Counting rational points of hyperelliptic curves over finite fields

Abstract

Given a finite field F_q with a prime power q, one can ask how many points an hyperelliptic curve of a large fixed "degree" d > 0 has. It is difficult to answer this question in general, so we can consider a probabilistic answer instead. Such an answer was previously obtained by Kurlberg and Rudnick, precisely when d goes to infinity. This was first generalized by Bucur, David, Feigon, and Lalin for p-fold cyclic covers of the line and later by Cheong, Wood, and Zaman. The two generalizations are different from each other because the limits are taken differently. A main goal of the thesis is a heuristic attempt to give a generalization of these two as a conjecture and solve more cases of it.