Modular curves and rigid analytic spaces

Brian Conrad (University of Michigan)

In the study of congruence properties of classical
modular forms, it is useful (following Katz) to work
with modular curves over p-adic fields --- but with
these curves viewed as rigid-analytic curves rather
than as algebraic curves. This point of view remains
of essential interest; for example, it was used in
work of Buzzard/Taylor on Galois representations
attached to weight 1 forms. In both the complex-analytic
and algebraic cases, the relevance of the geometry
of modular curves in the study of modular forms is
the moduli space property of these curves.

A natural question therefore arises: are the
rigid-analytic counterparts of algebraic modular
curves actually moduli spaces in the rigid-analytic
category? The answer is "yes", but the proof is
surprisingly non-trivial. In particular, the
method used in the complex-analytic case (exponential
uniformization and upper half-plane models) is useless
in the non-archimedean case, so we must instead exploit
Raynaud's theory of formal models and Grothendieck's
formal GAGA theorems. No prior knowledge of rigid
analytic geometry will be assumed.