30) American Mathematical Society, 2012) in the area of arithmetic dynamics. , Springer-Verlag, N.Y., 2007) and a research monograph ( His current research program in arithmetic dynamics includes work on arithmetic and dynamical degree. Among Professor Silverman's contributions are construction of global and local dynamical height functions (with Greg Call), finiteness of integral points in orbits, construction of dynamical units (with Patrick Morton), finiteness of rational periodic points for Henon maps, construction of dynamical moduli spaces, and the Morton-Silverman Conjecture on uniform boundedness of algebraic periodic points.
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Professor Silverman is one of the founders of this field and has many published research papers and graduate students in the area. The more recent field of arithmetic dynamics attempts to understand the arithmetic properties of iteration of polynomial or rational functions applied to rational or algebraic points on a variety.
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Professor Silverman has also written several books on arithmetic geometry at the advanced graduate and research level:Īdvanced Topics in the Arithmetic of Elliptic Curvesĭiscrete dynamical systems over the real and complex numbers have a long and distinguished history. Independence of Heegner points defined over distinct quadratic imaginary fields (joint work with Micheal Rosen). Upper and lower bounds for divisibility sequences on tori, elliptic curves, and abelian varieties and the relation of such bounds to Vojta's conjecture. Lower bounds for heights over abelian extensions and Lehmer's conjecture on abelian varieties (joint work with Matt Baker). Limit formulas for the variation of the canonical height in families of abelian varieties, with applications to the injectivity of the specialization map.Īpplication of the abc conjecture to Wieferich primes.Ī limit formula for the rank of an elliptic surface (joint work with Micheal Rosen) and application to the variation of the rank of specializations.
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This joint work with Marc Hindry answers questions raised by Serge Lang. Lower bounds for canonical heights and application to uniform bounds for the number of integral points on elliptic curves in terms of the rank of the Mordell-Weil group. A short list of some of his results includes: Professor Silverman's primary area of research is arithmetic geometry, especially the existence and distribution of integral and rational points on (elliptic) curves and on higher dimensional (abelian) varieties.