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Register a new userGenerators of the pro-p Iwahori and Galois representations
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Authors
Christophe Cornut
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For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct Galois extensions of Q with Galois group between the pro-p and the standard Iwahori subgroups of G.
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A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy.
In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem.
Let $mathcal{G}$ be a connected reductive almost simple group over the Witt ring $W(mathbb{F})$ for $mathbb{F}$ a finite field of characteristic $p$. Let $R$ and $R$ be complete noetherian local $W(mathbb{F})$ -algebras with residue field $mathbb{F}$. Under a mild condition on $p$ in relation to structural constants of $mathcal{G}$, we show the following results: (1) Every closed subgroup $H$ of $mathcal{G}(R)$ with full residual image $mathcal{G}(mathbb{F})$ is a conjugate of a group $mathcal{G}(A)$ for $Asubset R$ a closed subring that is local and has residue field $mathbb{F}$ . (2) Every surjective homomorphism $mathcal{G}(R)tomathcal{G}(R)$ is, up to conjugation, induced from a ring homomorphism $Rto R$. (3) The identity map on $mathcal{G}(R)$ represents the universal deformation of the representation of the profinite group $mathcal{G}(R)$ given by the reduction map $mathcal{G}(R)tomathcal{G}(mathbb{F})$. This generalizes results of Dorobisz and Eardley-Manoharmayum and of Manoharmayum, and in addition provides an abstract classification result for closed subgroups of $mathcal{G}(R)$ with residually full image. We provide an axiomatic framework to study this type of question, also for slightly more general $mathcal{G}$, and we study in the case at hand in great detail what conditions on $mathbb{F}$ or on $p$ in relation to $mathcal{G}$ are necessary for the above results to hold.
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