Journal of the Ramanujan Mathematical Society

Joseph H. Silverman ¹

Affiliations
1 Mathematics Department, Box 1917, Brown University, Providence, RI 02912, US

Abstract

Let φ: W→V be a dominant rational map between quasi-projective varieties of the same dimension. We give two proofs that h_V (φ(P))≫h_W(P) for all points P in a nonempty Zariski open subset of W. For dominant rational maps φ: Pⁿ→Pⁿ, we give a uniform estimate in which the implied constant depends only on n and the degree of φ. As an application, we prove a specialization theorem for equidimensional dominant rational maps to semiabelian varieties, providing a complement to Habegger’s recent theorem on unlikely intersections.

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Joseph H. Silverman ¹, Nelson Stephens ²

Affiliations
1 Mathematics Department, Box 1917, Brown University, Providence, RI 02912, US
2 Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, GB

Abstract

An elliptic divisibility sequence (EDS) is a sequence of integers (W_n)_n≥0 generated by the nonlinear recursion satisfied by the division polyomials of an elliptic curve. We give a formula for the sign of W_n for unbounded nonsingular EDS, a typical case being Sign (W_n)=(-1) ^└nβ┘ for an irrational number β∈ℝ. As an application, we show that the associated sequence of absolute values (|W_n|) cannot be realized as the fixed point counting sequence of any abstract dynamical system.

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