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

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      https://www.ias.ac.in/article/fulltext/pmsc/123/01/0001-0018

    • Keywords

       

      Divisibility; class numbers; quadratic extensions; function fields.

    • Abstract

       

      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 $\geq 3$, we show that for every $\epsilon > 0$, there are $\gg q^{L\left(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon\right)}$ polynomials $f\in \mathbb{F}_q[x]$ 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 $q^{L\left(\frac{1}{2}+\frac{1}{g}\right)}$ of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.

    • Author Affiliations

       

      Pradipto Banerjee1 Srinivas Kotyada2

      1. Indian Statistical Institute, Stat-Math Unit, 203 Barrackpore Trunk Road, Kolkata 700 108, India
      2. Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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