Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 13D / c

Choose a fundamental parallelogram with no pole on its boundary. Opposite sides are translates traversed in opposite directions, so periodicity givesThe residue theorem then givesA lone simple pole has a nonzero residue by definition of its order, so an elliptic function cannot have exactly one simple pole in a fundamental cell.