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Sep 18, 2013
09/13

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Florian Herzig

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We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n = 4 using automorphic inductions of Hecke characters.

Source: http://arxiv.org/abs/0803.0185v2

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Jul 19, 2013
07/13

by
Florian Herzig

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Let F be a finite extension of Q_p. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \bar F_p to be supersingular. We then give the classification of irreducible admissible smooth GL_n(F)-representations over \bar F_p in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel-Livne for n =...

Source: http://arxiv.org/abs/1005.1713v2

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64

Sep 19, 2013
09/13

by
Florian 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 Satake isomorphism for the Hecke algebra of compactly supported, K-biequivariant functions f: G(F) \to End V. These Hecke algebras were first considered by Barthel-Livne for GL_2. They play a role in the recent mod p and p-adic...

Source: http://arxiv.org/abs/0910.4570v2

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64

Sep 22, 2013
09/13

by
Florian Herzig; Jacques Tilouine

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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 modularity condition is formulated using the etale and the algebraic de Rham cohomology of Siegel modular varieties of level prime to p. We concentrate on the case when the Galois representation is ordinary at p and we give a corresponding list of Serre weights. When the representation is...

Source: http://arxiv.org/abs/0812.1525v4

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Sep 21, 2013
09/13

by
Matthew Emerton; Toby Gee; Florian Herzig

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We prove that for forms of U(3) which are compact at infinity and split at places dividing a prime p, in generic situations the Serre weights of a mod p modular Galois representation which is irreducible when restricted to each decomposition group above p are exactly those previously predicted by the third author. We do this by combining explicit computations in p-adic Hodge theory, based on a formalism of strongly divisible modules and Breuil modules with descent data which we develop in the...

Source: http://arxiv.org/abs/1106.4522v2

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Jun 28, 2018
06/18

by
Toby Gee; Florian Herzig; David Savitt

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We formulate a number of related generalisations of the weight part of Serre's conjecture to the case of GL(n) over an arbitrary number field, motivated by the formalism of the Breuil-M\'ezard conjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We generalise one of these conjectures to the case of connected reductive groups which are unramified over Q_p, and we also generalise the second author's previous conjecture for GL(n)/Q to this setting,...

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1509.02527

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6.0

Jun 30, 2018
06/18

by
Robert Guralnick; Florian Herzig; Pham Huu Tiep

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The notion of adequate subgroups was introduced by Jack Thorne [59]. It is a weakening of the notion of big subgroups used by Wiles and Taylor in proving automorphy lifting theorems for certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown in [22] and [23] that if the dimension is smaller than the characteristic then almost all absolutely irreducible representations are adequate. We extend the results by considering all...

Topics: Mathematics, Number Theory, Representation Theory, Group Theory

Source: http://arxiv.org/abs/1405.0043

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51

Jul 20, 2013
07/13

by
Robert Guralnick; Florian Herzig; Richard Taylor; Jack Thorne

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We study adequate subgroups of $GL_n$ over a finite field. This notion is useful in the study of automorphy lifting theorems. In particular, we give a sufficient condition for a subgroup to be adequate.

Source: http://arxiv.org/abs/1107.5993v1

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Jun 27, 2018
06/18

by
Toby Gee; Florian Herzig; Tong Liu; David Savitt

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We prove several results concerning the existence of potentially crystalline lifts with prescribed Hodge-Tate weights and inertial types of a given n-dimensional mod p representation of the absolute Galois group of K, where K/Q_p is a finite extension. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of the other results are global, making use of automorphy lifting theorems.

Topics: Number Theory, Mathematics

Source: http://arxiv.org/abs/1506.01050

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Jun 30, 2018
06/18

by
Noriyuki Abe; Guy Henniart; Florian Herzig; Marie-France Vigneras

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Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by...

Topics: Mathematics, Number Theory, Representation Theory

Source: http://arxiv.org/abs/1412.0737

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Jun 30, 2018
06/18

by
Noriyuki Abe; Guy Henniart; Florian Herzig; Marie-France Vigneras

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This is a list of questions raised by our joint work arXiv:1412.0737 and its sequels.

Topics: Number Theory, Representation Theory, Mathematics

Source: http://arxiv.org/abs/1703.02063