Deformation Rings Research Articles

Galois Deformation Rings and Modularity in the Residually Reducible Case

Galois Deformation Rings and Modularity in the Residually Reducible Case

Extremal weights and a tameness criterion for mod $p$ Galois representations

We study the weight part of Serre’s conjecture for generic n -dimensional mod p Galois representations. We first generalize Herzig’s conjecture to the case where the field is ramified at p and prove the weight elimination direction of the conjecture. We then introduce a new class of weights associated to n -dimensional local mod p representations which we call extremal weights . Using a “Levi reduction” property of certain potentially crystalline Galois deformation spaces, we prove the modularity of these weights. As a consequence, we deduce the weight part of Serre’s conjecture for unit groups of some division algebras in generic situations.

Arithmetic statistics for Galois deformation rings

Arithmetic statistics for Galois deformation rings

Colength one deformation rings

Let K / Q p K/\mathbb {Q}_p be a finite unramified extension, ρ ¯ : G a l ( Q ¯ p / K ) → G L n ( F ¯ p ) \overline {\rho }:\mathrm {Gal}(\overline {\mathbb {Q}}_p/K)\rightarrow \mathrm {GL}_n(\overline {\mathbb {F}}_p) a continuous representation, and τ \tau a tame inertial type of dimension n n . We explicitly determine, under mild regularity conditions on τ \tau , the potentially crystalline deformation ring R ρ ¯ η , τ R^{\eta ,\tau }_{\overline {\rho }} in parallel Hodge–Tate weights η = ( n − 1 , ⋯ , 1 , 0 ) \eta =(n-1,\cdots ,1,0) and inertial type τ \tau when the shape of ρ ¯ \overline {\rho } with respect to τ \tau has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre’s conjecture. Along the way we make unconditional the local-global compatibility results of Park and Qian [Mém. Soc. Math. Fr. (N.S.) 173 (2022), pp. vi+150].

WILES DEFECT OF HECKE ALGEBRAS VIA LOCAL-GLOBAL ARGUMENTS

Abstract In his work on modularity of elliptic curves and Fermat’s last theorem, A. Wiles introduced two measures of congruences between Galois representations and between modular forms. One measure is related to the order of a Selmer group associated to a newform $f \in S_2(\Gamma _0(N))$ (and closely linked to deformations of the Galois representation $\rho _f$ associated to f), whilst the other measure is related to the congruence module associated to f (and is closely linked to Hecke rings and congruences between f and other newforms in $S_2(\Gamma _0(N))$ ). The equality of these two measures led to isomorphisms $R={\mathbf T}$ between deformation rings and Hecke rings (via a numerical criterion for isomorphisms that Wiles proved) and showed these rings to be complete intersections. We continue our study begun in [BKM21] of the Wiles defect of deformation rings and Hecke rings (at a newform f) acting on the cohomology of Shimura curves over ${\mathbf Q}$ : It is defined to be the difference between these two measures of congruences. The Wiles defect thus arises from the failure of the Wiles numerical criterion at an augmentation $\lambda _f:{\mathbf T} \to {\mathcal O}$ . In situations we study here, the Taylor–Wiles–Kisin patching method gives an isomorphism $ R={\mathbf T}$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of the results in [BKM21] that compute the Wiles defect at $\lambda _f: R={\mathbf T} \to {\mathcal O}$ , and explain in an a priori manner why the answer in [BKM21] is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet–Takahashi that computes change of degrees of optimal parametrizations of elliptic curves over ${\mathbf Q}$ by Shimura curves as we vary the Shimura curve. The results we prove are not attainable using only the methods of Ribet–Takahashi.

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

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p=2 obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL(2): At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families.

On the Cohomology of GL(N) and Adjoint Selmer Groups

Abstract We prove under certain conditions (local-global compatibility and vanishing of modulo $p$ cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of $\operatorname{\textsf{GL}}(N)$ over a CM field; we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as the first Tate-Shafarevich group of this motive. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. By a result of Y. Cai generalizing previous results by Galatius-Venkatesh on the graded cohomology group of a locally symmetric space, our maps relate the (non-Eisenstein) localization of the graded cohomology group for a locally symmetric space to the exterior algebra of the Selmer group of the Tate dual of the adjoint representation. We generalize this to Hida families as well.

Derivatives of Beilinson–Flach classes, Gross–Stark formulas and a $p$-adic Harris–Venkatesh conjecture

We propose an alternative approach to the study of exceptional zeros from the point of view of Euler systems. As a first application, we give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the \mathcal{L} -invariant of the adjoint of a weight one modular form in terms of units and p -units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques, we discuss a new method exclusively using the properties of Beilinson–Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be seen as a counterpart to the earlier work of Loeffler, Venjakob, and Zerbes. In our second application, we illustrate how these techniques could lead to a better understanding of this setting by introducing a new motivic p -adic L -function whose special values encode information just about the unit of the adjoint (and not also the p -unit), in the spirit of the conjectures of Harris and Venkatesh. We further discuss conjectural connections with the arithmetic of triple products of Coleman families.

G-valued crystalline deformation rings in the Fontaine–Laffaille range

Let$G$be a split reductive group over the ring of integers in a$p$-adic field with residue field$\mathbf {F}$. Fix a representation$\overline {\rho }$of the absolute Galois group of an unramified extension of$\mathbf {Q}_p$, valued in$G(\mathbf {F})$. We study the crystalline deformation ring for$\overline {\rho }$with a fixed$p$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for$G$-valued representations. In particular, we give a root theoretic condition on the$p$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups.

Main conjectures for higher rank nearly ordinary families – I

In this article, we present the first half of our project on the Iwasawa theory of higher rank Galois deformations over deformations rings of arbitrary dimension. We develop a theory of Coleman maps for a very general class of coefficient rings, devise a dimension reduction procedure for locally restricted Euler systems and finally, put these into use in order to prove a divisibility in a 3-variable main conjecture for nearly ordinary families of Rankin-Selberg convolutions.

Rigidity of Automorphic Galois Representations Over CM Fields

Abstract We show the vanishing of adjoint Bloch–Kato Selmer groups of automorphic Galois representations over CM fields. This proves their rigidity in the sense that they have no deformations that are de Rham. In order for this to make sense, we also prove that automorphic Galois representations over CM fields are de Rham themselves. Our methods draw heavily from the 10 author paper, where these Galois representations were studied extensively. Another crucial piece of inspiration comes from the work of P. Allen who used the smoothness of certain local deformation rings in characteristic $0$ to obtain rigidity in the polarized case.

Singular equivalences of Morita type with level, Gorenstein algebras, and universal deformation rings

Let k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be an indecomposable finitely generated non-projective Gorenstein-projective left Λ-module whose stable endomorphism ring is isomorphic to k. In this article, we prove that the universal deformation rings R(Λ,V) and R(Λ,ΩΛV) are isomorphic, where ΩΛV denotes the first syzygy of V as a left Λ-module. We also prove the following result. Assume that Λ is also Gorenstein and that Γ is another Gorenstein k-algebra such that there exists ℓ≥0 and a pair of bimodules (XΛΓ,YΓΛ) that induces a singular equivalence of Morita type with level ℓ (as introduced by Z. Wang) between Λ and Γ. Then the left Γ-module X⊗ΛV is also Gorenstein-projective with stable endomorphism ring isomorphic to k, and the universal deformation ring R(Γ,X⊗ΛV) is isomorphic to R(Λ,V).

Universal deformation rings of modules for generalized Brauer tree algebras of polynomial growth

Let k be an arbitrary field, be a k-algebra and V be a -module. When it exists, the universal deformation ring of V is a k-algebra whose local homomorphisms to R parametrize the lifts of V up to , where R is any complete, local commutative Noetherian k-algebra with residue field k. Symmetric special biserial algebras, which coincide with Brauer graph algebras, can be viewed as generalizing the blocks of finite type p-modular group algebras. Bleher and Wackwitz classified the universal deformation rings for all modules for symmetric special biserial algebras with finite representation type. In this paper, we begin to address the tame case. Specifically, let be any 1-domestic, symmetric special biserial algebra. By viewing as generalized Brauer tree algebras and making use of a derived equivalence, we classify the universal deformation rings for those -modules V with stable endomorphism ring isomorphic to k. The latter is a natural condition, since it guarantees the existence of the universal deformation ring .

Explicit non-Gorenstein $$R={\mathbb {T}}$$ via rank bounds II: Computational aspects

This is the second in a pair of papers about residually reducible Galois deformation rings with non-optimal level. In the first paper, we proved a Galois-theoretic criterion for the deformation ring to be as small as possible. This paper focuses on the computations needed to verify this criterion. We adapt a technique developed by Sharifi to compute number fields with twisted-Heisenberg Galois group and prescribed ramification, and compute the splitting behavior of primes in these extensions.

On the $\mu$ equals zero conjecture for fine Selmer groups in Iwasawa theory

We study the Iwasawa theory of the fine Selmer group associated to certain Galois representations. The vanishing of the $\mu$-invariant is shown to follow in some cases from a natural property satisfied by Galois deformation rings. We outline conditions under which the $\mu=0$ conjecture is shown to hold for various Galois representations of interest.

On local Galois deformation rings

Abstract We show that framed deformation rings of mod p representations of the absolute Galois group of a p-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application, we prove density results of loci with prescribed p-adic Hodge theoretic properties.

Local models for Galois deformation rings and applications

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of \(\mathbb {Q}_p\) with small regular Hodge–Tate weights. We establish several significant facts about their geometry including a unibranch property at special points and a representation theoretic description of the irreducible components of their special fibers. We derive from these geometric results a number of local and global consequences: the Breuil–Mézard conjecture in arbitrary dimension for tamely potentially crystalline deformation rings with small Hodge–Tate weights (with appropriate genericity conditions), the weight part of Serre’s conjecture for U(n) as formulated by Herzig (for global Galois representations which satisfy the Taylor–Wiles hypotheses and are sufficiently generic at p), and an unconditional formulation of the weight part of Serre’s conjecture for wildly ramified representations.

Friction Performance Analysis of Reactor Coolant Pump Shaft Seal Based on Sensor and Computer Simulation

In recent years, the state has put forward a grand plan to develop nuclear power in order to control environmental pollution and carbon emissions. In nuclear power plants, mechanical seals for reactor coolant pump play an important role in nuclear power safety production. However, China cannot independently produce such mechanical seals, and they are blocked by foreign related technologies, which has seriously affected China’s nuclear power development plan and the overall safety of China’s nuclear power operation. Under this background, this paper studies the friction performance of the shaft seal of the reactor coolant pump, uses the sensor to obtain the data of friction factors such as low-pressure leakage, effectively monitors the low-pressure leakage through calculation and simulation modeling, truly reflects the operating state of the reactor coolant pump, and provides a new research direction and experimental basis for further analysis of the friction performance of the shaft seal of the reactor coolant pump. The results show that under the joint action of the pressure difference and the force generated by the deformation of the moving ring plate, the cone angle formed on the seal end face is 1300.9 μ convergence gap of rad. When the inlet water temperature is 65°C, the leakage rate is 1867.8 L/h. The deformation of the moving ring deformation ring plate can hinder the increase of the deformation cone angle of the moving ring. The greater the thickness of the moving ring deformation ring plate, the greater the seal leakage rate. The inlet temperature of the sealing medium and the friction performance of the material also have an impact on the sealing performance.

Arbeitsgemeinschaft: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups

The purpose of the workshop was to study derived generalizations of Mazur’s deformation ring of Galois representations, and the relationship of such a derived deformation ring to the homology of arithmetic groups.

Effect of increasing the ramification on pseudo-deformation rings

Given a continuous, odd, semi-simple $2$-dimensional representation of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$ and a prime $\ell$ not dividing $Np$, we study the relation between the universal deformation rings of the corresponding pseudo-representation for the groups $G_{\mathbb{Q},N\ell p}$ and $G_{\mathbb{Q},Np}$. 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 Bockle for the reduced pseudo-deformation rings. We improve these results when the pseudo-representation is unobstructed and $p$ does not divide $\ell^2-1$. When the pseudo-representation is unobstructed and $p$ divides $\ell+1$, we prove that the universal deformation rings in characteristic $0$ and $p$ of the pseudo-representation for $G_{\mathbb{Q},N\ell p}$ are not local complete intersection rings. As an application of our main results, we prove a big $R=\mathbb{T}$ theorem.