Spiegelungssatz: a combinatorial proof for the 4-rank
  The Spiegelungssatz is an inequality between the (4)-ranks of the narrow ideal class groups of the quadratic fields (\mathbb{Q}(\sqrt{D})) and (\mathbb{Q}(\sqrt{-D})). We provide a combinatorial proof of this inequality. Our interpretation gives an affine system of equations that allows to describe precisely some equality cases. 
Laurent Habsieger (ICJ), Emmanuel Royer
math.NT