Summary: Class number in totally imaginary extensions
of totally real function fields
Abstract. We show that, up to isomorphism, there are only finitely
many totally real function fields which have any totally imaginary extension
of a given ideal class number.
The case of imaginary quadratic extensions of Fq(X) with ideal class
number one is quite known : we know that there are only four (see [M]). The
real quadratic case is completely different since there are infinitely many
such fields (see [S]). The regulator in the last case is a hard parameter
to deal with. The situation that we are interested in, is totally imaginary
extensions of totally real extensions of the rational function field Fq(X).
In our situation, the problem of each regulator subsists but we can easily
compute the quotient of them : we show that it is essentially the index of
units of their rings of integers. After that, we prove the divisibilty in the
general case of the divisor class numbers in a finite separable extension of
function fields. This result, with the Riemann Hypothesis allow us to show
the finiteness of the number of such function fields if we fix the ideal class
number of the imaginary field.