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SUMMARY:The arithmetic of dynamical systems - Professor J. Silverman (Brow
 n University)
DTSTART:20070220T170000Z
DTEND:20070220T180000Z
UID:TALK6455@talks.cam.ac.uk
CONTACT:4993
DESCRIPTION:Classical discrete dynamics is the study of iteration of self-
 maps of a real or complex object\, for example iteration of a morphism $\\
 f:V(\\CC)\\to V(\\CC)$ on the complex points of an algebraic variety. In t
 his setting it turns out that there are natural dynamical analogues to man
 y fundamentals theorems and conjectures in the theory of Diophantine equat
 ions and arithmetic geometry.  The purpose of this talk is to describe thi
 s Diophantine/dynamical analogy and to highlight recent progress and open 
 questions in this comparatively new subject\, concentrating for concretene
 ss on the case of a projective morphism $ \\f:\\PP^N \\to \\PP^N $ defined
  over $\\QQ$.  Among the topics are:\n\n* Rationality of Periodic Points.\
 nTo what extent can the periodic or preperiodic points of~$\\f$ be $\\QQ$-
 rational?  The Diophantine analogue is $\\QQ$-rationality of torsion point
 s on elliptic curves (Mazur\, Merel) and abelian varieties.\n\n* Integral 
 Points in Orbits.\nTo what extent can the forward $\\f$-orbit of a point c
 ontain infinitely many integer points? The Diophantine analogue is integra
 l points on curves (Siegel) and on higher dimensional varieties (Faltings\
 , Vojta).\n\n* Dynamical Moduli Spaces.\nTo what extent can the set of mor
 phisms $\\PP ^N \\to \\PP^N$ modulo "dynamical isomorphism" be given  a (n
 ice) algebraic structure.\nThe Diophantine analogues are elliptic modular 
 curves and moduli spaces for abelian varieties.\n
LOCATION:Wolfson Room (MR 2) Centre for Mathematical Sciences\, Wilberforc
 e Road\, Cambridge
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