A classification of ECM-friendly families of elliptic curves using modular curves

2021 ◽  
Author(s):  
Razvan Barbulescu ◽  
Sudarshan Shinde
Keyword(s):  
Elliptic Curves ◽  
Modular Curves

scholarly journals On the singularity structure of Kahan discretizations of a class of quadratic vector fields

2021 ◽  
Author(s):  
René Zander
Keyword(s):  
Elliptic Curves ◽  
Vector Fields ◽  
Singularity Structure ◽  
Quadratic Vector Fields ◽  
Parameter Values

AbstractWe discuss the singularity structure of Kahan discretizations of a class of quadratic vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic curves.

scholarly journals Neron Classification of Elliptic Curves Where the Residual Characteristics Equal 2 Or 3

1993 ◽  
Vol 44 (2) ◽  
pp. 119-152 ◽  
Author(s):  
I. Papadopoulos
Keyword(s):  
Elliptic Curves

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.

Derived Category and Canonical Bundle – I

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Elliptic Curves ◽  
Derived Categories ◽  
Derived Category ◽  
Canonical Bundle ◽  
Smooth Curves ◽  
Coherent Sheaves

This chapter is devoted to results by Bondal and Orlov which show that for varieties with ample (anti-)canonical bundle, the bounded derived category of coherent sheaves determines the variety. Except for the case of elliptic curves, this settles completely the classification of derived categories of smooth curves. The complexity of the derived category is reflected by its group of autoequivalences. This is studied by means of ample sequences.

scholarly journals A MODULI INTERPRETATION FOR THE NON-SPLIT CARTAN MODULAR CURVE

2017 ◽  
Vol 60 (2) ◽  
pp. 411-434 ◽  
Author(s):  
MARUSIA REBOLLEDO ◽  
CHRISTIAN WUTHRICH
Keyword(s):  
Elliptic Curves ◽  
Half Plane ◽  
London Math ◽  
Number Fields ◽  
Arithmetic Geometry ◽  
Modular Curves ◽  
Torsion Points ◽  
Modular Curve ◽  
Upper Half Plane ◽  
Cartan Subgroups

AbstractModular curves likeX0(N) andX1(N) appear very frequently in arithmetic geometry. While their complex points are obtained as a quotient of the upper half plane by some subgroups of SL2(ℤ), they allow for a more arithmetic description as a solution to a moduli problem. We wish to give such a moduli description for two other modular curves, denoted here byXnsp(p) andXnsp+(p) associated to non-split Cartan subgroups and their normaliser in GL2(𝔽p). These modular curves appear for instance in Serre's problem of classifying all possible Galois structures ofp-torsion points on elliptic curves over number fields. We give then a moduli-theoretic interpretation and a new proof of a result of Chen (Proc. London Math. Soc.(3)77(1) (1998), 1–38;J. Algebra231(1) (2000), 414–448).

scholarly journals Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

2010 ◽  
Vol 130 (9) ◽  
pp. 1964-1968 ◽  
Author(s):  
Filip Najman
Keyword(s):  
Elliptic Curves ◽  
Complete Classification ◽  
Cyclotomic Fields

Cubic forms, powers of primes and classification of elliptic curves

2021 ◽  
Author(s):  
Karolina Chałupka
Keyword(s):  
Elliptic Curves ◽  
Cubic Forms

scholarly journals Quadratic points on non-split Cartan modular curves

2021 ◽  
pp. 1-23
Author(s):  
Philippe Michaud-Rodgers
Keyword(s):  
Elliptic Curves ◽  
Rational Points ◽  
Quadratic Fields ◽  
Modular Curves ◽  
Symmetric Powers

In this paper, we study quadratic points on the non-split Cartan modular curves [Formula: see text], for [Formula: see text] and [Formula: see text]. Recently, Siksek proved that all quadratic points on [Formula: see text] arise as pullbacks of rational points on [Formula: see text]. Using similar techniques for [Formula: see text], and employing a version of Chabauty for symmetric powers of curves for [Formula: see text], we show that the same holds for [Formula: see text] and [Formula: see text]. As a consequence, we prove that certain classes of elliptic curves over quadratic fields are modular.

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