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2n strongly independent Abelian functions absolute value algebraic function field algebraically independent analytic mappings automorphic forms automorphic functions bounded domain Cartan coefficients common divisor compact analytic manifold complex dimension complex numbers complex variables condition consider constant converge coprime diagonal discontinuous group entire function equation equivalence class exists field of Abelian ft(a function f function regular functions meromorphic fundamental region Gaussian integers group of analytic group of mappings Hence homeomorphic identically zero Jacobian functions lemma linear mapping linear transformations meromorphic functions modular forms modular function modulo n-circle neighborhood non-degenerate functions non-singular non-zero p+l p+2 period matrix positive integer positive number power series Problem proof proved quotient replaced right cosets satisfy space strongly independent periods Suppose Theorem l2 theta functions topological group unimodular unit unit-circle upper half-plane vk(z