scholarly journals Modular invariants and isogenies

2019 ◽  
Vol 15 (03) ◽  
pp. 569-584
Author(s):  
Fabien Pazuki
Keyword(s):  
Elliptic Curves ◽  
Abelian Varieties ◽  
Classical Estimate ◽  
Modular Invariants ◽  
Explicit Bounds ◽  
The Difference ◽  
Faltings Height ◽  
Modular Polynomials

We provide explicit bounds on the difference of heights of the [Formula: see text]-invariants of isogenous elliptic curves defined over [Formula: see text]. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman’s inequality and isogeny estimates by Gaudron and Rémond. We give applications in the study of Vélu’s formulas and of modular polynomials.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

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 Average of the first invariant factor of the reductions of abelian varieties of CM type

2016 ◽  
Vol 12 (02) ◽  
pp. 445-463 ◽  
Author(s):  
Sungjin Kim
Keyword(s):  
Abelian Group ◽  
Asymptotic Formula ◽  
Prime Ideal ◽  
Elliptic Curves ◽  
Short Range ◽  
Abelian Varieties ◽  
Prime Ideals ◽  
Good Reduction ◽  
Invariant Factor ◽  
Field Of Definition

For a field of definition [Formula: see text] of an abelian variety [Formula: see text] and prime ideal [Formula: see text] of [Formula: see text] which is of a good reduction for [Formula: see text], the structure of [Formula: see text] as abelian group is: [Formula: see text] where [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text]. We are interested in finding an asymptotic formula for the number of prime ideals [Formula: see text] with [Formula: see text], [Formula: see text] has a good reduction at [Formula: see text], [Formula: see text]. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.

scholarly journals Logarithmic abelian varieties, III: Logarithmic elliptic curves and modular curves

2013 ◽  
Vol 210 ◽  
pp. 59-81 ◽  
Author(s):  
Takeshi Kajiwara ◽  
Kazuya Kato ◽  
Chikara Nakayama
Keyword(s):  
Elliptic Curves ◽  
Abelian Varieties ◽  
Modular Curves

AbstractWe illustrate the theory of log abelian varieties and their moduli in the case of log elliptic curves.

scholarly journals Computing Modular Polynomials

2005 ◽  
Vol 8 ◽  
pp. 195-204 ◽  
Author(s):  
Denis Charles ◽  
Kristin Lauter
Keyword(s):  
Number Theory ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Probabilistic Algorithm ◽  
Computational Number Theory ◽  
Modular Polynomials

AbstractThis paper presents a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves, and are useful in many aspects of computational number theory and cryptography. The algorithm presented here has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. The need to compute the exponentially large integral coefficients is avoided by working directly modulo a prime, and computing isogenies between elliptic curves via Vélu's formulas.

scholarly journals Elliptic curves on Abelian varieties

2015 ◽  
Vol 59 (2) ◽  
pp. 319-336
Author(s):  
Robert Auffarth II
Keyword(s):  
Elliptic Curves ◽  
Abelian Varieties
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