Successive maxima of the non-genus part of class numbers

96 pages including large numerical tables and PARI programs ; Some PARI programs have bring out a property for the non-genus part of the class number of imaginary quadratic fields , with respect to (√D)^ε, where D is the absolute value of the discriminant and ε in ]0, 1[, in relation with the ε-conjecture. The general Conjecture 3.1, restricted to quadratic fields, states that, for ε in ]0, 1[, the successive maxima, as D increases, of H/(2^(N-1)√D^ε), 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. We prove, under an assumption (true for p=2, 3), that the sequence of successive maxima of H/(p^(N-1)√D^ε) is infinite (Theorem 2.5). 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 H/√D^ε are also analyzed.