scholarly journals A deformation problem for Galois representations over imaginary quadratic fields

2009 ◽  
Vol 8 (4) ◽  
pp. 669-692 ◽  
Author(s):  
Tobias Berger ◽  
Krzysztof Klosin
Keyword(s):  
Valuation Ring ◽  
Quadratic Field ◽  
Galois Representations ◽  
Discrete Valuation Ring ◽  
Quadratic Fields ◽  
Sufficient Condition ◽  
Imaginary Quadratic Fields ◽  
Field Extensions ◽  
Deformation Problem ◽  
Deformation Ring

AbstractWe prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

scholarly journals LIFTING TORSION GALOIS REPRESENTATIONS

2015 ◽  
Vol 3 ◽  
Author(s):  
CHANDRASHEKHAR KHARE ◽  
RAVI RAMAKRISHNA
Keyword(s):  
Valuation Ring ◽  
Fourier Coefficients ◽  
Galois Representations ◽  
Finite Extension ◽  
Discrete Valuation Ring ◽  
Ring Of Integers ◽  
Ramification Index ◽  
Complete Discrete Valuation Ring ◽  
Free Level ◽  
Deformation Ring

Let $p\geqslant 5$ be a prime, and let ${\mathcal{O}}$ be the ring of integers of a finite extension $K$ of $\mathbb{Q}_{p}$ with uniformizer ${\it\pi}$. Let ${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$ have modular mod-${\it\pi}$ reduction $\bar{{\it\rho}}$, be ordinary at $p$, and satisfy some mild technical conditions. We show that ${\it\rho}_{n}$ can be lifted to an ${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when $K$ is a ramified extension of $\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring ${\mathcal{O}}$ is the weight-$2$ deformation ring for $\bar{{\it\rho}}$ for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable $\bar{{\it\rho}}$ of weight 2 can have arbitrarily large ramification index at $p$.

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 Truncated Euler Systems over Imaginary Quadratic Fields

2009 ◽  
Vol 195 ◽  
pp. 97-111
Author(s):  
Soogil Seo
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Abelian Extension ◽  
Iwasawa Theory ◽  
Quadratic Fields ◽  
Galois Module ◽  
Imaginary Quadratic Fields ◽  
Euler Systems ◽  
Graded Module ◽  
Elliptic Units

AbstractLet K be an imaginary quadratic field and let F be an abelian extension of K. It is known that the order of the class group ClF of F is equal to the order of the quotient UF/ElF of the group of global units UF by the group of elliptic units ElF of F. We introduce a filtration on UF/ElF made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

scholarly journals NOTE ON THE DIVISIBILITY OF THE CLASS NUMBER OF CERTAIN IMAGINARY QUADRATIC FIELDS

2009 ◽  
Vol 51 (1) ◽  
pp. 187-191 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Diophantine Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe prove that the class number of the imaginary quadratic field $\Q(\sqrt{2^{2k}-3^n})$ is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

scholarly journals Gorenstein tiled orders

2014 ◽  
Vol 6 (2) ◽  
pp. 260-281
Author(s):  
V. Kiricenko ◽  
M. Khibina ◽  
L. Mashchenko ◽  
M. Plakhotnyk ◽  
V. Zhuravlev
Keyword(s):  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Main Diagonal ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The aim of this article is to describe exponent matrices of Gorenstein tiled orders. The necessary and sufficient condition for possibility such to construct a Gorenstein tiled order which has given orders over some discrete valuation ring (the unique for the whole main diagonal) on the main diagonal and its permutation be a product of correspond cycles with given cyclic Gorenstein tiled order is considered.

scholarly journals Similar Sublattices of Planar Lattices

2011 ◽  
Vol 63 (6) ◽  
pp. 1220-1537 ◽  
Author(s):  
Michael Baake ◽  
Rudolf Scharlau ◽  
Peter Zeiner
Keyword(s):  
Dirichlet Series ◽  
Generating Functions ◽  
Quadratic Field ◽  
Zeta Functions ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Planar Lattice ◽  
Planar Lattices

AbstractThe similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields.

The Square Sieve and the Lang–Trotter Conjecture

2005 ◽  
Vol 57 (6) ◽  
pp. 1155-1177 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Etienne Fouvry ◽  
M. Ram Murty
Keyword(s):  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Upper Bounds ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Frobenius Endomorphism ◽  
Square Sieve

AbstractLet E be an elliptic curve defined over ℚ and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which ℚ(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, under a generalized Riemann hypothesis we show that this number is OE(x17/18 log x), and unconditionally we show that this number is We also prove that the number of imaginary quadratic fields K, with −disc K ≤ x and of the form K = ℚ(πp), is ≫E log log log x for x ≥ x0(E). These results represent progress towards a 1976 Lang–Trotter conjecture.

scholarly journals The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields

2015 ◽  
Vol 11 (06) ◽  
pp. 1961-2017 ◽  
Author(s):  
Rodney Lynch ◽  
Patrick Morton
Keyword(s):  
Nontrivial Solution ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Explicit Formulas ◽  
Odd Degree ◽  
Class Fields ◽  
Fermat Equation ◽  
Hilbert Class Field ◽  
Integral Solutions

It is shown that the quartic Fermat equation x4 + y4 = 1 has nontrivial integral solutions in the Hilbert class field Σ of any quadratic field [Formula: see text] whose discriminant satisfies -d ≡ 1 (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in [Formula: see text], for p (> 7) a prime congruent to 7 (mod 8), but does have a nontrivial solution in the odd degree extension Σ of K. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in detail and the results are applied to prove several properties of the Weber singular moduli introduced by Yui and Zagier.

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