G-valued local deformation rings and global lifts

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.

We prove that the universal unramified deformation ring [math] of a continuous Galois representation [math] (for a totally real field [math] and finite field [math] ) is finite over [math] in many cases. We also prove (under similar hypotheses) that the universal deformation ring [math] is finite over the local deformation ring [math] .

Deformations and the rigidity method

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

We prove the vanishing of the geometric Bloch-Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor-Wiles-Kisin patching method together with Kisin's technique of analyzing the generic fibre of universal deformation rings. Along the way we give a characterization of smooth closed points on the generic fibre of Kisin's potentially semistable local deformation rings in terms of their Weil-Deligne representations.

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.

Let F be a finite field of characteristic ' > 0, F a number field, GF the absolute Galois group of F and let ¯ : GF ! GLN(F) be an absolutely irreducible continuous representation. Suppose S is a finite set of places containing all places above ' and above 1 and all those at whichramifies. Let O be a complete discrete valuation ring of characteristic zero with residue field F. In such a situation one may consider all deformations ofto O-algebras which are unramified outside S and satisfy certain local deformation conditions at the places in S. This was first studied by Mazur, (12), and under rather general hypotheses, the existence of a universal deformation ring was proven. In (2) I studied, among other things, the number of generators needed for an ideal I in a presentations of such a universal deformation ring as a quotient of a power series ring over O by I. The present manuscript is an update of this part of (2). The proofs have been simplified, the results slightly generalized. We also treat ' = 2, more general groups than GLN, and cases where not all relations are local. The results in (2) and hence also in the present manuscript are one of the (many) tools used in the recent attacks on Serre's conjecture by C. Khare and others.

Universal deformation rings need not be complete intersections

We answer a question of M. Flach by showing that there is a linear representation of a profinite group whose (unrestricted) universal deformation ring is not a complete intersection. We show that such examples arise in arithmetic in the following way. There are infinitely many real quadratic fields F for which there is a mod 2 representation of the Galois group of the maximal unramified extension of F whose universal deformation ring is not a complete intersection. Finally, we discuss bounds on the singularities of universal deformation rings of representations of finite groups in terms of the nilpotency of the associated defect groups.

Given a continuous, odd, semi-simple 2 -dimensional representation of G ℚ , N p over a finite field of odd characteristic p and a prime ℓ not dividing N p , we study the relation between the universal deformation rings of the corresponding pseudo-representations for the groups G ℚ , N ℓ p and G ℚ , N p . As a related problem, we investigate when the universal pseudo-representation arises from an actual representation over the universal deformation ring. Under some hypotheses, we prove analogues of theorems of Boston and Böckle for the reduced pseudo-deformation rings. We improve these results when the pseudo-representation is unobstructed and p does not divide ℓ 2 - 1 . When the pseudo-representation is unobstructed and p divides ℓ + 1 , we prove that the universal deformation rings in characteristic 0 and p of the pseudo-representation for G ℚ , N ℓ p are not local complete intersection rings. As an application of our main results, we prove a big R = 𝕋 theorem.

Conjecturally, the Galois representations that are attached to essentially self-dual regular algebraic cuspidal automorphic representations are Zariski-dense in a polarized Galois deformation ring. We prove new results in this direction in the context of automorphic forms on definite unitary groups over totally real fields. This generalizes the infinite fern argument of Gouvêa–Mazur and Chenevier and relies on the construction of nonclassical p-adic automorphic forms and the computation of the tangent space of the space of trianguline Galois representations. This boils down to a surprising statement about the linear envelope of intersections of Borel subalgebras.

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\bar {\rho }$. If $\bar {\rho }$ is scalar at some primes dividing the discriminant of the quaternion algebra, then the Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight-2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda _f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda _f$ by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho _f$ which $f$ gives rise to by the Eichler–Shimura construction. One of the main tools used in the proof is Taylor–Wiles–Kisin patching.

We fix a primep. In this paper, starting from a given Galois representation ϕ having values inp-adic points of a classical groupG, we study the adjoint action of ϕ on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(ϕ) of ϕ. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(ϕ))/L for each number fieldL. We scrutinize the behavior of Sel(Ad(ϕ))/E∞ as an Iwasawa module for a fixed ℤp-extensionE∞/E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kahler differentials of the universal nearly ordinary deformation ring of ϕ. WhenG=GL(2), ϕ is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.

We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\overline{\sigma}$ with $\overline{\sigma}^{{\rm ss}}=\chi^{k-2}\oplus\rho\oplus\chi^{k-1}$, where $\chi$ is the mod $p$ cyclotomic character and $\rho$ is an absolutely irreducible reduction of the Galois representation $\rho_f$ attached to a cusp form $f$ of weight $2k-2$. We show that if the Bloch-Kato Selmer groups $H^1_f({\bf Q},\rho_f(1-k)\otimes{\bf Q}_p/{\bf Z}_p)$ and $H^1_f({\bf Q},\rho(2-k))$ have order $p$, and there exists a characteristic zero absolutely irreducible deformation of $\overline{\sigma}$ then the universal deformation ring is a dvr. When $k=2$ this allows us to establish the modularity part of the Paramodular Conjecture in cases when one can find a suitable congruence of Siegel modular forms. As an example we prove the modularity of an abelian surface of conductor 731. When $k>2$, we obtain an $R^{{\rm red}}=T$ theorem showing modularity of all such deformations of $\overline{\sigma}$.

We investigate the case of deformations of even Galois representations. Our methods are the group theoretic ones mainly developed by Nigel Boston to study odd representations. We present conditions for Borel and tame cases under which the universal deformation ring is isomorphic to ℤp[[T]] and where we compute the universal deformation explicitly. Furthermore we produce a family of examples of totally real S3 extensions which satisfy the above conditions in the tame case and we give examples in the Borel case. Finally we study the change of the deformation space under enlarging the ramification and thus give an example of an even representation that is not twist‐finite.