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.