Reduction theorems for the Brauer conjecture on the number of characters in a $p$-block

Theorems are proved that reduce the proof of the Brauer conjecture for finite groups $G$ with a $p$-soluble centralizer of a $p$-element to the evaluation of the minimum of a suitable positive definite quadratic form, whose matrix is given in terms of the Cartan matrix of a $p$-block of a group of simpler structure than $G$.