ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11, 47 Uetake, Yoichi, Adam Mickiewicz University, Poznań, Poland A weak resolvent representation of a scattering matrix for automorphic functions A weak resolvent of a bounded operator A on a Hilbert space H is the complex-valued function of the form c((zI − A)^{−1}b), where b ∈ H and c ∈ H^{∗}. We represent a scattering matrix of a discrete-time Lax-Phillips scattering system, in this form, and show that this representation has some cyclic properties. We also treat the case where the incoming and outgoing subspaces D_{−} and D_{+} are not necessarily orthogonal. We examine the scattering matrix for an automorphic wave equation on the fundamental domain P SL(2, Z) \ H. In this case, it can be shown that the zeros of the Riemann ζfunction correspond to the eigenvalues of A. Then the Riemann Hypothesis is equivalently expressed in terms of cyclicity.