Successive maxima of the non-genus part of class numbers

TL;DR

This paper investigates the behavior of the non-genus part of class numbers in quadratic and higher degree fields, proposing conjectures and providing computational evidence for the occurrence of successive maxima related to prime discriminants.

Contribution

It introduces a conjecture about the maxima of class number ratios in quadratic fields and extends the analysis to cyclic and abelian fields of various degrees, with proofs under certain assumptions.

Findings

Successive maxima occur only for prime discriminants in quadratic fields.

The sequence of maxima of the class number ratio is infinite under certain conditions.

Computational evidence supports the conjecture across various field degrees.

Abstract

Some PARI programs have bringed out a property for the non-genus part of the class number of the imaginary quadratic fields, with respect to $(\sqrt D\,)^{\varepsilon}$, where $D$ is the absolute value of the discriminant and $\varepsilon \in ]0, 1[$, in relation with the $\varepsilon$-conjecture. The general Conjecture 3.1, restricted to quadratic fields, states that, for $\varepsilon \in ]0, 1[$, the successive maxima, as $D$ increases, of $\frac{H}{2^{N-1} \cdot (\sqrt D\,)^{\varepsilon}}$, where $H$ is the class number and $N$ the number of ramified primes, occur only for prime discriminants (i.e., $H$ odd); we perform computations giving some obviousness in the selected intervals. For degree $p>2$ cyclic fields, we define a "mean value" of the non-genus parts of the class numbers of the fields having the same conductor and obtain an analogous property on the successive maxima. In Theorem 2.5 we prove, under an assumption (true for $p=2, 3$), that the sequence of successive maxima of $\frac{H}{p^{N-1} \cdot (\sqrt D\,)^{\varepsilon}}$ is infinite. Finally we consider cyclic or non-cyclic abelian fields of degrees $4, 8, 6, 9, 10, 30$ to test the Conjecture 3.1. The successive maxima of $\frac{H}{(\sqrt D\,)^{\varepsilon}}$ are also analyzed.