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.