Zeta-functions of binary Hermitian forms and special values of Eisenstein series on three-dimensional hyperbolic space

Abstract

The present work contains a systematic investigation of zeta-functions of binary Hermitian forms over the ring o of integers of an imaginary quadratic number field K = tl~(]/~) with discriminant D < 0. The resulting relations between these zetafunctions have numerous applications to the computation of special values of the Eisenstein series for the group PSL(2,o) acting on the three-dimensional hyperbolic space I-I. In particular we reprove many known formulae for representation numbers for quaternary quadratic forms over Z, and we obtain many explicit new results for such representation numbers. In Sect. 2 we develop a theory of representations of numbers by binary Hermitian forms over o which parallels the classical theory of representations of integers by binary quadratic forms over ~. A key result is a certain bijection described in Theorem 2.3. This bijection means that the sum of the numbers of representations [modulo SL(2, o)-units] of an integer k # 0 by a set of representatives of the SL(2, o)-classes of binary Hermitian forms over o with discriminant A is equal to the number of cosets 2 +ko with 2~ o, 22-+ A = 0 m o d k . The number of these cosets is explicitly computed in Sect. 2 in all cases. Hence a certain mean value of the representation numbers is known whereas simple formulae for the individual representation numbers do not exist. We apply the formulae of Sect. 2 to the study of zeta-functions of (definite or indefinite) Hermitian forms. For simplicity we restrict in this introduction to the case of positive definite forms over o although the results of Chap. I are valid in the indefinite case as well. Let f be a positive definite binary Hermitian form over o with SL(2, o)-unit group gl(f) . Then we define