The following question occurred to me while writing lectures last night and distracted me from productive work all morning. The least I can do is distract you all in turn.
Let be an elliptic curve over . Let and be two points of such that is not torsion. It is well known that there are no nonconstant meromorphic functions on which have neither poles nor zeroes in .
Are there any nonconstant holomorphic functions at at all, where we allow essential singularities at and ?