Vol. 1, No. 1, 2008

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Divisibility of class numbers of imaginary quadratic function fields

Adam Merberg

Vol. 1 (2008), No. 1, 47–58

We consider applications to function fields of methods previously used to study divisibility of class numbers of quadratic number fields. Let K be a quadratic extension of Fq(x), where q is an odd prime power. We first present a function field analog to a Diophantine method of Soundararajan for finding quadratic imaginary function fields whose class groups have elements of a given order. We also show that this method does not miss many such fields. We then use a method similar to Hartung to show that there are infinitely many imaginary K whose class numbers are indivisible by any odd prime distinct from the characteristic.

number theory, quadratic function fields, class numbers, class groups, divisibility
Mathematical Subject Classification 2000
Primary: 11R29
Secondary: 11R11
Received: 3 August 2007
Revised: 28 October 2007
Accepted: 28 October 2007
Published: 28 February 2008

Communicated by Ken Ono
Adam Merberg
Department of Mathematics
Brown University
151 Thayer Street
Providence, RI 02912
United States