Gourt > Science > Math > Number Theory > Elliptic Curves and Modular Forms (2)
In mathematics, an elliptic curve is an algebraic curve defined by an equation of the form
- y2 = x3 + a x + b,
If y2 = P(x), where P is any polynomial of degree three or four in x with no repeated roots, then we obtain a nonsingular plane curve of genus one, which is often also called an elliptic curve. Even more generally, an algebraic curve of genus one, for example from the intersection of two three-dimensional quadric surfaces, is called an elliptic curve.
One finds that elliptic curves correspond to embeddings of the torus into the complex projective plane; such embeddings generalize to arbitrary fields, and so it is said that elliptic curves are non-singular projective algebraic curves of genus 1 over a field K, together with a distinguished point defined over K. The natural group structure of a torus manifests itself in a curious geometric way on an elliptic curve; the set of points of the curve form an abelian group.
More on [ Elliptic curve ]
Fermat's Last Theorem :: Diophantine Equations
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Primality Proving :: Primality Tests
Modular forms and elliptic curves!
mereghost (Marcello Rocha) Wed, 02 Dec 2009 22:34:09 -0000
Modular forms and elliptic curves!
14H52: Elliptic Curves - From the Known Math series.
A Proof of the Full Shimura-Taniyama-Weil Conjecture - PDF-format article by Henri Darmon on the completion of the proof by Wiles, Breuil, Conrad, Diamond and Taylor.
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Algorithms for Modular Elliptic Curves - Book by John Cremona, with introduction, tables and software.
An Elementary Introduction to Elliptic Curves - By Len Charlap, David Robbins and Raymond Coley. Downloadable text in PostScript (.ps) format.
Arithmetic of Cuves - Papers and surveys by Ed Schaefer.
Bibliography for Automorphic and Modular Forms, L-Functions, Representations, and Number Theory - Compiled by Paul Garrett, 1996.
Counting Points on Elliptic Curves - Robert Harley, Pierrick Gaudry, François Morain and Mireille Fouquet have established new records for point counting in characteristic 2, using a new algorithm by to Takakazu Satoh.
Course Notes - Full notes as .dvi, .pdf, and .ps files for all the advanced courses J. S. Milne taught between 1986 and 1999.
ECDL Project - Elliptic Curve Discrete Logarithms Project. They solved ECC2K-108 in April 2000. History and related papers.
Meta Description: [ ECDL Project front page ]
ECMNET - The ECMNET Project to find large factors by the Elliptic Curve Method, mainly Cunningham numbers.
Elliptic Curves - Links to research papers maintained by Stéfane Fermigier.
Elliptic Curves and Elliptic Functions - Introductory notes by Charles Daney.
Elliptic Curves and Formal Groups - Lecture notes from a seminar J. Lubin, J.-P. Serre and J. Tate.
Elliptic Curves and Right Triangles - Slides (GIF) of lectures by Karl Rubin at Stanford University.
Elliptic Curves and Their Applications to Cryptography - Web text by Andreas Enge.
Meta Description: [ Elliptic Curves and Their Applications to Cryptography: An Introduction ]
Elliptic Curves Handout - Syllabus and detailed reading list by Miles Reid, University of Warwick.
Elliptic Curves II - Lecture notes by Johan P. Hansen.
Elliptic Curves with H. A. Verrill - Lecture notes and resources by Helena Verrill, Louisiana State University, 2004.
Elliptic Divisibility Sequences - Articles and links, compiled by Graham Everest.
Elliptic Functions and Elliptic Curves - Lecture notes by Jan Nekovář (PS/PDF).
Elliptical Curve Cryptography - Explains the difference between an elliptical curve and an ellipse. Discusses fields, applications, choosing a fixed point, and related topics.
Explicit Approaches to Modular Abelian Varieties - William Stein, Ph.D. thesis, Berkeley, 2000.
History of Elliptic Curve Rank Records - A table up to rank 24 compiled by Andrej Dujella.
Meta Description: [ Table of elliptic curve rank records ]
Iwasawa Theory of Elliptic Curves - Lecture notes and surveys by Ralph Greenberg, University of Washington (PS).
Joseph Silverman - Includes errata for his books Rational Points on Elliptic Curves and Advanced Topics in the Arithmetic of Elliptic Curves.
Kolyvagin Seminar - A semester-long seminar studying Kolyvagin's application of Euler systems to elliptic curves. Includes extensive lecture notes in PostScript or DVI format.
Mathematical Things - Tom Womack's pages address many elliptic curve subjects, including curves of given rank and small conductor, Mordell curves of large rank, and interesting torsion groups.
Modular Forms and Hecke Operators - Notes by William A. Stein of a course by Ken Ribet.
Modular Forms Course - Notes of a 1996 Berkeley course of Ken Ribet's on modular forms and Hecke operators.
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Modular Forms Example Sheets - From a course on modular forms.
Monstrous Moonshine - the surprising and mysterious connections between the monster (and also other finite sporadic simple groups) and modular functions.
Moonshine Bibliography - Books and papers relating to the Conway-Norton-Thompson Moonshine conjecture, proved by Richard Borcherds.
On 5 and 7 Descents for Elliptic Curves - Tom Fisher's Ph.D. thesis (Cambridge, 2000) in DVI and PS format.
Papers by Richard Borcherds - Including proof of the Moonshine Conjecture (TeX,DVI,PDF).
Prime Values of Elliptic Divisibility Sequences - By Graham Everest.
Rational Points on Elliptic Curves - A course by Jerrold Tunnell. An introduction to rational points on elliptic curves through examples.
Recent Progress in the Theory of Elliptic Curves - An abstract to Henri Darmon's and Bertolini's work, which approaches a p-adic variant of the Birch - Swinnerton-Dyer conjecture, for curves of rank higher than one.
Richard Taylor - Publications including the joint paper with Andrew Wiles which completed the proof of Fermat's Last Theorem.
The Birch and Swinnerton-Dyer Conjecture - A Clay Mathematics Institute Prize problem, with description by Andrew Wiles [PDF] and lecture by Fernando Rodriguez-Villegas [.ram].
Torsion Points on Elliptic Curves - Elementary introduction and brief explanation of some well-known results.
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