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SUMMARY:An explicit Chebotarev Density Theorem on average and applications
DTSTART:20230228T150000
DTEND:20230228T160000
DTSTAMP:20260407T145919Z
UID:c549f71af7602b9c2e95e97b823cce51fe88c286f0cff897bda4d110
CATEGORIES:Conferences - Seminars
DESCRIPTION:Ilaria Viglino\, ETHZ\nThe study of specific families of polyn
 omials and their splitting fields can provide useful examples and evidence
 s for conjectures regarding invariants related to number field extensions 
 and related objects. The main example that can illustrate the relevance of
  the work\, is the family P0 n\,N of degree n monic polynomials f with int
 eger coefficients\, so that the maximum of the absolute values of the coef
 ficients is less or equal than N\, and the splitting field Kf over the rat
 ionals Q is the full symmetric group Sn. We let N → +∞. With a little 
 work\, this can actually be generalized to polynomials with integral coeff
 icients in a fixed number field of degree d over Q. Let π_f\,r(x) be the 
 function counting the primes less or equal than x such that f ∈ P0n\,N h
 as a fixed square-free splitting type r modulo p. It turns out that the qu
 antity (π_f\,r(x)−δ(r)π(x)((δ(r)−δ(r)2)π(x))^(−1/2) is distrib
 uted like a normal distribution with mean 0 and variance 1\, whenever x is
  small compared to N\, e.g. x = N1/ log logN. Here δ(r) is the coefficien
 t in the asymptotic predicted by the classical Chebotarev theorem. This re
 sult leads to interesting applications\, as finding upper bounds for the t
 orsion part of the class number in terms of the absolute discriminant\, as
  it was done for other infinite families by Ellenberg\, Venkatesh and Heat
 h-Brown\, Pierce.
LOCATION:MA A3 30 https://plan.epfl.ch/?room==MA%20A3%2030
STATUS:CONFIRMED
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