scholarly journals Deformations of reducible Galois representations to Hida-families

2021 ◽  
pp. 1-24
Author(s):  
Anwesh Ray
Keyword(s):  
Deformation Theory ◽  
Galois Representations ◽  
Congruence Class ◽  
Representation Formula ◽  
Theoretic Approach ◽  
Hida Families ◽  
Global Deformation

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that [Formula: see text] lifts to a Hida line for which the weights range over a congruence class modulo-[Formula: see text]. The advantage of the purely Galois theoretic approach is that it allows us to construct [Formula: see text]-adic families of Galois representations lifting the actual representation [Formula: see text], and not just the semisimplification.

Locally indecomposable Galois representations with full residual image

2017 ◽  
Vol 13 (05) ◽  
pp. 1191-1211
Author(s):  
Devika Sharma
Keyword(s):  
Modular Forms ◽  
Deformation Theory ◽  
Galois Representations ◽  
Galois Representation ◽  
Mild Conditions ◽  
Residual Image ◽  
Hida Families

We consider certain [Formula: see text]-ordinary non-CM Hida families with full residual Galois representation and give mild conditions under which every arithmetic point in these families is locally indecomposable when [Formula: see text]. The proof uses methods from deformation theory and mostly works for any odd prime [Formula: see text], but ultimately relies on the existence of a weight [Formula: see text] form in an auxiliary family which is available only for [Formula: see text]. We end by giving several non-trivial examples of [Formula: see text]-ordinary non-CM locally indecomposable modular forms of small level with full residual Galois representation.

scholarly journals The geometry of Hida families II: -adic -modules and -adic Hodge theory

2018 ◽  
Vol 154 (4) ◽  
pp. 719-760
Author(s):  
Bryden Cais
Keyword(s):  
Galois Representations ◽  
Hodge Theory ◽  
Geometric Proof ◽  
De Rham Cohomology ◽  
Etale Cohomology ◽  
The Family ◽  
Étale Cohomology ◽  
De Rham ◽  
Hida Families ◽  
And Control

We construct the $\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the $\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s $\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [$\unicode[STIX]{x1D6F7}$–$\unicode[STIX]{x1D6E4}$modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable $\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.

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 Congruences with Eisenstein series and -invariants

2019 ◽  
Vol 155 (5) ◽  
pp. 863-901 ◽  
Author(s):  
Joël Bellaïche ◽  
Robert Pollack
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Eisenstein Series ◽  
Galois Representations ◽  
Selmer Groups ◽  
Main Conjecture ◽  
The Family ◽  
Eisenstein Ideal ◽  
Hida Families

We study the variation of $\unicode[STIX]{x1D707}$-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $p$-adic zeta function. This lower bound forces these $\unicode[STIX]{x1D707}$-invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $U_{p}-1$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $p$-adic $L$-function is simply a power of $p$ up to a unit (i.e. $\unicode[STIX]{x1D706}=0$). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

Global Deformation Theory

2018 ◽  
pp. 111-267
Author(s):  
Gert-Martin Greuel ◽  
Christoph Lossen ◽  
Eugenii Shustin
Keyword(s):  
Deformation Theory ◽  
Global Deformation

Local torsion on abelian surfaces with real multiplication by ${\bf Q}(\sqrt{5})$

2014 ◽  
Vol 10 (07) ◽  
pp. 1807-1827
Author(s):  
Adam Gamzon
Keyword(s):  
Elliptic Curve ◽  
Deformation Theory ◽  
Galois Representations ◽  
Finite Extension ◽  
Text Structure ◽  
Torsion Point ◽  
Abelian Surfaces ◽  
Real Multiplication

Fix an integer d ≥ 1. In 2008, David and Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This paper proves an analogous averaging result for principally polarized abelian surfaces A over Q with real multiplication by [Formula: see text] and a level-[Formula: see text] structure. Furthermore, we indicate how the result on abelian surfaces with real multiplication relates to the deformation theory of modular Galois representations.

scholarly journals Images of Galois representations in mod p Hecke algebras

2020 ◽  
pp. 1-21
Author(s):  
Laia Amorós
Keyword(s):  
Finite Field ◽  
Commutative Algebra ◽  
Galois Representations ◽  
Residue Field ◽  
Representation Formula ◽  
Galois Representation ◽  
Number Fields ◽  
Abelian Extensions ◽  
Hecke Eigenform ◽  
Coefficient Ring

Let [Formula: see text] denote the mod [Formula: see text] local Hecke algebra attached to a normalized Hecke eigenform [Formula: see text], which is a commutative algebra over some finite field [Formula: see text] of characteristic [Formula: see text] and with residue field [Formula: see text]. By a result of Carayol we know that, if the residual Galois representation [Formula: see text] is absolutely irreducible, then one can attach to this algebra a Galois representation [Formula: see text] that is a lift of [Formula: see text]. We will show how one can determine the image of [Formula: see text] under the assumptions that (i) the image of the residual representation contains [Formula: see text], (ii) [Formula: see text] and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain [Formula: see text]-elementary abelian extensions of big non-solvable number fields.

scholarly journals Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

2015 ◽  
Vol 218 ◽  
pp. 125-173
Author(s):  
Tadashi Ochiai ◽  
Kazuma Shimomoto
Keyword(s):  
System Theory ◽  
Deformation Theory ◽  
Galois Representations ◽  
Hyperplane Section ◽  
Euler System ◽  
Local Rings ◽  
Local Domain ◽  
Strong Version ◽  
Mixed Characteristic ◽  
Torsion Modules

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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