Articles de revues : « Galois deformation rings »

1

Galatius, S., et A. Venkatesh. « Derived Galois deformation rings ». Advances in Mathematics 327 (mars 2018) : 470–623. http://dx.doi.org/10.1016/j.aim.2017.08.016.

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2

Kim, Wansu. « Galois deformation theory for norm fields and flat deformation rings ». Journal of Number Theory 131, n^o 7 (juillet 2011) : 1258–75. http://dx.doi.org/10.1016/j.jnt.2011.01.008.

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Booher, Jeremy, et Stefan Patrikis. « $G$-valued Galois deformation rings when $\ell \neq p$ ». Mathematical Research Letters 26, n^o 4 (2019) : 973–90. http://dx.doi.org/10.4310/mrl.2019.v26.n4.a2.

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Calegari, Frank, Søren Galatius et Akshay Venkatesh. « Arbeitsgemeinschaft : Derived Galois Deformation Rings and Cohomology of Arithmetic Groups ». Oberwolfach Reports 18, n^o 2 (24 août 2022) : 1001–46. http://dx.doi.org/10.4171/owr/2021/18.

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Böckle, Gebhard, Chandrashekhar B. Khare et Jeffrey Manning. « Wiles defect for Hecke algebras that are not complete intersections ». Compositio Mathematica 157, n^o 9 (16 août 2021) : 2046–88. http://dx.doi.org/10.1112/s0010437x21007454.

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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.

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Boston, Nigel, et Stephen V. Ullom. « Representations related to CM elliptic curves ». Mathematical Proceedings of the Cambridge Philosophical Society 113, n^o 1 (janvier 1993) : 71–85. http://dx.doi.org/10.1017/s0305004100075770.

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In [10], Mazur showed that the p-adic lifts of a given absolutely irreducible representation are parametrized by a universal deformation ξ:Gℚ, S → GL2() where has the form . (Here Gℚ, S is the Galois group over ℚ of a maximal algebraic extension unramified outside a finite set S of rational primes.) In [1, 3, 10], situations were investigated where the universal deformation ring turned out to be ℚp[[T1T2, T3]] (i.e. r = 3, I = (0)). In [2], the tame relation of algebraic number theory led to more complicated universal deformation rings, ones whose prime spectra consist essentially of four-dimensional sheets.

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Berger, Tobias, et Krzysztof Klosin. « On deformation rings of residually reducible Galois representations and R = T theorems ». Mathematische Annalen 355, n^o 2 (29 février 2012) : 481–518. http://dx.doi.org/10.1007/s00208-012-0793-1.

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Booher, Jeremy, et Brandon Levin. « G-valued crystalline deformation rings in the Fontaine–Laffaille range ». Compositio Mathematica 159, n^o 8 (17 juillet 2023) : 1791–832. http://dx.doi.org/10.1112/s0010437x23007297.

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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.

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Ochiai, Tadashi, et Kazuma Shimomoto. « Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals) ». Nagoya Mathematical Journal 218 (juin 2015) : 125–73. http://dx.doi.org/10.1215/00277630-2891620.

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AbstractIn this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

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Ochiai, Tadashi, et Kazuma Shimomoto. « Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals) ». Nagoya Mathematical Journal 218 (juin 2015) : 125–73. http://dx.doi.org/10.1017/s0027763000027045.

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AbstractIn this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

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11

Guiraud, David-Alexandre. « Unobstructedness of Galois deformation rings associated to regular algebraic conjugate self-dual cuspidal automorphic representations ». Algebra & ; Number Theory 14, n^o 6 (30 juillet 2020) : 1331–80. http://dx.doi.org/10.2140/ant.2020.14.1331.

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Previato, Emma. « Multivariable Burchnall–Chaundy theory ». Philosophical Transactions of the Royal Society A : Mathematical, Physical and Engineering Sciences 366, n^o 1867 (22 juin 2007) : 1155–77. http://dx.doi.org/10.1098/rsta.2007.2064.

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Burchnall & Chaundy (Burchnall & Chaundy 1928 Proc. R. Soc. A 118 , 557–583) classified the (rank 1) commutative subalgebras of the algebra of ordinary differential operators. To date, there is no such result for several variables. This paper presents the problem and the current state of the knowledge, together with an interpretation in differential Galois theory. It is known that the spectral variety of a multivariable commutative ring will not be associated to a KP-type hierarchy of deformations, but examples of related integrable equations were produced and are reviewed. Moreover, such an algebro-geometric interpretation is made to fit into A.N. Parshin's newer theory of commuting rings of partial pseudodifferential operators and KP-type hierarchies which uses higher local fields.

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Böckle, Gebhard, Ashwin Iyengar et Vytautas Paškūnas. « On local Galois deformation rings ». Forum of Mathematics, Pi 11 (2023). http://dx.doi.org/10.1017/fmp.2023.25.

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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.

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14

Ray, Anwesh, et Tom Weston. « Arithmetic statistics for Galois deformation rings ». Ramanujan Journal, 19 mai 2024. http://dx.doi.org/10.1007/s11139-024-00839-0.

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Böckle, Gebhard, Ashwin Iyengar et Vytautas Paškūnas. « On local Galois deformation rings – CORRIGENDUM ». Forum of Mathematics, Pi 12 (2024). http://dx.doi.org/10.1017/fmp.2024.3.

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16

Calegari, Frank, Matthew Emerton et Toby Gee. « GLOBALLY REALIZABLE COMPONENTS OF LOCAL DEFORMATION RINGS ». Journal of the Institute of Mathematics of Jussieu, 3 septembre 2020, 1–70. http://dx.doi.org/10.1017/s1474748020000195.

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Let $n$ 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’.

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A’Campo, Lambert. « Rigidity of Automorphic Galois Representations Over CM Fields ». International Mathematics Research Notices, 18 mai 2023. http://dx.doi.org/10.1093/imrn/rnad087.

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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.

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Le, Daniel, Bao V. Le Hung, Brandon Levin et Stefano Morra. « Local models for Galois deformation rings and applications ». Inventiones mathematicae, 3 octobre 2022. http://dx.doi.org/10.1007/s00222-022-01163-4.

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LE, DANIEL, BAO V. LE HUNG, BRANDON LEVIN et STEFANO MORRA. « SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE ». Forum of Mathematics, Pi 8 (2020). http://dx.doi.org/10.1017/fmp.2020.1.

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We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $p$ . This is a generalization to $\text{GL}_{3}$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $n$ -dimensional Galois representations’, Duke Math. J. 149(1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math. 212(1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $\text{GL}_{3}(\mathbb{F}_{q})$ .

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20

BARTLETT, ROBIN. « ON THE IRREDUCIBLE COMPONENTS OF SOME CRYSTALLINE DEFORMATION RINGS ». Forum of Mathematics, Sigma 8 (2020). http://dx.doi.org/10.1017/fms.2020.12.

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We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_{K}$ for a finite extension $K/\mathbb{Q}_{p}$ . This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For $K$ unramified over $\mathbb{Q}_{p}$ and Hodge–Tate weights in $[0,p]$ , we study the geometry of this space. As a consequence, we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_{p}$ , with Hodge–Tate weights in $[0,p]$ , are potentially diagonalizable.

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Böckle, Gebhard, Chandrashekhar B. Khare et Jeffrey Manning. « WILES DEFECT OF HECKE ALGEBRAS VIA LOCAL-GLOBAL ARGUMENTS ». Journal of the Institute of Mathematics of Jussieu, 25 avril 2024, 1–81. http://dx.doi.org/10.1017/s1474748024000021.

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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.

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Deo, Shaunak V. « On Density of Modular Points in Pseudo-Deformation Rings ». International Mathematics Research Notices, 16 mars 2023. http://dx.doi.org/10.1093/imrn/rnad037.

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Abstract Given a continuous, odd, reducible, and semi-simple $2$-dimensional representation $\bar \rho _{0}$ of $G_{{\mathbb{Q}},Np}$ over a finite field of odd characteristic $p$, we study the relation between the universal deformation ring of the pseudo-representation corresponding to $\bar \rho _{0}$ (pseudo-deformation ring) and the big $p$-adic Hecke algebra to prove that the maximal reduced quotient of the pseudo-deformation ring is isomorphic to the local component of the big $p$-adic Hecke algebra corresponding to $\bar \rho _{0}$ if a certain global Galois cohomology group has dimension $1$. This partially extends the results of Böckle to the case of residually reducible representations. We give an application of our main theorem to the structure of Hecke algebras modulo $p$. As another application of our methods and results, we prove a result about non-optimal levels of newforms lifting $\bar \rho _{0}$ in the spirit of Diamond–Taylor. This also gives a partial answer to a conjecture of Billerey–Menares.

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Hellmann, Eugen, Christophe M. Margerin et Benjamin Schraen. « Density of automorphic points in deformation rings of polarized global Galois representations ». Duke Mathematical Journal -1, n^o -1 (1 janvier 2022). http://dx.doi.org/10.1215/00127094-2021-0080.

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24

Iyengar, Srikanth B., Chandrashekhar B. Khare, Jeffrey Manning et Eric Urban. « Congruence modules in higher codimension and zeta lines in Galois cohomology ». Proceedings of the National Academy of Sciences 121, n^o 17 (19 avril 2024). http://dx.doi.org/10.1073/pnas.2320608121.

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This article builds on recent work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main results are a criterion for detecting regularity of local rings in terms of congruence modules, and a more refined version of a result tracking the change of congruence modules under deformation. Number theoretic applications include the construction of canonical lines in certain Galois cohomology groups arising from adjoint motives of Hilbert modular forms.

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