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Let N q (g) denote the maximal number of F q-rational points on any curve of genus g over F q. Ihara (for square q) and Serre (for general q) proved that lim sup g→∞ N q (g)/g > 0 for any fixed q. In their proofs they constructed curves with many points in infinitely many genera; however, their sequences of genera are somewhat sparse. In this paper, we(More)
For any elements a, c of a number field K, let Γ(a, c) denote the backwards orbit of a under the map fc : C → C given by fc(x) = x 2 + c. We prove an upper bound on the number of elements of Γ(a, c) whose degree over K is at most some constant B. This bound depends only on a, [K : Q], and B, and is valid for all a outside an explicit finite set. We also(More)
We study the orbits of a polynomial f ∈ C[X], namely the sets {α, f (α), f (f (α)),. .. } with α ∈ C. We prove that if two nonlinear complex polynomials f, g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C d with a d-tuple of orbits of nonlinear polynomials, and we(More)