Finiteness and the Counting Problem for Galais representations

点击翻译按钮获取中文摘要

(1) We proved the finiteness and non-existence of mod p Galois representations for primes p with 2 【less than or equal】 p 【less than or equal】 31 when the "reduced Serre weight" and the Artin conductor outside p are small. These results seem to be best possible as results obtained by our methods.(2) We proved the potentially abelian case of the Finiteness conjecture of Fontaine-Mazur. Namely, there exist only finitely many isomorphism classes of semisimple potentially abelian geometric p-adic representations of a number field of finite degree over the rationals with a given dimension, bounded inertial level, and given Hodge-Tate type.(3) We proved the induction formula for mod l Galois representations.(4) We improved the p-adic method by T.Satoh of the fast computation of the number of rational points of an elliptic curve over a finite field F_<pN>. Our algorithm works in O(N^<2μ+0.5>) time and O(N^2) memory.