Exponents of Class Groups of Quadratic Function Fields over Finite Fields

http://dx.doi.org/10.4153/CMB-2001-040-0
Canad. Math. Bull. 44(2001), 398-407

  Published:2001-12-01
 Printed: Dec 2001

We find a lower bound on the number of imaginary quadratic extensions of the function field $\F_q(T)$ whose class groups have an element of a fixed order. More precisely, let $q \geq 5$ be a power of an odd prime and let $g$ be a fixed positive integer $\geq 3$. There are $\gg q^{\ell (\frac{1}{2}+\frac{1}{g})}$ polynomials $D \in \F_q[T]$ with $\deg(D) \leq \ell$ such that the class groups of the quadratic extensions $\F_q(T,\sqrt{D})$ have an element of order~$g$.