On the distribution of the number of points on algebraic curves in extensions of finite fields

**Speaker**: Dr. Omran Ahmadi (UCD)

**Time**: 4:00PM**Date**: Wed 7th April 2010

**Location**: Mathematical Sciences Seminar Room

**Abstract**

Let C be a smooth absolutely irreducible curve of genus g greater than or equal to 1 defined over F_q, the finite field of q elements. Let #C(F_{q^n}) be the number of F_{q^n}-rational points on C. Under a certain multiplicative independence condition on the roots of the zeta-function of C, we derive an asymptotic formula for the number of n =1, ..., N such that (#C(F_{q^n}) - q^n 1)/2gq^{n/2} belongs to a given interval I which is a subset of [-1,1]. This can be considered as an analogue of the Sato-Tate distribution which covers the case when the curve E is defined over Q and considered modulo consecutive primes p, although in our scenario the distribution function is different. This talk is based on a joint work with Igor Shparlinski.

(This talk is part of the K-Theory, Quadratic Forms and Number Theory series.)

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