Compatibility of arithmetic and algebraic local constants (the case )

2015 ◽  
Vol 151 (9) ◽  
pp. 1626-1646 ◽  
Author(s):  
Jan Nekovář
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Selmer Groups ◽  
Real Multiplication ◽  
Special Cases ◽  
Congruent Modulo ◽  
Multiplication Theorem ◽  
Parity Conjecture

We show that arithmetic local constants attached by Mazur and Rubin to pairs of self-dual Galois representations which are congruent modulo a prime number $p>2$ are compatible with the usual local constants at all primes not dividing $p$ and in two special cases also at primes dividing $p$. We deduce new cases of the $p$-parity conjecture for Selmer groups of abelian varieties with real multiplication (Theorem 4.14) and elliptic curves (Theorem 5.10).

scholarly journals Multiplicative isogeny estimates

1998 ◽  
Vol 64 (2) ◽  
pp. 178-194 ◽  
Author(s):  
David Masser
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Abelian Varieties

AbstractThe theory of isogeny estimates for Abelian varieties provides ‘additive bounds’ of the form ‘d is at most B’ for the degrees d of certain isogenies. We investigate whether these can be improved to ‘multiplicative bounds’ of the form ‘d divides B’. We find that in general the answer is no (Theorem 1), but that sometimes the answer is yes (Theorem 2). Further we apply the affirmative result to the study of exceptional primes ℒ in connexion with modular Galois representations coming from elliptic curves: we prove that the additive bounds for ℒ of Masser and Wüstholz (1993) can be improved to multiplicative bounds (Theorem 3).

scholarly journals Determinants of subquotients of Galois representations associated with abelian varieties

2013 ◽  
Vol 13 (3) ◽  
pp. 517-559 ◽  
Author(s):  
Eric Larson ◽  
Dmitry Vaintrob
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Riemann Hypothesis ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Number Fields ◽  
Torsion Points ◽  
Prime Degree ◽  
Rho A

AbstractGiven an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell $, the ${\ell }^{n} $-torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.Applying our results in dimension $g= 1$, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

scholarly journals Level raising mod 2 and arbitrary 2-Selmer ranks

2016 ◽  
Vol 152 (8) ◽  
pp. 1576-1608 ◽  
Author(s):  
Bao V. Le Hung ◽  
Chao Li
Keyword(s):  
Elliptic Curves ◽  
Abelian Varieties ◽  
Galois Representation ◽  
Selmer Groups

We prove a level raising mod $\ell =2$ theorem for elliptic curves over $\mathbb{Q}$. It generalizes theorems of Ribet and Diamond–Taylor and also explains different sign phenomena compared to odd $\ell$. We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.

scholarly journals Non-openness of v-adic Galois representation for A-motives

2020 ◽  
pp. 1-21
Author(s):  
Maike Ella Elisabeth Frantzen
Keyword(s):  
Elliptic Curves ◽  
Function Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Galois Representation ◽  
Drinfeld Modules ◽  
Ground Field ◽  
Rank One

Drinfeld modules and [Formula: see text]-motives are the function field analogs of elliptic curves and abelian varieties. For both Drinfeld modules and [Formula: see text]-motives, one can construct their [Formula: see text]-adic Galois representations and ask whether the images are open. For Drinfeld modules, this question has been answered by Richard Pink and his co-authors; however, this question has not been addressed for [Formula: see text]-motives. Here, we clarify the rank-one case for [Formula: see text]-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field.

Images of mod p Galois Representations Associated to Elliptic Curves

2001 ◽  
Vol 44 (3) ◽  
pp. 313-322 ◽  
Author(s):  
Amadeu Reverter ◽  
Núria Vila
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Torsion Points ◽  
Galois Action ◽  
Mod P

AbstractWe give an explicit recipe for the determination of the images associated to the Galois action on p-torsion points of elliptic curves. We present a table listing the image for all the elliptic curves defined over without complex multiplication with conductor less than 200 and for each prime number p.

scholarly journals Variation of Tamagawa numbers of semistable abelian varieties in field extensions

2018 ◽  
Vol 166 (3) ◽  
pp. 487-521
Author(s):  
L. ALEXANDER BETTS ◽  
VLADIMIR DOKCHITSER ◽  
V. DOKCHITSER ◽  
A. MORGAN
Keyword(s):  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Complete Classification ◽  
Local Fields ◽  
Selmer Groups ◽  
Genus 2 ◽  
Parity Conjecture ◽  
Tamagawa Numbers ◽  
Principally Polarised Abelian Varieties

AbstractWe investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on thep-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the formy2=f(x), under some simplifying hypotheses.

scholarly journals Integral Tate modules and splitting of primes in torsion fields of elliptic curves

2016 ◽  
Vol 12 (01) ◽  
pp. 237-248 ◽  
Author(s):  
Tommaso Giorgio Centeleghe
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Prime Number ◽  
Galois Representations ◽  
Number Fields ◽  
Tate Module ◽  
Explicit Procedure ◽  
Polynomial Formula ◽  
Class Polynomials

Let [Formula: see text] be an elliptic curve over a finite field [Formula: see text], and [Formula: see text] a prime number different from the characteristic of [Formula: see text]. In this paper, we consider the problem of finding the structure of the Tate module [Formula: see text] as an integral Galois representations of [Formula: see text]. We indicate an explicit procedure to solve this problem starting from the characteristic polynomial [Formula: see text] and the [Formula: see text]-invariant [Formula: see text] of [Formula: see text]. Hilbert Class Polynomials of imaginary quadratic orders play an important role here. We give a global application to the study of prime-splitting in torsion fields of elliptic curves over number fields.

scholarly journals On the parity of ranks of Selmer groups IV. With an appendix by Jean-Pierre Wintenberger

2009 ◽  
Vol 145 (6) ◽  
pp. 1351-1359 ◽  
Author(s):  
Jan Nekovář
Keyword(s):  
Real Number ◽  
Elliptic Curves ◽  
Large Class ◽  
Number Fields ◽  
Selmer Groups ◽  
Parity Conjecture ◽  
Totally Real

AbstractWe prove the parity conjecture for the ranks of p-power Selmer groups (p⁄=2) of a large class of elliptic curves defined over totally real number fields.

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