DEFORMATIONS OF GALOIS REPRESENTATIONS AND THE THEOREMS OF SATO–TATE AND LANG–TROTTER

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

For every prime number p ≥ 3 p\geq 3 and every integer m ≥ 1 m\geq 1 , we prove the existence of a continuous Galois representation ρ : G Q → G l m ( Z p ) \rho : G_\mathbb {Q} \rightarrow Gl_m(\mathbb {Z}_p) which has open image and is unramified outside { p , ∞ } \{p,\infty \} if p ≡ 3 p\equiv 3 mod 4 4 and is unramified outside { 2 , p , ∞ } \{2,p,\infty \} if p ≡ 1 p \equiv 1 mod 4 4 . We also revisit the question of the lifting of residual Galois representations in terms of embedding problems; that allows us to produce Galois representations with open image in the group of upper triangular matrices with diagonal entries equal to 1 1 , unramified outside { p , ∞ } \{p,\infty \} , for m m “small” comparing to p p .

Deformations and the rigidity method

Let p ≥ 7 p\geq 7 be a prime and n > 1 n>1 be a natural number. We show that there exist infinitely many Galois representations ϱ : Gal ⁡ ( Q ¯ / Q ) → GL n ⁡ ( Z p ) \varrho :\operatorname {Gal}(\bar {\mathbb {Q}}/\mathbb {Q})\rightarrow \operatorname {GL}_{n}(\mathbb {Z}_p) which are unramified outside { p , ∞ } \{p, \infty \} with large image. More precisely, the Galois representations constructed have image containing the kernel of the mod- p t p^t reduction map SL n ⁡ ( Z p ) → SL n ⁡ ( Z / p t Z ) \operatorname {SL}_n(\mathbb {Z}_p)\rightarrow \operatorname {SL}_n(\mathbb {Z}/p^t\mathbb {Z}) , where t ≔ 8 ( n 2 − n ) ( 3 + ⌊ log p ⁡ ( 2 n + 1 ) ⌋ ) + 8 t≔8(n^2-n)\left (3+\lfloor \operatorname {log}_p(2^n+1)\rfloor \right )+8 . The results are proven via a purely Galois theoretic lifting construction. When p ≡ 1 mod 4 p\equiv 1\mod {4} , our results are conditional since in this case, we assume a very weak version of Vandiver’s conjecture.

We study compatible families of three-dimensional Galois representations constructed in the étale cohomology of a smooth projective variety. We prove a theorem asserting that the residual images will be generically large if certain easy-to-check conditions are satisfied. We only consider representations with coefficients in an imaginary quadratic field. For primes inert in this field, the residual representations (when irreducible) are unitary. We apply our result to an example constructed by van Geemen and Top, obtaining a family of special linear groups and one of special unitary groups as Galois groups over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. We also consider the case of cohomological modular forms for a congruence subgroup of SL [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]. Assuming Clozel's conjecture stating that a geometric family of three-dimensional Galois representations can be attached to them, we verify for three examples the conditions guaranteeing generically large images and explicitly bound the finite set of primes with non-maximal image. We also discuss what intrinsic conditions a modular form should verify to guarantee that the images of the attached Galois representations will be generically large.

We consider certain [Formula: see text]-ordinary non-CM Hida families with full residual Galois representation and give mild conditions under which every arithmetic point in these families is locally indecomposable when [Formula: see text]. The proof uses methods from deformation theory and mostly works for any odd prime [Formula: see text], but ultimately relies on the existence of a weight [Formula: see text] form in an auxiliary family which is available only for [Formula: see text]. We end by giving several non-trivial examples of [Formula: see text]-ordinary non-CM locally indecomposable modular forms of small level with full residual Galois representation.

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.

In this work we generalise the main result of [1] to the family of hyperelliptic curves with potentially good reduction over a p-adic field which have genus $g=({p-1})/{2}$ and the largest possible image of inertia under the $\ell$ -adic Galois representation associated to its Jacobian. We will prove that this Galois representation factors as the tensor product of an unramified character and an irreducible representation of a finite group, which can be either equal to the inertia image (in which case the representation is easily determined) or a $C_2$ -extension of it. In this second case, there are two suitable representations and we will describe the Galois action explicitly in order to determine the correct one.

For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Bockle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvea-Mazur. We generalize Bockle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under assumptions necessary to apply a small $R = \mathbb{T}$ theorem and an assumption on the local mod $p$ representation, we prove that every irreducible component of the universal polarized deformation space contains an automorphic point. When combined with work of Chenevier, this implies new results on the Zariski density of automorphic points in polarized deformation space in dimension three.

We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into GLn. Existing theorems require that the residual representation have ‘big’ image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor–Wiles method which allows one to weaken this hypothesis.

(K), as is explained, for example, in [W1] and [W2].Fontaine and Mazur [FM] conjectured that this is always the case. Significantprogress on this conjecture was achieved [W3] by proving particular instancesof the following “lifting conjecture”:Conjecture 1.1. Suppose that l is odd and that the residual represen-tation ρ¯ attached to ρ is modular. Then ρ itself is modular.Conjecture 1.1 is proved in [W3] and [TW] when K = Q and the restriction ofρ to the decomposition groups at the primes above l are semistable in the senseof [DDT, §2.4]. This is enough (using the primes l = 3 and 5) to establish theShimura-Taniyama conjecture for semistable elliptic curves, thanks to a keyresult of Langlands and Tunnell. Progressively stronger cases of Conjecture 1.1were subsequently proved by [Di], [CDT], [Fu], and [SW1]; in [SW1], Skinnerand Wiles obtain quite general results in the context where K is any totallyreal field, the principal assumption being that ρ is ordinary at the primesabove l.In this note we consider Galois representations which occur in “rigid fam-ilies”, and establish their modularity under Conjecture 1.1. This implies themodularity (over suitable real abelian extensions) of the Galois representationsoccurring in the cohomology of the curvesy

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parametrizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

Starting with a 2-dimensional mod p Galois representation, we construct a deformation to a power series ring in infinitely many variables over the p-adics. The image of this representation is full in the sense that it contains SL2 of this power series ring. Furthermore, all Zp specializations of this deformation are potentially semistable at p.

Modular units and the surjectivity of a Galois representation

We explore the question of how big the image of a Galois representation at- tached to a -adic modular form with no complex multiplication is and show that for a generic set of -adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all but a density 0 subset have large image.