Were those great planes anything like the ground we have here in mathoverflow?

Since the relevance is for proving Leopoldt, may I ask the opinion of people about the following paper? arxiv.org/abs/0905.1274

In your statement there is no dependence anywhere on an underlying field. What is the connection with fields? Is this the von Staudt-Clausen for some $p$-cyclotomic field?

@Pete Clark. Perhaps for the situation of moduli of elliptic curves the expose of Deligne-Rapoport ought to be more accessible than Katz-Mazur.

@Buzzard. I plus-ed your comment for the remark on Pete Clark waking up.

So what is the other Bloch-Kato?

It was indeed this Bloch-Kato that I had in mind. I heard it in a seminar, but didn't absorb neither the complete statement nor the references.

But is the Tate curve an elliptic curve(ie a curve of genus one with a fixed point)?

The Tate Curve! :)

Also I have referred to the simpler books of Miranda and Narasimhan, with the questioner in mind. These books use only complex analysis of one variable, and the definition of genus from topology.

Not to mention that Riemann-Roch would give the group law over any field/ring/scheme. You have done some handwaving with the the Lefschetz principle for transporting to over other fields. This is not quite enough. Riemann-Roch is much neater.

Riemann-Roch would tell you why there is a group law precisely on curves of genus 1. Also, Riemann-Roch is obvious more profound than just looking at a torus or Weierstrass elliptic functions on the complex plane.

@Anton. I imagine, the translation in future instances would be easier for common users by capturing the LaTeX from the original post. How do you do this? Please explain on meta.

You missed the whole central idea which the the deepest: The Riemann-Roch theorem gives the group law.

I suppose, you can use $n! + 1$ instead in the exponent. Then the derivative does not even converge on the boundary. This makes things simpler.