Proves that lattices inside locally algebraic types from U(2)-arithmetic manifold cohomology depend only on the Galois representation at p for higher weights small relative to p, and that patched modules with irreducible cosocle are cyclic.
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Breuil's Lattice Conjecture for GL2(K)
Proves that lattices inside locally algebraic types from U(2)-arithmetic manifold cohomology depend only on the Galois representation at p for higher weights small relative to p, and that patched modules with irreducible cosocle are cyclic.