Paper 2, Section II, 23G

Riemann Surfaces
Part II, 2011

Let Λ\Lambda be a lattice in C\mathbb{C} generated by 1 and τ\tau, where τ\tau is a fixed complex number with non-zero imaginary part. Suppose that ff is a meromorphic function on C\mathbb{C} for which the poles of ff are precisely the points in Λ\Lambda, and for which f(z)1/z20f(z)-1 / z^{2} \rightarrow 0 as z0z \rightarrow 0. Assume moreover that f(z)f^{\prime}(z) determines a doubly periodic function with respect to Λ\Lambda with f(z)=f(z)f^{\prime}(-z)=-f^{\prime}(z) for all zC\Λz \in \mathbb{C} \backslash \Lambda. Prove that:

(i) f(z)=f(z)f(-z)=f(z) for all zC\Λz \in \mathbb{C} \backslash \Lambda.

(ii) ff is doubly periodic with respect to Λ\Lambda.

(iii) If it exists, ff is uniquely determined by the above properties.

(iv) For some complex number A,fA, f satisfies the differential equation f(z)=6f(z)2+Af^{\prime \prime}(z)=6 f(z)^{2}+A.