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In trying to get into the topic of moduli spaces of elliptic curves, the following question arises:

What is the state of the art in the topic right now?

Deligne and Rapoport describes how the moduli problems with level-$N$ structure each have a model over $\mathbb Z[1/N]$. This is according to Igusa. Next, in the preface to Katz and Mazur's book, some work of Drinfeld is mentioned and the book's aim is to apparently re-write the Deligne-Rapoport construction based on Drinfeld's work.

Here I am a bit confused. What does this given in addition? There is already a smooth model over $\mathbb Z[1/N]$; so what does Katz-Mazur give in addition?

Next, are there any advances in this topic after Katz-Mazur?

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  • $\begingroup$ Perhaps better suited to mathoverflow.net ? $\endgroup$ – lhf Oct 16 '13 at 3:37
  • $\begingroup$ @lhf: I would rather not go over there, having had very bad personal experiences. You always dread castigation and fear that your question will be closed. This site is much more open, friendly and receptive. Not to mention, many of the very same experts from there frequent this site too. This site is supposed to welcome questions at all levels, is it not? Or was it just lip service? $\endgroup$ – Ellipseman Oct 16 '13 at 3:45
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Your sketch of the history is rather misleading: you make it sound like everything in Katz--Mazur is a rewrite of Deligne--Rapoport, and everything in Deligne--Rapoport is a rewrite of Igusa! As far as I (not an expert) understand, the main contribution of Katz--Mazur was to define models over $\mathbf{Z}_p$ for $p$ dividing the level, which are not smooth of course, but still have some good moduli interpretation (in terms of Drinfeld level structures). Deligne--Rapoport don't do this: they do define a $\mathbf{Z}_p$-model for $X_0(p)$ but it doesn't represent a functor in any obvious way, and they don't do higher $p$-power levels.

As for current work in the area: there is a lot of literature on finding semistable models of modular curves (e.g. Coleman and McMurdy have a series of papers on this); and there is a lot of very deep work extending the results from modular curves to more general Shimura varieties, e.g. Kai-Wen Lan's Harvard thesis (published as a monograph "Arithmetic compactifications of PEL Shimura varities").

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