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SUMMARY:Galois Groups of D-finite Series modulo Primes [MPIM]
DTSTART:20250521T123000Z
DTEND:20250521T133000Z
DTSTAMP:20260514T102100Z
UID:indico-event-415@math-events.uni-bonn.de
DESCRIPTION:Speakers: Florian Fürnsinn (Universität Wien)\n\nNumber theo
 ry lunch seminar\nA power series with rational number coefficients is call
 ed D-finite if it satisfies a linear differential equation with polynomial
  coefficients over the rationals. Taking the reduction of such a D-finite 
 power series modulo a prime number\, one often obtains an algebraic power 
 series. For example\, for diagonals of multivariate rational functions thi
 s was observed by Furstenberg in 1967\, and for hypergeometric functions t
 his can be deduced from results by Christol\, and was made precise in rece
 nt work by Vargas-Montoya. In these cases\, one can consider the (usual\, 
 algebraic) Galois group of the reduction over the field of rational functi
 ons.In this talk\, I will showcase many examples of D-finite series of dif
 ferent nature for which we are (almost) able to compute said Galois groups
  for all prime numbers for which they are defined. I will then collect the
 se observations to raise questions on the general behavior of these groups
 : Is there some uniformity of the Galois groups of the reductions of a giv
 en D-finite series with respect to the prime number\, and does there exist
  an object in characteristic zero governing their behavior\, like the diff
 erential Galois group of the minimal equation for the series? Much like th
 e Grothendieck p-curvature conjecture\, we are looking for a local-global 
 principle\, but we are concerned with only a single solution of a differen
 tial equation\, instead of a full basis.This talk is based on joint work w
 ith X. Caruso and D. Vargas-Montoya.\n\nhttps://math-events.uni-bonn.de/ev
 ent/415/
LOCATION:MPIM\, Vivatsgasse\,  7 - Lecture Hall (Max Planck Institute for 
 Mathematics)
URL:https://math-events.uni-bonn.de/event/415/
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