scholarly journals Oddness of residually reducible Galois representations

2018 ◽  
Vol 14 (05) ◽  
pp. 1329-1345 ◽  
Author(s):  
Tobias Berger
Keyword(s):  
Automorphic Forms ◽  
Galois Representations ◽  
Selmer Group ◽  
Parity Condition

We show that suitable congruences between polarized automorphic forms over a CM field always produce elements in the Selmer group for exactly the ±-Asai (aka tensor induction) representation that is critical in the sense of Deligne. For this, we relate the oddness of the associated polarized Galois representations (in the sense of the Bellaïche-Chenevier sign being [Formula: see text]) to the parity condition for criticality. Under an assumption similar to Vandiver’s conjecture this also provides evidence for the Fontaine–Mazur conjecture for residually reducible polarized Galois representations.

scholarly journals On the Eisenstein ideal for imaginary quadratic fields

2009 ◽  
Vol 145 (03) ◽  
pp. 603-632 ◽  
Author(s):  
Tobias Berger
Keyword(s):  
Lower Bound ◽  
Automorphic Forms ◽  
Quadratic Field ◽  
Galois Representations ◽  
Hecke Algebra ◽  
Quadratic Fields ◽  
Selmer Group ◽  
Imaginary Quadratic Fields ◽  
Eisenstein Ideal ◽  
Eisenstein Cohomology

AbstractFor certain algebraic Hecke charactersχof an imaginary quadratic fieldFwe define an Eisenstein ideal in ap-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the specialL-valueL(0,χ). We further prove that its index is bounded from above by thep-valuation of the order of the Selmer group of thep-adic Galois character associated toχ−1. This uses the work of R. Tayloret al. on attaching Galois representations to cuspforms of GL2/F. Together these results imply a lower bound for the size of the Selmer group in terms ofL(0,χ), coinciding with the value given by the Bloch–Kato conjecture.

scholarly journals Arithmetic gauge theory: A brief introduction

2018 ◽  
Vol 33 (29) ◽  
pp. 1830012 ◽  
Author(s):  
Minhyong Kim
Keyword(s):  
Moduli Spaces ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Arithmetic Geometry ◽  
Quantum Field ◽  
Principal Bundles ◽  
Langlands Program ◽  
Effective Version ◽  
Diophantine Geometry ◽  
Higher Genus

Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

Lifting n-Dimensional Galois Representations

2008 ◽  
Vol 60 (5) ◽  
pp. 1028-1049 ◽  
Author(s):  
Spencer Hamblen
Keyword(s):  
Characteristic Zero ◽  
Galois Representations ◽  
Galois Groups ◽  
Geometric Shape ◽  
General Linear ◽  
Linear Groups ◽  
Selmer Group ◽  
General Linear Groups

AbstractWe investigate the problem of deforming n-dimensional mod p Galois representations to characteristic zero. The existence of 2-dimensional deformations has been proven under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of n-dimensional deformations.We then examine under which conditions we may place restrictions on the shape of our deformations at p, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over ℚ of certain infinite subgroups of p-adic general linear groups.

scholarly journals On some dimension formula for automorphic forms of weight one I

1982 ◽  
Vol 85 ◽  
pp. 213-221 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Fuchsian Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
The Other ◽  
Dimension Formula ◽  
Linearly Independent ◽  
Other Hand ◽  
Algebraic Properties ◽  
Lecture Notes ◽  
Weight One

Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d0 the number of linearly independent automorphic forms of weight 1 for Γ. It would be interesting to have a certain formula for d0. But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d0 using only the basic algebraic properties of Γ. On the other hand, Serre has given such a formula of d0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.

scholarly journals Galois representations attached to automorphic forms on over fields

2014 ◽  
Vol 150 (4) ◽  
pp. 523-567 ◽  
Author(s):  
Chung Pang Mok
Keyword(s):  
Automorphic Forms ◽  
Galois Representations ◽  
Suitable Condition ◽  
Two Dimensional ◽  
Automorphic Representations ◽  
Cuspidal Automorphic Representations ◽  
Central Characters

AbstractIn this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

scholarly journals (p,p)-Galois representations attached to automorphic forms on GLn

2011 ◽  
Vol 252 (2) ◽  
pp. 379-406 ◽  
Author(s):  
Eknath Ghate ◽  
Narasimha Kumar
Keyword(s):  
Automorphic Forms ◽  
Galois Representations

scholarly journals On mod local-global compatibility for in the ordinary case

2017 ◽  
Vol 153 (11) ◽  
pp. 2215-2286 ◽  
Author(s):  
Florian Herzig ◽  
Daniel Le ◽  
Stefano Morra
Keyword(s):  
Unitary Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Serre's Conjecture ◽  
Ordinary Case ◽  
Serre’S Conjecture ◽  
The Way

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$. Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$-action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

scholarly journals Raising the level for

2014 ◽  
Vol 2 ◽  
Author(s):  
JACK A. THORNE
Keyword(s):  
Automorphic Forms ◽  
Galois Representations

AbstractWe prove a simple level-raising result for regular algebraic, conjugate self-dual automorphic forms on$\mathrm{GL}_n$. This gives a systematic way to construct irreducible Galois representations whose residual representation is reducible.

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