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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 class number, quadratic function field
MSC Classifications: 11R58, 11R29 show english descriptions Arithmetic theory of algebraic function fields [See also 14-XX]
Class numbers, class groups, discriminants
11R58 - Arithmetic theory of algebraic function fields [See also 14-XX]
11R29 - Class numbers, class groups, discriminants
 

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