Exponents of Class Groups of Quadratic Function Fields over Finite Fields

Cardon, David A. ; Murty, M. Ram

Canadian mathematical bulletin, Tome 44 (2001) no. 4, pp. 398-407

Résumé

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

Détail

DOI : 10.4153/CMB-2001-040-0

Mots-clés : 11R58, 11R29, class number, quadratic function field