The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations

By F. Herzig. 2008

121 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d21c91398fc010c5caf9df9fa7b4cc45ad83afb2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. C. Breuil
2. T. Gee
3. M. Emerton
4. Nicolas Bergeron
5. Akshay Venkatesh
6. David Geraghty

In Geometry and Topology

1. T. Gee
2. M. Emerton
3. F. Herzig
4. David Geraghty
5. C. Breuil
6. Akshay Venkatesh
7. Nicolas Bergeron
8. Peter Scholze

In Mathematical Physics

1. T. Gee
2. M. Emerton
3. F. Herzig
4. C. Breuil
5. David Geraghty
6. Akshay Venkatesh
7. Nicolas Bergeron
8. M. Kisin

Citations in the corpus

• The asymptotic growth of torsion homology for arithmetic groups (2010, background): “ slight generalization of [2] (split semisimple group, as opposed to SLn); we don’t address the difficult question of pinning down all the weights in which a given Galois representation occurs (see e.g [24, 34, 35] and an unpublished manuscript by Clozel and Belabas where they make precise computation”
• Weight cycling and Serre-type conjectures for unitary groups (2011, methodology): “ulations of Section 3 to establish the elimination result that we need; that is, to show that in generic situations, if ¯ris modular of some weight then that weight is one of the weights predicted in [Her09]. We carry out these arguments in the setting of Section 4. In Section 6 we combine the resul”
• A geometric perspective on the Breuil–Mézard conjecture (2011, background): “)Naτ ⊗k,τF. The representations Faare absolutely irreducible and pairwise non-isomorphic, and every irreducible F-representation of GLn(k) is of the form Fa for some a(see for example the appendix to [Her09]). We say that an element λ∈ (Zn +) HomQp (K,E) is a lift of a Serre weight a∈ (Zn +) Hom(k,F”
• The emerging p-adic Langlands programme (2011, methodology): “Then the “Serre conjecture” of [39] and [33] states that the Serre weights of W(ρ|Gal(Qp/Qp)) should be exactly those Serre weights”
• COMPANION FORMS FOR UNITARY AND SYMPLECTIC GROUPS (2010, background): “l GLn(k(˜v)) are tensor products of irreducible representations of the GLn(k(˜v)). From the standard classification of the irreducible Fl-representations of GLn(k(˜v)) (see for example the appendix to [Her09]), one sees that: (1) There is an irreducible Fl-representation Fλ of Q v∈Sl GLn(k(˜v)) for e”
• The classification of irreducible admissible mod p representations of a p-adic GLn (2010, background): “This goes back to Steinberg if G is semisimple; in general, see Proposition 1.3 in the appendix to [ 16 ].”