A Quick Introduction to Algebraic Geometry and Elliptic Curves

2003 ◽  
pp. 5-31
Author(s):  
D. S. Nagaraj ◽  
B. Sury
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves

scholarly journals Mazur’s rational torsion result for pointless genus one curves: examples

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Arjan Dwarshuis ◽  
Majken Roelfszema ◽  
Jaap Top
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Rational Point ◽  
Torsion Points ◽  
Arbitrary Genus ◽  
Mazur’S Theorem ◽  
Genus One

AbstractThis note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

scholarly journals Vector-valued modular forms and the modular orbifold of elliptic curves

2016 ◽  
Vol 13 (01) ◽  
pp. 39-63 ◽  
Author(s):  
Luca Candelori ◽  
Cameron Franc
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Vector Bundles ◽  
Projective Line ◽  
Free Module ◽  
Weighted Projective Line ◽  
Holomorphic Vector Bundles ◽  
Vector Valued ◽  
Standard Techniques

This paper presents the theory of holomorphic vector-valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector-valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of [Formula: see text] play in the holomorphic theory of vector-valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line [Formula: see text].

Hodge Theory and Complex Algebraic Geometry II

2003 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Algebraic Geometry ◽  
Hodge Theory

Elliptic Curves and Big Galois Representations

2008 ◽  
Author(s):  
Daniel Delbourgo
Keyword(s):  
Elliptic Curves ◽  
Galois Representations

Elliptic Curves

1997 ◽  
Author(s):  
Henry McKean ◽  
Victor Moll
Keyword(s):  
Elliptic Curves

A First Course in Computational Algebraic Geometry

2013 ◽  
Author(s):  
Wolfram Decker ◽  
Gerhard Pfister
Keyword(s):  
Algebraic Geometry ◽  
Computational Algebraic Geometry
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