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SUMMARY:Daniel Vargas-Montoya (University Lyon 1)
DTSTART:20210531T111500Z
DTEND:20210531T121500Z
DTSTAMP:20260423T041624Z
UID:WarsawNT/33
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/WarsawNT/33/
 ">Algebraicity modulo $p$ of G functions\, hypergeometric series and stron
 g Frobenius structure</a>\nby Daniel Vargas-Montoya (University Lyon 1) as
  part of Warsaw Number Theory Seminar\n\nLecture held in currently online.
 \n\nAbstract\nB. Dwork in his work about zeta function of a hypersurface o
 ver finite fields introduced the notion of strong Frobenius structure. In 
 this talk we are going to take up this notion for the study of algebraicit
 y modulo $p$ of  Siegel G functions\, where $p$ is a prime number. \nFirst
 ly\, we are going to see that if $f(t)$ is a power series (or Siegel G fun
 ction) with coefficients in the ring of integers $\\mathbb{Z}$ and if $f(t
 )$ is solution of a differential operator $L$ having strong Frobenius stru
 cture for $p$ of period $h$\, then the reduction of $f$ modulo $p$ is alge
 braic over $\\mathbb{F}_p(t)$ and its algebraicity degree is bounded by $p
 ^{n^2h}$\, where $n$ is the order of L and $\\mathbb{F}_p$ is the field of
  $p$ elements. Secondly\, we are going to show that\, under reasonable hyp
 otheses\, rigid differential operators have a  strong Frobenius structure 
 for almost every prime number $p$.\nFinally\,  we are going to illustrate 
 our results with several examples coming of hypergeometric series of type 
 ${}_nF_n-1$.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/33/
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