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

Successive maxima of the non-genus part of class numbers