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

By I. I Piatetskii-Shapiro

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Kaehler differentials

This e-book is predicated on a lecture direction that I gave on the college of Regensburg. the aim of those lectures used to be to give an explanation for the function of Kahler differential types in ring thought, to organize the line for his or her software in algebraic geometry, and to guide as much as a little research difficulties The textual content discusses nearly completely neighborhood questions and is accordingly written within the language of commutative alge- algebra.

Non-commutative Algebraic Geometry

This direction used to be learn within the division of arithmetic on the college of Washington in spring and fall 1999.

Additional resources for Automorphic Functions and the Geometry of Classical Domains (Mathematics and Its Applications)

Sample text

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.