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# Exponents of Class Groups of Quadratic Function Fields over Finite Fields

Published:2001-12-01
Printed: Dec 2001
• David A. Cardon
• M. Ram Murty
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## Abstract

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~$g$.
 Keywords: class number, quadratic function field
 MSC Classifications: 11R58 - Arithmetic theory of algebraic function fields [See also 14-XX] 11R29 - Class numbers, class groups, discriminants