Capitulation problem for global function fields

Abstract

Let q be a power of an odd prime number p, k = F(q)(t) be the rational function field over the finite field Fq. In this paper, we construct infinitely many real (resp. imaginary) quadratic extensions K over k whose ideal class group capitulates in a proper subfield of the Hilbert class field of K. The proof of the infinity of such fields K relies on an estimation of certain character sum over finite fields.