Indivisibility of class numbers of imaginary quadratic function fields

Abstract

We show that for an odd prime number l, there are infinitely many imaginary quadratic extensions F over the rational function field K = Fq(T ) such that the class number of F is not divisible by l. This work is published in Acta Arithmetica 132.4 (2008). Let p be an odd prime number, q a power of p and Fq the finite field with cardinality q. Let T be an indeterminate and K = Fq(T ) the rational function field. Let l be an odd prime number. Friesen [3], Cardon and Murty [1], respectively, proved that there are infinitely many real and imaginary, respectively, quadratic extensions F over K such that the class number of F is divisible by l. In [6], Kimura proved that there are infinitely many quadratic extensions F over K such that the class number of F is not divisible by 3. For an odd prime number l, Ichimura [5] constructed infinitely many imaginary quadratic extensions F over K such that the class number of F is not divisible by l, when the order of q mod l in the multiplicative group (Z/lZ)∗ is odd or l = p. In this talk, we shall show the following theorem. Theorem 0.1 Let l be an odd prime number. Then there are infinitely many imaginary quadratic extensions F over K such that the class number of F is not divisible by l. Theorem 0.1 is a function field analogue of Hartung’s work [4] on the imaginary quadratic number fields. To prove this theorem, following Hartung’s