P-adic Teichmüller Theory

In mathematics, ''p''-adic Teichmüller theory describes the "uniformization" of ''p''-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfaces and their moduli. It was introduced and developed by .

The first problem is to reformulate the Fuchsian uniformization of a complex Riemann surface (an isomorphism from the upper half plane to a universal covering space of the surface) in a way that makes sense for ''p''-adic curves. The existence of a Fuchsian uniformization is equivalent to the existence of a canonical indigenous bundle over the Riemann surface: the unique indigenous bundle that is invariant under complex conjugation and whose monodromy representation is quasi-Fuchsian. For ''p''-adic curves, the analogue of complex conjugation is the Frobenius endomorphism, and the analogue of the quasi-Fuchsian condition is an integrality condition on the indigenous line bundle. So in ''p''-adic Teichmüller theory, the ''p''-adic analogue the Fuchsian uniformization of Teichmüller theory, is the study of integral Frobenius invariant indigenous bundles.

See also

*Inter-universal Teichmüller theory
*Anabelian geometry
* nilcurve

References

*
*
*

Algebraic geometry
Number theory
p-adic numbers

{{numtheory-stub