Finiteness and the Counting Problem for Galais representations

(1)证明了当“简化的Serre权”和p外的Artin导体都很小时，2 <或等于p <或等于31的素数p的模p伽罗瓦表示的有限性和不存在性。这些结果似乎是用我们的方法得到的最好结果。(2)证明了Fontaine-Mazur有限猜想的潜在阿贝尔情形。即，在给定维数、有界惯性水平和给定Hodge-Tate型的有序数上，有限次数域的半简单潜在阿贝尔几何p进表示的同构类只存在有限多个。(3)证明了模1伽罗瓦表示的归纳公式。(4)改进了T.Satoh在有限域F_<pN>上快速计算椭圆曲线有理点数的p进方法。算法运行时间为0 (N^<2μ+0.5>)，内存为0 （N^2）。

(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.