A note on Galois representations valued in reductive groups with open image

In this paper, we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$ . In particular, under certain technical conditions applicable to any root datum, we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\operatorname {\mathrm {GL}}_n$ -valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm {G}_2$ -valued representations.

Let X X be a complete smooth variety defined over a number field K K and let i i be an integer. The absolute Galois group G a l K \mathrm {Gal}_K of K K acts on the i i th étale cohomology group H e ´ t i ( X K ¯ , Q ℓ ) H^i_{\mathrm {\acute {e}t}}(X_{\bar K},\mathbb {Q}_\ell ) for all primes ℓ \ell , producing a system of ℓ \ell -adic representations { Φ ℓ } ℓ \{\Phi _\ell \}_\ell . The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ ℓ \Phi _\ell admits a reductive Q \mathbb {Q} -form that is independent of ℓ \ell if X X is projective. Denote by Γ ℓ \Gamma _\ell and G ℓ \mathbf {G}_\ell respectively the monodromy group and the algebraic monodromy group of Φ ℓ s s \Phi _\ell ^{\mathrm {ss}} , the semisimplification of Φ ℓ \Phi _\ell . Assuming that G ℓ 0 \mathbf {G}_{\ell _0} satisfies some group theoretic conditions for some prime ℓ 0 \ell _0 , we construct a connected quasi-split Q \mathbb {Q} -reductive group G Q \mathbf {G}_{\mathbb {Q}} which is a common Q \mathbb {Q} -form of G ℓ ∘ \mathbf {G}_\ell ^\circ for all sufficiently large ℓ \ell . Let G Q s c \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}} be the universal cover of the derived group of G Q \mathbf {G}_{\mathbb {Q}} . As an application, we prove that the monodromy group Γ ℓ \Gamma _\ell is big in the sense that Γ ℓ s c ≅ G Q s c ( Z ℓ ) \Gamma _\ell ^{\mathrm {sc}}\cong \mathbf {G}_{\mathbb {Q}}^{\mathrm {sc}}(\mathbb {Z}_\ell ) for all sufficiently large ℓ \ell .

These are the notes of a five-lecture course presented at the Computations with Modular Forms summer school, aimed at graduate students in number theory and related areas. Sections 1–4 give a sketch of the theory of reductive algebraic groups over Q, and of Gross’s purely algebraic definition of automorphic forms in the special case when G(R) is compact. Sections 5–9 describe how these automorphic forms can be explicitly computed, concentrating on the case of definite unitary groups; and Sects. 10 and 11 describe how to relate the results of these computations to Galois representations, and present some examples where the corresponding Galois representations can be identified, giving illustrations of various instances of Langlands’ functoriality conjectures.KeywordsModular FormAlgebraic GroupReductive GroupCompact SubgroupAutomorphic FormThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp4 over totally real fields.

We extend the methods of Wiles and of Taylor and Wiles from GL 2 to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), minimally ramified, l -adic lifts of certain automorphic mod l Galois representations of any dimension. We also make a conjecture about the structure of mod l automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for GL 2 . Following Wiles' method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.

Given an equivariant oriented cohomology theory $h$, a split reductive group $G$, a maximal torus $T$ in $G$, and a parabolic subgroup $P$ containing $T$, we explain how the $T$-equivariant oriented cohomology ring $h_T(G/P)$ can be identified with the dual of a coalgebra defined using exclusively the root datum of $(G,T)$, a set of simple roots defining $P$ and the formal group law of $h$. In two papers [Push-pull operators on the formal affine Demazure algebra and its dual, arXiv:1312.0019] and [A coproduct structure on the formal affine Demazure algebra, arXiv:1209.1676], we studied the properties of this dual and of some related operators by algebraic and combinatorial methods, without any reference to geometry. The present paper can be viewed as a companion paper, that justifies all the definitions of the algebraic objects and operators by explaining how to match them to equivariant oriented cohomology rings endowed with operators constructed using push-forwards and pull-backs along geometric morphisms. Our main tool is the pull-back to the $T$-fixed points of $G/P$ which injects the cohomology ring in question into a direct product of a finite number of copies of the $T$-equivariant oriented cohomology of a point.

Let ${\mathfrak o}$ be the ring of integers in a finite extension field of ${\mathbb Q}_p$, let $k$ be its residue field. Let $G$ be a split reductive group over ${\mathbb Q}_p$, let ${\mathcal H}(G,I_0)$ be its pro-$p$-Iwahori Hecke ${\mathfrak o}$-algebra. In \cite{dfun} we introduced a general principle how to assign to a certain additionally chosen datum $(C^{(\bullet)},\phi,\tau)$ an exact functor $M\mapsto{\bf D}(\Theta_*{\mathcal V}_M)$ from finite length ${\mathcal H}(G,I_0)$-modules to $(\varphi^r,\Gamma)$-modules. In the present paper we concretely work out such data $(C^{(\bullet)},\phi,\tau)$ for the classical matrix groups. We show that the corresponding functor identifies the set of (standard) supersingular ${\mathcal H}(G,I_0)\otimes_{{\mathfrak o}}k$-modules with the set of $(\varphi^r,\Gamma)$-modules satisfying a certain symmetry condition.

Let G be a split reductive group over a local field K, and let G((t)) be the corresponding loop group. In [1], we have introduced the notion of a representation of (the group of K-points) of G((t)) on a pro-vector space. In addition, we have defined an induction procedure, which produced G((t))-representations from usual smooth representations of G. We have conjectured that the induction of a cuspidal irreducible representation of G is irreducible. In this paper, we prove this conjecture for G=SL2.

Let G be an isotropic reductive algebraic group over a commutative ring R. Assume that the elementary subgroup E(R) of group of points G(R) is correctly defined. Then E(R) is perfect, except for the well-known cases of a split reductive group of type C_2 or G_2.

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space \({\cal S}(X)\) (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of \( {\cal S}(X)^I \), which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of \( {\cal S}(X) \), which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.

Let X X be a smooth curve over a finite field of characteristic p p , let E E be a number field, and let L = { L λ } \mathbf {L} = \{\mathcal {L}_\lambda \} be an E E -compatible system of lisse sheaves on the curve X X . For each place λ \lambda of E E not lying over p p , the λ \lambda -component of the system L \mathbf {L} is a lisse E λ E_\lambda -sheaf L λ \mathcal {L}_\lambda on X X , whose associated arithmetic monodromy group is an algebraic group over the local field E λ E_\lambda . We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E E -compatible system L \mathbf {L} is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ \lambda ”. More precisely, after replacing E E by a finite extension, there exists a connected split reductive algebraic group G 0 G_0 over the number field E E such that for every place λ \lambda of E E not lying over p p , the identity component of the arithmetic monodromy group of L λ \mathcal {L}_\lambda is isomorphic to the group G 0 G_0 with coefficients extended to the local field E λ E_\lambda .

We give a complete classification of 2-dimensional complex Galois representations of dihedral type over a local field Qp, including their conductors and determinants on inertia. This explicit classification complements earlier works [2], [5] on these types of representations.

We prove modularity lifting theorems for ℓ-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine–Laffaille condition at ℓ. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor–Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GLn.

We revisit the paper [Automorphy lifting for residually reducible$l$-adic Galois representations, J. Amer. Math. Soc. 28 (2015), 785–870] by the third author. We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.

We give a description of the arboreal Galois representation of a quadratic polynomial $$f(x)=x^2-(4k+1)$$, where $$k\in {\mathbb {N}}$$ and $$4k+1$$ is not a square in $${\mathbb {Z}}$$.