Counting rational points of hyperelliptic curves over finite fields

finite fields;rational points;hyperelliptic curves;zeta functions;510

자연과학대학 수리과학부 ;

University:서울대학교 대학원

关键词: finite fields;    rational points;    hyperelliptic curves;    zeta functions;    510;   

Others  :  http://s-space.snu.ac.kr/bitstream/10371/131498/1/000000028726.pdf

美国|英语

来源: Seoul National University Open Repository

【 摘 要 】

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.