Class number formulas for certain bicyclic biquadratic number fields as finite sums involving Jacobi elliptic functions (Q2909033)

\textit{P. G. L. Dirichlet} [J. Reine Angew. Math. 24, 291--371 (1842; ERAM 024.0732cj)] transferred Gauss's theory of binary quadratic forms to forms with coefficients from the ring of Gaussian integers. In modern terms, one of his main results was the class number formula for extensions \(K = \mathbb Q(i,\sqrt{m}\,)\), which, up to a simple factor depending on the unit group, is the product of the class numbers of the quadratic subfields \(\mathbb Q(\sqrt{m}\,)\) and \(\mathbb Q(\sqrt{-m}\,)\). Dirichlet also pointed out that, just as the class number \(h\) of a real quadratic number field with fundamental unit \(\varepsilon\) satisfies an equation giving \(\varepsilon^h\) as a finite product of cyclotomic factors, there is a similar formula for the class number \(H\) of \(K\) in terms of factors depending on the division of the lemniscate. The necessary investigations were later published by \textit{P. Bachmann} [Math. Ann. 16, 537--550 (1880; JFM 12.0126.01)] and \textit{P. S. Nazimoff} (Nazimov) [Application of the theory of elliptic functions to number theory. (Russian) Moscow (1885; JFM 17.0139.01)]. In this article, the authors derive corresponding class number formulas for quadratic extensions of complex quadratic number fields with class number \(1\).