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We provide a new aspect of the -adic Teichmüller theory established by Mochizuki. The formal stack classifying -adic canonical liftings of ordinary nilpotent indigenous bundles embodies a -adic analogue of uniformization of hyperbolic Riemann surfaces, as well as a hyperbolic analogue of Serre–Tate theory of ordinary abelian varieties. We prove a comparison theorem for the canonical symplectic structure on the cotangent bundle of this formal stack and Goldman’s symplectic structure. This result may be thought of as a -adic analogue of comparison theorems in the theory of projective structures on Riemann surfaces proved by Kawai and other mathematicians.
We investigate the irreducible cuspidal -representations of a reductive -adic group over a field of characteristic different from . In all known cases, such a representation is the compactly induced representation from a smooth -representation of a compact modulo centre subgroup of . When is algebraically closed, for many groups , a list of pairs has been produced, such that any irreducible cuspidal -representation of has the form , for a pair unique up to conjugation. We verify that those lists are stable under the action of field automorphisms of , and we produce similar lists when is no longer assumed algebraically closed. Our other main result concerns supercuspidality. This notion makes sense for the irreducible cuspidal -representations of , but also for the representations above, which involve representations of finite reductive groups. In most cases we prove that is supercuspidal if and only if is supercuspidal.
We prove a microlocal partition of energy for solutions to linear half-wave or Schrödinger equations in any space dimension. This extends well-known (local) results valid for the wave equation outside the wave cone, and allows us in particular, in the case of even dimension, to generalize the radial estimates due to Côte, Kenig and Schlag to nonradial initial data.
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