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SUMMARY:Matthew de Courcy-Ireland (EPFL)
DTSTART:20220210T223000Z
DTEND:20220210T233000Z
DTSTAMP:20260423T024019Z
UID:JNTS/34
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/JNTS/34/">Ma
 rkoff graphs mod p</a>\nby Matthew de Courcy-Ireland (EPFL) as part of Col
 umbia CUNY NYU number theory seminar\n\n\nAbstract\nWe discuss a family of
  cubic surfaces defined by $$x^2+y^2+z^2=xyz+k$$ modulo prime numbers. The
  solutions form a graph\, where each vertex $(x\,y\,z)$ is joined to the o
 ther solution of the same quadratic in any of the three variables. These m
 oves are related to a nonlinear action of the modular group PGL(2\,$\\math
 bb{Z}$) on the surface. We outline some ways these equations arise\, and h
 ow we became interested in showing that the associated graphs cannot be em
 bedded in the plane. We describe constructions showing that the graphs for
  $k=0$ are not planar if the prime is congruent to 1 modulo 4\, or congrue
 nt to a quadratic residue 1\, 2\, or 4 modulo 7. We also sketch a proof of
  non-planarity for all sufficiently large primes.\n
LOCATION:https://researchseminars.org/talk/JNTS/34/
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