Deformation Rings Research Articles

Semi-stable deformation rings in even Hodge–Tate weights: The residually reducible case

Let [Formula: see text] be a prime number and [Formula: see text] a positive even integer less than [Formula: see text]. In this paper, we find the strongly divisible modules corresponding to the Galois stable lattices in each 2-dimensional semi-stable non-crystalline representation of [Formula: see text] with Hodge–Tate weights [Formula: see text] whose mod-[Formula: see text] reductions are corresponding to nontrivial extensions of two distinct characters. We use these results to construct the irreducible components of the semi-stable deformation rings in Hodge–Tate weights [Formula: see text] of the non-split reducible residual representations of [Formula: see text].

Generic local deformation rings when

We determine the local deformation rings of sufficiently generic mod $l$ representations of the Galois group of a $p$-adic field, when $l \neq p$, relating them to the space of $q$-power-stable semisimple conjugacy classes in the dual group. As a consequence, we give a local proof of the $l \neq p$ Breuil–Mézard conjecture of the author, in the tame case.

The universal ordinary deformation ring associated to a real quadratic field

The universal ordinary deformation ring associated to a real quadratic field

Connectedness of Kisin varieties associated to absolutely irreducible Galois representations

AbstractWe consider the Kisin variety associated to ann-dimensional absolutely irreducible modpGalois representationρ¯{\bar{\rho}}of ap-adic fieldKtogether with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin’s conjecture holds ifKis totally ramified withn=3{n=3}or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated toρ¯{\bar{\rho}}of given Hodge–Tate weights. We also give counterexamples to show Kisin’s conjecture does not hold in general.

Density of automorphic points in deformation rings of polarized global Galois representations

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.

On the automorphy of 2-dimensional potentially semistable deformation rings of Gℚp

On the automorphy of 2-dimensional potentially semistable deformation rings of Gℚp

Wiles defect for Hecke algebras that are not complete intersections

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.

On a Deformation Theory of Finite Dimensional Modules Over Repetitive Algebras

Let Λ be a basic finite dimensional algebra over an algebraically closed field \(\Bbbk \), and let \(\widehat {\Lambda }\) be the repetitive algebra of Λ. In this article, we prove that if \(\widehat {V}\) is a left \(\widehat {\Lambda }\)-module with finite dimension over \(\Bbbk \), then \(\widehat {V}\) has a well-defined versal deformation ring \(R(\widehat {\Lambda },\widehat {V})\), which is a local complete Noetherian commutative \(\Bbbk \)-algebra whose residue field is also isomorphic to \(\Bbbk \). We also prove that \(R(\widehat {\Lambda }, \widehat {V})\) is universal provided that \(\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk \) and that in this situation, \(R(\widehat {\Lambda }, \widehat {V})\) is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over \(\mathbb {P}^{1}_{\Bbbk }\).

Galois deformation spaces with a sparsity of automorphic points

Let $$k/\mathbb {F}_p$$ denote a finite field. For any split connected reductive group G/W(k) and a CM number field F, we deform nearly ordinary Galois representations $$\overline{\rho }:{{\,\mathrm{Gal}\,}}(\overline{F}/F) \rightarrow G(k)$$ to analytic families $$X_{\overline{\rho }}$$ of Galois representations $$\Gamma _F \rightarrow G(\overline{\mathbb {Q}}_p)$$ lifting $$\overline{\rho }$$ such that the set of points of $$X_{\overline{\rho }}$$ which are geometric (in the sense of the Fontaine–Mazur conjecture) with parallel Hodge–Tate weights is contained in an analytic subvariety of $$X_{\overline{\rho }}$$ with positive codimension. Thus, the set of points in $$X_{\overline{\rho }}$$ which could (conjecturally) be associated to automorphic forms is sparse. This generalizes a result of Calegari and Mazur for $$F/\mathbb {Q}$$ quadratic imaginary and $$G = {{\,\mathrm{GL}\,}}_2$$ . The sparsity of automorphic points for a CM field F contrasts with the situation when F is a totally real field, where automorphic points are often provably dense.

Class groups and local indecomposability for non-CM forms

In the late 1990s, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those $p$-ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at $p$. It is expected that such $p$-ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman–Greenberg’s question using Galois deformation theory. In particular, for $p$-ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the $p$-indivisibility of a certain class group.

Universal Deformation Rings of Modules over Self-injective Cluster-Tilted Algebras Are Trivial

Let [Formula: see text] be a fixed algebraically closed field of arbitrary characteristic, let [Formula: see text] be a finite dimensional self-injective [Formula: see text]-algebra, and let [Formula: see text] be an indecomposable non-projective left [Formula: see text]-module with finite dimension over [Formula: see text]. We prove that if [Formula: see text] is the Auslander–Reiten translation of [Formula: see text], then the versal deformation rings [Formula: see text] and [Formula: see text] (in the sense of F.M. Bleher and the second author) are isomorphic. We use this to prove that if [Formula: see text] is further a cluster-tilted [Formula: see text]-algebra, then [Formula: see text] is universal and isomorphic to [Formula: see text].

Patching and multiplicity 2k for Shimura curves

We use the Taylor-Wiles-Kisin patching method to investigate the multiplicities with which Galois representations occur in the mod $\ell$ cohomology of Shimura curves over totally real number fields. Our method relies on explicit computations of local deformation rings done by Shotton, which we use to compute the Weil class group of various deformation rings. Exploiting the natural self-duality of the cohomology groups, we use these class group computations to precisely determine the structure of a patched module in many new cases in which the patched module is not free (and so multiplicity one fails). Our main result is a "multiplicity $2^k$" theorem in the minimal level case (which we prove under some mild technical hypotheses), where $k$ is a number that depends only on local Galois theoretic information at the primes dividing the discriminant of the Shimura curve. Our result generalizes Ribet's classical multiplicity 2 result and the results of Cheng, and provides progress towards the Buzzard-Diamond-Jarvis local-global compatibility conjecture. We also prove a statement about the endomorphism rings of certain modules over the Hecke algebra, which may have applications to the integral Eichler basis problem.

Derived Beilinson–Flach elements and the arithmetic of the adjoint of a modular form

Kings, Lei, Loeffler and Zerbes constructed in \[LLZ], \[KLZ1] a three-variable Euler system $\kappa(\textbf{g},\textbf{h})$ of Beilinson–Flach elements associated to a pair of Hida families $(\textbf{g},\textbf{h})$ and exploited it to obtain applications to the arithmetic of elliptic curves, extending the earlier work \[BDR]. The aim of this article is to show that this Euler system also encodes arithmetic information concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when $\textbf{g}$ and $\textbf{h}$ specialize in weight 1 to $p$-stabilizations of eigenforms such that one is dual to the other. We encounter an exceptional zero phenomenon which forces the specialization of $\kappa(\textbf{g},\textbf{h})$ to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of the main conjecture of \[DLR4] on iterated integrals and the main conjecture of \[DR1] for Beilinson–Flach elements in the adjoint setting. The main point of this paper is that the methods of \[DLR1], \[DLR4] and \[CH], where the above conjectures are proved when the weight 1 eigenforms have CM, do not apply to our setting and new ideas are required. In the previous works, a crucial ingredient is a factorization of $p$-adic $L$-functions, which in our scenario is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and $p$-adic $L$-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the Elliptic Stark Conjecture and of its generalizations that appear in the literature.

Arithmetic representations of fundamental groups, II: Finiteness

Let X be a smooth curve over a finitely generated field k, and let l be a prime different from the characteristic of k. We analyze the dynamics of the Galois action on the deformation rings of mod l representations of the geometric fundamental group of X. Using this analysis, we prove several finiteness results for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey–Mazur conjecture for function fields in characteristic 0. For example, we show that if X is a normal, connected variety over C, then the (typically infinite) set of representations of π1(Xan) into GLn(Ql‾), which come from geometry, has no limit points. As a corollary, we deduce that if L is a finite extension of Ql, then the set of representations of π1(Xan) into GLn(L), which arise from geometry, is finite.

On the Hilbert eigenvariety at exotic and CM classical weight 1 points

Let F be a totally real number field and let f be a classical cuspidal p-regular Hilbert modular eigenform over F of parallel weight 1. Let x be the point on the p-adic Hilbert eigenvariety $${\mathcal {E}}$$ corresponding to an ordinary p-stabilization of f. We show that if the p-adic Schanuel conjecture is true, then $${\mathcal {E}}$$ is smooth at x if f has CM. If we additionally assume that $$F/\mathbb {Q}$$ is Galois, we show that the weight map is etale at x if f has either CM or exotic projective image (which is the case for almost all cuspidal Hilbert modular eigenforms of parallel weight 1). We prove these results by showing that the completed local ring of the eigenvariety at x is isomorphic to a universal nearly ordinary Galois deformation ring.

Codimension two cycles in Iwasawa theory and tensor product of Hida families

The purpose of this paper is to build on results in higher codimension Iwasawa theory. The setting of our results involves Galois representations arising as cyclotomic twist deformations associated to (1) the tensor product of two cuspidal Hida families F and G, and (2) the tensor product of three cuspidal Hida families F, G and H. On the analytic side, we consider (1) a pair of 3-variable Rankin–Selberg p-adic L-functions constructed by Hida and (2) a balanced 4-variable p-adic L-function (due to Hsieh and Yamana) and an unbalanced 4-variable p-adic L-function (whose existence is currently conjectural). In each of these setups, when the two p-adic L-functions generate a height two ideal in the corresponding deformation ring, we use codimension two cycles of that ring to relate them to a pair of pseudo-null modules.

On deformations of Gorenstein-projective modules over Nakayama and triangular matrix algebras

Let k be a fixed field of arbitrary characteristic and let Λ be a basic connected Nakayama k -algebra without simple projective modules. In this article we prove that if V is an indecomposable finitely generated Gorenstein-projective left Λ-module, then the versal deformation ring R ( Λ , V ) (in the sense of F. M. Bleher and the author) is universal and stable after taking syzygies . We also prove the following result. Let Σ = ( Λ B 0 Γ ) be a triangular matrix finite dimensional Gorenstein k -algebra with B projective as a left Λ-module and with Γ of finite global dimension. If ( V W ) f is a finitely generated Gorenstein-projective left Σ-module with End _ Σ ( ( V W ) f ) = k , then V is also a finitely generated Gorenstein-projective left Λ-module with End _ Λ ( V ) = k , and the versal deformation rings R ( Σ , ( V W ) f ) and R ( Λ , V ) are both universal and isomorphic.

GLOBALLY REALIZABLE COMPONENTS OF LOCAL DEFORMATION RINGS

AbstractLet $n$ be either $2$ or an odd integer greater than $1$, and fix a prime $p>2(n+1)$. Under standard ‘adequate image’ assumptions, we show that the set of components of $n$-dimensional $p$-adic potentially semistable local Galois deformation rings that are seen by potentially automorphic compatible systems of polarizable Galois representations over some CM field is independent of the particular global situation. We also (under the same assumption on $n$) improve on the main potential automorphy result of Barnet-Lamb et al. [Potential automorphy and change of weight, Ann. of Math. (2) 179(2) (2014), 501–609], replacing ‘potentially diagonalizable’ by ‘potentially globally realizable’.

On the modularity of 2-adic potentially semi-stable deformation rings

Using $p$-adic local Langlands correspondence for $\operatorname{GL}_2(\mathbb{Q}_2)$ and an ordinary $R = \mathbb{T}$ theorem, we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi-stable deformation ring. This gives a new proof of the Breuil-M\'{e}zard conjecture for 2-dimensional representations of the absolute Galois group of $\mathbb{Q}_2$, which is new in the case $\overline{r}$ a twist of an extension of the trivial character by itself. As a consequence, a local restriction in Pa\v{s}k\=unas' proof of Fontaine-Mazur conjecture is removed.

Unobstructedness of Galois deformation rings associated to regular algebraic conjugate self-dual cuspidal automorphic representations

Let F be a CM field and let (rπ,λ)λ be the compatible system of residual 𝒢n-valued representations of GalF attached to a regular algebraic conjugate self-dual cuspidal (RACSDC) automorphic representation π of GLn(𝔸), as studied by Clozel, Harris and Taylor (2008) and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to rπ,λ are unobstructed for all places λ in a subset of Dirichlet density 1, continuing the investigations of Mazur, Weston and Gamzon. During the proof, we develop a general framework for proving unobstructedness (with future applications in mind) and an R=T-theorem, relating the universal crystalline deformation ring of rπ,λ and a certain unitary fixed-type Hecke algebra.