Relevant bibliographies by topics / Galois deformation rings

Academic literature on the topic 'Galois deformation rings'

Author: Grafiati
Published: 13 July 2024

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Journal articles on the topic "Galois deformation rings"

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

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Kim, Wansu. "Galois deformation theory for norm fields and flat deformation rings." Journal of Number Theory 131, no. 7 (July 2011): 1258–75. http://dx.doi.org/10.1016/j.jnt.2011.01.008.

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Booher, Jeremy, and Stefan Patrikis. "$G$-valued Galois deformation rings when $\ell \neq p$." Mathematical Research Letters 26, no. 4 (2019): 973–90. http://dx.doi.org/10.4310/mrl.2019.v26.n4.a2.

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Calegari, Frank, Søren Galatius, and Akshay Venkatesh. "Arbeitsgemeinschaft: Derived Galois Deformation Rings and Cohomology of Arithmetic Groups." Oberwolfach Reports 18, no. 2 (August 24, 2022): 1001–46. http://dx.doi.org/10.4171/owr/2021/18.

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Böckle, Gebhard, Chandrashekhar B. Khare, and Jeffrey Manning. "Wiles defect for Hecke algebras that are not complete intersections." Compositio Mathematica 157, no. 9 (August 16, 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, and Stephen V. Ullom. "Representations related to CM elliptic curves." Mathematical Proceedings of the Cambridge Philosophical Society 113, no. 1 (January 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, and Krzysztof Klosin. "On deformation rings of residually reducible Galois representations and R = T theorems." Mathematische Annalen 355, no. 2 (February 29, 2012): 481–518. http://dx.doi.org/10.1007/s00208-012-0793-1.

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Booher, Jeremy, and Brandon Levin. "G-valued crystalline deformation rings in the Fontaine–Laffaille range." Compositio Mathematica 159, no. 8 (July 17, 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, and Kazuma Shimomoto. "Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)." Nagoya Mathematical Journal 218 (June 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, and Kazuma Shimomoto. "Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)." Nagoya Mathematical Journal 218 (June 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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Dissertations / Theses on the topic "Galois deformation rings"

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Park, Chol. "Semi-Stable Deformation Rings in Hodge-Tate Weights (0,1,2)." Diss., The University of Arizona, 2013. http://hdl.handle.net/10150/293444.

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In this dissertation, we study semi-stable representations of G(Q(p)) and their mod p-reductions, which is a part of the problem in which we construct deformation spaces whose characteristic 0 closed points are the semi-stable lifts with Hodge-Tate weights (0, 1, 2) of a fixed absolutely irreducible residual representation ρ : G(Q(p)) → GL₃(F(p)). We first classify the isomorphism classes of semi-stable representations of G(Q(p)) with regular Hodge-Tate weights, by classifying admissible filtered (phi,N)-modules with Hodge-Tate weights (0, r, s) for 0 < r < s. We also construct a Galois stable lattice in some irreducible semi-stable representations with Hodge-Tate weights (0, 1, 2), by constructing strongly divisible modules, which is an analogue of Galois stable lattices on the filtered (ɸ, N)-module side. We compute the reductions mod p of the corresponding Galois representations to the strongly divisible modules we have constructed, by computing Breuil modules, which is, roughly speaking, mod p-reduction of strongly divisible modules. We also determine which Breuil modules corresponds to irreducible mod p representations of G(Q(p)).
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Guiraud, David-Alexandre [Verfasser], and Gebhard [Akademischer Betreuer] Böckle. "A framework for unobstructedness of Galois deformation rings / David-Alexandre Guiraud ; Betreuer: Gebhard Böckle." Heidelberg : Universitätsbibliothek Heidelberg, 2016. http://d-nb.info/1180610385/34.

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Moon, Yong Suk. "Galois Deformation Ring and Barsotti-Tate Representations in the Relative Case." Thesis, Harvard University, 2016. http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493581.

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In this thesis, we study finite locally free group schemes, Galois deformation rings, and Barsotti-Tate representations in the relative case. We show three independent but related results, assuming p > 2. First, we give a simpler alternative proof of Breuil’s result on classifying finite flat group schemes over the ring of integers of a p-adic field by certain Breuil modules [5]. Second, we prove that the locus of potentially semi-stable representations of the absolute Galois group of a p-adic field K with a specified Hodge-Tate type and Galois type cuts out a closed subspace of the generic fiber of a given Galois deformation ring, without assuming that K/Qp is finite. This is an extension of the corresponding result of Kisin when K/Qp is finite [19]. Third, we study the locus of Barsotti-Tate representations in the relative case, via analyzing certain extendability of p-divisible groups. We prove that when the ramification index is less than p-1, the locus of relative Barsotti-Tate representations cuts out a closed subspace of the generic fiber of a Galois deformation ring, if the base scheme is 2-dimensional satisfying some conditions. When the ramification index is greater than p-1, we show that such a result does not hold in general.
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Book chapters on the topic "Galois deformation rings"

1

Hida, Haruzo. "HECKE ALGEBRAS AS GALOIS DEFORMATION RINGS." In Hilbert Modular Forms and Iwasawa Theory, 162–285. Oxford University Press, 2006. http://dx.doi.org/10.1093/acprof:oso/9780198571025.003.0003.

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