Automorphic Functions and the Geometry of Classical Domains by I. I Piatetskii-Shapiro

By I. I Piatetskii-Shapiro

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Now let w1 = (Zl' Ul' t 1 ) and W2 = (Z2' U2, t 2) be two arbitrary points in S. We will now prove that there is an analytic automorphism of our domain S that maps one of these points onto the other. , an automorphism of the form (4). The theorem is proved. We will now find the form of an element dv of an invariant volume. (z, u,t) dx dy dU 1 dU2 dtl dt 2, Li2 = Imu, dx = dXl ... dx ll , tl = Ret, t2 = Imt, dy = dYl ... dYIl' (21) 39 SIEGEL DOMAINS The existence of "parallel translations" implies that A(Z, u, t) = A(Im Z - Re Lt(u, u), t).

We have proved that any Siegel domain of genus 2 is analytically equivalent to a bounded domain in C"+ m• For the domain S, the part of the boundary that consists of the points (z, u) such that 1m Z = F(u, u) is called the skeleton. It is possible to show that every function that is analytic in S and whose modulus has a maximum in S has at least one maximum-modulus point on the skeleton. On the the other hand, it is easy to construct an example of function whose modulus has a maximum at a preassigned point on the skeleton.

These domains appear as a result of the following fact. , it contains analytic "subvarieties" of various dimensions. In the theory of automorphic functions of several complex variables it is important to consider the passage to the limit that results in a point inside a domain approaching a boundary point belonging to some analytic "subvariety". Siegel domains of genus 3 are very convenient for studying such a transition to the limit. Chapter 5 contains applications of Siegel domains of genus 3 to the theory of bounded homogeneous domains.

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