Elliptic and abelian period spaces
In this article we determine the dimension of elliptic period spaces and period spaces for abelian surfaces. This continues work started by Th. Schneider in the thirties and having a long history since then. In particular we give an answer to a problem of Schneider. One of the key results—and this is the actual cause for writing this paper—is an application to so-called $\alpha $-curvature lines and geodesics on the three-axial ellipsoid. There irrationality of quotients of periods of the third kind on elliptic curves and of quotients of periods of the first kind on a hyperelliptic curve of genus 2 determines whether the geodesic flows are compact or not. Our main results show that the period spaces have the expected dimension, which gives the irrationality results needed. The methods used are a detailed study of extensions of elliptic curves by commutative linear algebraic groups, the study of the endomorphism algebra attached to such an extension and an appeal to the classification of endomorphism algebras in the abelian case. Crucial to the proofs is the use of the analytic subgroup theorem applied to groups which depend on the endomorphism structure.