scholarly journals GLOBALLY REALIZABLE COMPONENTS OF LOCAL DEFORMATION RINGS

2020 ◽  
pp. 1-70
Author(s):  
Frank Calegari ◽  
Matthew Emerton ◽  
Toby Gee
Keyword(s):  
Galois Representations ◽  
Local Deformation ◽  
Deformation Rings ◽  
Galois Deformation

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’.

scholarly journals $G$-valued Galois deformation rings when $\ell \neq p$

2019 ◽  
Vol 26 (4) ◽  
pp. 973-990
Author(s):  
Jeremy Booher ◽  
Stefan Patrikis
Keyword(s):  
Deformation Rings ◽  
Galois Deformation

A local-to-global principle for deformations of Galois representations

1999 ◽  
Vol 1999 (509) ◽  
pp. 199-236 ◽  
Author(s):  
Gebhard Böckle
Keyword(s):  
Finite Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Local Deformation ◽  
Galois Groups ◽  
Equivalence Classes ◽  
Deformation Space ◽  
Finite Set ◽  
Global Deformation ◽  
Global Principle

Abstract Given an absolutely irreducible Galois representation : GE → GLN (k), E a number field, k a finite field of characteristic l > 2, and a finite set of places Q of E containing all places above l and ∞ and all where ∞ ramifies, there have been defined many functors representing strict equivalence classes of deformations of such a representation, e.g. by Mazur or Wiles in [15] or [26], with various conditions on the behaviour of the deformations at the places in Q and with the condition that the deformations are unramified outside Q. Those functors are known to be representable. For as above, our goal is to present a rather general class of global deformation functors that satisfy local deformation conditions and to investigate for those, under what conditions the global deformation functor is determined by the local deformation functors. We will give precise conditions under which the local functors for all places in Q are sufficient to describe the global functor, first in a coarse form, then in a refined form using auxiliary primes as done by Taylor and Wiles in [24]. This has several consequences. The strongest is that one can derive ring theoretic results for the universal deformation space by Mazur if one uses results of Diamond and Wiles, cf. [11] and [26], and if one has a good understanding of all local situations. Furthermore it is easier to understand what happens under increasing the ramification as done by Boston and Ramakrishna in [6] and [20], [21]. Finally we shall reinterpret the results in the case of a tame representation by directly considering presentations of certain pro-l Galois groups and revisiting the prime-to-adjoint principle of Boston, cf. [5].

scholarly journals G-valued local deformation rings and global lifts

2019 ◽  
Vol 13 (2) ◽  
pp. 333-378
Author(s):  
Rebecca Bellovin ◽  
Toby Gee
Keyword(s):  
Local Deformation ◽  
Deformation Rings

scholarly journals SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

2020 ◽  
Vol 8 ◽  
Author(s):  
DANIEL LE ◽  
BAO V. LE HUNG ◽  
BRANDON LEVIN ◽  
STEFANO MORRA
Keyword(s):  
Representation Theory ◽  
Galois Representations ◽  
Galois Representation ◽  
Modular Representation ◽  
Duke Math ◽  
Modular Representation Theory ◽  
Deformation Rings ◽  
Invent Math

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 $\text{GL}_{3}(\mathbb{F}_{q})$ .

scholarly journals Derived Galois deformation rings

2018 ◽  
Vol 327 ◽  
pp. 470-623 ◽  
Author(s):  
S. Galatius ◽  
A. Venkatesh
Keyword(s):  
Deformation Rings ◽  
Galois Deformation
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