Elliptic Curves and Modular Forms Number Theory Math

Elliptic Curves and Modular Forms Number Theory Math

Elliptic Curves and Modular Forms

Elliptic Curves are related to the solutions to equations y^2 = x^3 + A x + B in the field of rationals, algebraic extensions of the rationals, p-adic rational numbers, or a finite field. They are used in factorization of integers and also played a role in the recent resolution of the conjecture known as Fermat's Last Theorem.

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See Also:

- Slides (GIF) of lectures by Karl Rubin at Stanford University.
- Elliptic Curve Discrete Logarithms Project. They solved ECC2K-108 in April 2000. History and related papers.
- Notes by William A. Stein of a course by Ken Ribet.
- Introductory notes by Charles Daney.
- Publications including the joint paper with Andrew Wiles which completed the proof of Fermat's Last Theorem.
- Books and papers relating to the Conway-Norton-Thompson Moonshine conjecture, proved by Richard Borcherds.
- By Len Charlap, David Robbins and Raymond Coley. Downloadable text in PostScript (.ps) format.
- Compiled by Paul Garrett, 1996.
- 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.
- Links to research papers maintained by Stéfane Fermigier.
- From a course on modular forms.
- PDF-format article by Henri Darmon on the completion of the proof by Wiles, Breuil, Conrad, Diamond and Taylor.
- Papers and surveys by Ed Schaefer.
- Workshop, MSRI Berkeley, 6-10 December 1999.
- From the Known Math series.
- Full notes as .dvi, .pdf, and .ps files for all the advanced courses J. S. Milne taught between 1986 and 1999.
- A Clay Mathematics Institute Prize problem, with description by Andrew Wiles [PDF] and lecture by Fernando Rodriguez-Villegas [.ram].
- Lecture notes by Jan Nekovář (PS/PDF).
- Book by John Cremona, with introduction, tables and software.
- A course by Jerrold Tunnell. An introduction to rational points on elliptic curves through examples.
- Tom Fisher's Ph.D. thesis (Cambridge, 2000) in DVI and PS format.
- Marc Joye's list of elliptic curve resources includes people, books, and links. Many preprints are available from the site.
- Syllabus and detailed reading list by Miles Reid, University of Warwick.
- Web text by Andreas Enge.
- Elementary introduction and brief explanation of some well-known results.
- Lecture notes by Johan P. Hansen.
- A semester-long seminar studying Kolyvagin's application of Euler systems to elliptic curves. Includes extensive lecture notes in PostScript or DVI format.
- the surprising and mysterious connections between the monster (and also other finite sporadic simple groups) and modular functions.
- The ECMNET Project to find large factors by the Elliptic Curve Method, mainly Cunningham numbers.
- Lecture notes from a seminar J. Lubin, J.-P. Serre and J. Tate.
- Notes of a 1996 Berkeley course of Ken Ribet's on modular forms and Hecke operators.
- Explains the difference between an elliptical curve and an ellipse. Discusses fields, applications, choosing a fixed point, and related topics.
- Lecture notes and surveys by Ralph Greenberg, University of Washington (PS).
- Including proof of the Moonshine Conjecture (TeX,DVI,PDF).
- Articles and links, compiled by Graham Everest.
- A table up to rank 24 compiled by Andrej Dujella.
- 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.
- William Stein, Ph.D. thesis, Berkeley, 2000.
- By Graham Everest.
- 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.
- Includes errata for his books Rational Points on Elliptic Curves and Advanced Topics in the Arithmetic of Elliptic Curves.

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