Integral Binary Hermitian Forms

Abstract

Here we include some classical results from the theory of binary hermitian forms which originate from Hermite (1854). We discuss the reduction theory of binary hermitian forms as described for example in Bianchi (1892). Eventually our considerations lead to Humbert’s computation of the covolume of SL(2, Thuong) where Thuong is the ring of integers in an imaginary quadratic number field. The work of Humbert on hermitian forms is contained in his papers (1915), (1919a)—(1919e). It contains an interesting error, we correct it in Section 9.6. We also develop a theory of representation numbers of binary hermitian forms which is analogous to the theory of binary quadratic forms as in Landau (1927).