Meromorphic functions admitting an algebraic addition theorem (Q1976582)

The author proves a statement about meromorphic functions which was already formulated by Weierstraß, but it seems that a concise proof has not yet been published. Let \(\mathcal{M}\)\((\mathbb C^{n})\) be the meromorphic function field of \(\mathbb C^{n}\) and \(K\) a subfield of \(\mathcal M\)\((\mathbb C^{n})\) of transcendence degree \(n\) over \(\mathbb C\). Assume that \(K\) admits an algebraic addition theorem in the following sense: For at least one system \(f_{0},f_{1},\dots,f_{n}\) of generators of \(K\) (i.e. for every such system) there exist non-zero rational functions \(R_{j}\) such that \[ f_{j}(x+y)=R_{j}(f_{0}(x),f_{1}(x),\dots,f_{n}(x),f_{0}(y),f_{1}(y),\dots,f_{n}(y)) \] for all \(x,y\in \mathbb C^{n}\) and \(j=0,1,\dots,n\). Assume moreover that \(K\) contains an element which depends on no less than \(n\) variables, even after an arbitrary invertible linear transformation of \(\mathbb C^{n}\). Then \(K\) is the meromorphic function field of a projective variety which contains an abelian complex Lie group \(\mathbb C^{n}/\Gamma=\mathbb C^{p}\times(\mathbb C^{*})^{q}\times A\), \(A\) an abelian variety, as a Zariski dense open subset. Moreover every element of \(K\) can be extended to a meromorphic function on \(\mathbb P_{1}^{p+q}\times A\). The problem can be formulated as an extension problem for holomorphic line bundles on quasi-abelian varieties. For his solution the author uses a detailed description of those bundles by their generalized theta factors.