By T. E. Venkata Balaji
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Extra info for An Introduction to Families, Deformations and Moduli
2) shows M is of the form C/L(τ ). Therefore our classification problem for elliptic curves may be stated as follows. Parametrize all possible conformally distinct complex structures on the underlying topological space of a fixed complex torus. Given τ1 , τ2 ∈ U , it is not hard to find a diffeomorphism between the tori they define. Hence we may also say that we want to parametrize all possible conformally distinct complex structures on the underlying differentiable manifold of a fixed complex torus.
It turns out that if we consider the (local) multiplicities with which p takes a value in B, then p also takes each value in B, counting multiplicities, exactly n times. Hence such a proper nonconstant holomorphic map p : M −→ M is called an n-sheeted ramified holomorphic covering map. Let us apply the results above to a nonconstant meromorphic function on a compact Riemann surface. Incidentally, every known proof of the existence of nonconstant meromorphic functions on an arbitrary Riemann surface is nontrivial.
G has for its fundamental domain the 2n-dimensional real closed “parallelopiped” given by 2n F = (tj wj ) ; 0 ≤ tj ≤ 1 . j=1 F is clearly compact and connected. The canonical quotient φ : Cn −→ Cn /G is a covering of Lie groups. Tωn := Cn /G is an n-dimensional compact connected commutative complex Lie group called an n-dimensional complex torus and it has universal covering space Cn with deck transformation group (covering group) G. Note that π1 (Tωn ) ∼ = G. The n × 2n complex matrix whose entries are given by the components of the vectors in the R-basis ω, is called the period matrix associated to Tωn .