A message from the department of mindless q-staring. Here is what looks to me like the paper I have been waiting for a long time: Tim Huber: A theory of theta functions to the quintic base http://arxiv.org/abs/1304.0684 The AGM relations are given on p.11. Two more fine papers: http://arxiv.org/find/math/1/au:+Huber_T/0/1/0/all/0/1 Best, jj