3 The minus class-number criterion
This chapter proves the analytic formula for the relative class number,
\[ h^- = 2p \prod _{\chi \ \mathrm{odd}} \Bigl(-\tfrac 12 B_{1,\chi ^{-1}}\Bigr), \]
and reduces it modulo \(p\) to the Bernoulli-numerator criterion for \(p \mid h^-\). The reduction proceeds in two steps: the odd characters modulo \(p\) are indexed by odd powers of the Teichmüller character, and each generalised Bernoulli value \(B_{1,\omega ^j}\) is replaced by the congruent classical quotient \(B_{j+1}/(j+1)\).