Topics in Complex Function Theory, Abelian Functions and Modular Functions of Several Variables
Develops the higher parts of function theory in a unified presentation. Starts with elliptic integrals & functions & uniformization theory, continues with automorphic functions & the theory of abelian integrals & ends with the theory of abelian functions & modular functions in several variables. The last topic originates with the author & appears here for the first time in book form.
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The preparation theorem
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Abelian functions Abelian varieties abs CW Akad algebraic variety Amer arbitrary assume automorphic functions bounded Chapter 3 Volume coefficients columns compact condition constant coordinates corresponding defined degree denote diagonal divisor Eisenstein series Ek(W elements elliptic modular entire function equation exponent system factor system finitely fixed follows formula function field fundamental region Furthermore given Hence Hermitian implies inequalities integral intersection invertible Jacobian functions lattice linear transformation mapping Math meromorphic function modular forms modular functions modular group modular matrix modular substitution Modulformen Modulfunktionen multiplication n-disk natural number neighborhood nondegenerate obtain period matrix Picard variety polynomial positive number power series proof of Theorem proved quotient rational functions regular function relation relatively prime respect Riemann rows satisfied Section space sufficiently small suitable symplectic theory theta functions Uber unimodular uniquely determined unit vanish identically view of Theorem yields zero