F. Herzig

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Suppose that G is a connected reductive group over a p-adic field F , that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Sa-take isomorphism for the Hecke algebra of compactly supported, K-biequivariant functions f :(More)
We present a Serre-type conjecture on the modularity of four-dimensional symplectic mod p Galois representations. We assume that the Galois representation is irreducible and odd (in the symplectic sense). The modu-larity condition is formulated using thé etale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p. We(More)
Let l be a prime, and let Γ be a finite subgroup of GL n (F l) = GL(V). With these assumptions we say that Condition (C) holds if for every irreducible Γ-submodule W ⊂ ad 0 V there exists an element g ∈ Γ with an eigenvalue α such that tr e g,α W = 0. Here, e g,α denotes the projection to the generalised α-eigenspace of g. This condition arises in the(More)
The notion of adequate subgroups was introduced by Jack Thorne [42]. It is a weakening of the notion of big subgroups used in generalizations of the Taylor-Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] that if the dimension is(More)