Congruences dyadiques entre nombres de classes de corps quadratiques

Abstract

We prove here a general mod 16 congruence between the class numbers of the two fields $$\mathbb{Q}(\sqrt d )$$ and $$\mathbb{Q}(\sqrt { - d} )$$ , for any square free positive integer d. Similarly we obtain congruences mod 64 relating the class numbers of the quadratic extensions corresponding to the integers d, 2d, -d, -2d. We use the p-adic analytic class number formula, which leads to study the difference of the values at 0 and 1 of the KUBOTA-LEOPOLDT's p-adic L-function (for p=2, and for the quadratic character associated to the real quadratic field). We use then IWASAWA's series, conveniently described for our purpose by Barsky. This gives a new proof for mod 16 congruences formally or simultaneously obtained by various authors. Moreover we get new results mod 16 and mod 64. Part A) presents the method and results, Part B) the proofs. All the details can be found in [4].