Divisibility of Class Numbers of Imaginary Quadratic Function Fields by a Fixed Odd Number

Proceedings Mathematical Sciences

Srinivas Kotyada²  Pradipto Banerjee¹ 

[1] Indian Statistical Institute, Stat-Math Unit, 0 Barrackpore Trunk Road, Kolkata 00 0, India$$;Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 00 , India$$

关键词: Divisibility;    class numbers;    quadratic extensions;    function fields.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $mathbb{F}_q(x)$ whose class groups have elements of a fixed odd order. More precisely, for 𝑞, a power of an odd prime, and 𝑔 a fixed odd positive integer ≥ 3, we show that for every $epsilon > 0$, there are $gg q^{Lleft(frac{1}{2}+frac{3}{2(g+1)}-epsilonight)}$ polynomials $fin mathbb{F}_q[x]$ with $deg f=L$, for which the class group of the quadratic extension $mathbb{F}_q(x,sqrt{f})$ has an element of order 𝑔. This sharpens the previous lower bound $q^{Lleft(frac{1}{2}+frac{1}{g}ight)}$ of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.

【 授权许可】

Unknown