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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)
In his Ph.D. Thesis of 1897, Dickson introduced certain permutation polynomials whose Galois groups are essentially the dihedral groups. These are now called Dickson polynomials of the rst kind, to distinguish them from their variations introduced by Schur in 1923 which are now called Dickson polynomials of the second kind. In the last few decades there(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 resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c q with the following property: for every integer g ≥ 0, there is a genus-g curve over F q with at least c q g rational points over F q. Moreover, we show that(More)