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)
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)
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)