields. That so-called Cohen-Lenstra heu
 ristic was later extended to other families of number fields\, at which po
 int much less naturally looking probability weights started occurring. It 
 turns out that if instead of class groups\, one talks about Arakelov class
  groups\, then the original heuristic holds in great generality\, provided
  one can make sense of "inverse proportional to #Aut(A)" in cases where th
 e automorphism group is infinite. In this talk I will present a theory of 
 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 th
 e end. This is joint work with Hendrik Lenstra. 
LOCATION:MR13
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