Commensurability of automorphism groups, and number theoretic applications

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• Alex Bartel (Warwick University)
• Tuesday 24 November 2015, 14:15-15:15
• MR13.

If you have a question about this talk, please contact Jack Thorne.

There is a general philosophy that if a family of algebraic objects behaves randomly, then the probability that an object from this family 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 groups of imaginary quadratic number fields. That so-called Cohen-Lenstra heuristic was later extended to other families of number fields, at which point 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 the 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 the end. This is joint work with Hendrik Lenstra.

This talk is part of the Number Theory Seminar series.

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