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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 14H52: Elliptic Curves

Introduction

This is a fascinating area of algebraic geometry dealing with nonsingular curves of genus 1 -- in English, solutions to equations y^2 = x^3 + A x + B. It turns out to have important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography). Also, what appear to be simple Diophantine equations often lead to elliptic curves. Through Riemann surfaces it has connections to topology; through modular forms and zeta functions to analysis. Elliptic curves also played a role in the recent resolution of the conjecture known as Fermat's Last Theorem.

History

Applications and related fields

See also 11G05, 11G07, 14Kxx

Subfields

Parent field: 14H - Algebraic Curves

Textbooks, reference works, and tutorials

Online texts on elliptic curves by James Milne and Ian Connell

Software and tables

If you have Maple, get Connell's APECS package, which does nearly everything anyone knows an algorithm for on an elliptic curve. Here's a link to APECS.

If you need to compute ranks of elliptic curves over the rationals, you definitely need Cremona's MWRANK. See his book or his web site for more information.

Others:

• SIMATH is particularly useful for finding all integral points on curves.
• "Eliptic", a public domain implementation of elliptic curves public key cryptography. [not tested]
• For comic relief you might want this UBASIC program written to look for rational points on a certain elliptic curve with a square x coordinate. (Turns out to be none)

Other web sites with this focus

• Comprehensive elliptic curves site

Selected topics at this site

• Historical introduction to elliptic curves.
• More on the historical introduction to elliptic curves.
• Descriptions of the method to put a curve in Weierstrass form.
• Example of a transformation of an elliptic curve into Weierstrass form
• An example of transformation to normal form for an elliptic curve.
• Maple input file to force a curve with rational point into normal form.
• Maple input file for reducing a curve y^2 = quartic to normal form, too.)
• Putting an elliptic into Weierstrass canonical form
• Can one parameterize the points on an elliptic curve (e.g. y^2=x^3+x, y^2=quartic in x)? (no)
• Elliptic curves must have an identity element. (Example: 3x^3+4y^3=5)
• A derivation of the explicit formula for the group law .
• The group structure of curves over Z/pZ
• Deciding whether a point on an elliptic curve is the double of another point
• Summary of Mazur's theorem on the possible torsion subgroups of E(Q)
• Possible torsion in elliptic curves over number fields
• Points of order 13 on elliptic curves
• Recent research has looked for elliptic curves over Q with high rank.
• Elliptic curves with high rank? A summary
• Elliptic curves with high rank? Another summary
• Henri Cohen on curves with high rank.
• Elliptic curves of rank 5
• Elliptic curves of rank=8 with torsion group=Z/2Z*Z/2Z
• Example of a curve of rank 23
• Comparison of generic v. specific ranks of families of elliptic curves
• People have looked for curves with large Sha (if you have to ask what Sha is, you don't want to know)
• What is the Shafarevich group Sha
• Long summary of modularity and other terms used. (It's a good intro to the theory).
• One answer to "What does it mean for a curve to be modular?".
• Connections of Dirichlet series and related topics to modularity of elliptic curves.
• Some ways of phrasing the Taniyama conjecture, an important case of which was solved by Wiles.
• Some comments on the constructiveness of Wiles' proof.
• Effective calculations of discriminant (etc) without having normal forms.
• Worked-out analysis of one equation worth money!
• We ask (not answer) the question, "for which quadratic extensions of Q does the curve x^3+y^3=z^3 have positive rank?"
• Examination of (x+y+z)^3=xyz
• Examination of x^2+y^3=z^6
• Solutions to x^2 + 2 = y^3
• Solutions to x^3 + y^3 = a z^3?
• "The sum of first n squares can not be an perfect square, except for n=24."
• When is the sum of consecutive cubes again a cube? (Solve x*y*(x^2+y^2-1)=z^3)
• The Times puzzle: find rational solutions to x^3+y^3=6.
• Which integers may be written as the sum of two rational cubes?
• Integer points on y^2 = x^4+x^3+x^2+x+1
• Some quartic equations in two variables (exercises from Rosen)
• Solutions to a x^2 + b y^3 = z^6 and the elliptic curves y^2=x^3+D
• Multiple integer solutions to x^3+y^3=N (the taxicab example) and the elliptic curves y^2=x^3+D
• Integral points on the elliptic curves y^2 = x^3 +- 2909 *2^k ?
• Integers satisfying two Pellian equations (leads to curves y^2 = x^3 - D x )
• There are no four squares in arithmetic progression
• Triangular number which is also a square-pyramidal number
• Use of elliptic curves for factoring.
• Two papers by Noam Elkies on the study of lattices arising as Mordell-Weil groups [Now a link to his site]
• Application of isogeny to a question about fields.
• Finding integer points on curves (e.g. y^2=x^3+17). Mention of SIMATH.
• Extreme examples concerning Hall's conjecture: no "big" solutions to y^2 = x^3 + k
• Reducing the search for integer points on y^2=quartic to Thue equations.
• Finding a provably complete set of integral points on an elliptic curve.
• Solving {a^2+b^2=square, a^2+(2b)^2=square} by infinite descent.
• Long example showing how to use APECS to analyze an elliptic curve.
• Example of use of the program MWRANK to examine sample elliptic curves.
• Right triangles with integer area and integer (or rational) sides (includes the "congruent number problem")
• How many triangles with rational sides and a given rational area?
• Looking for points on curves of the form y^2=x(x^2-d^2) (Tunnel's theorem).
• Two (unstructured) equations equations in three unknowns lead to an elliptic curve (although integer points are not fully known).
• Small integer values of |x^3-y^2|
• Using the Jacobian of y^2=quartic to transform the curve to Weierstrass form
• Parameterize the curve where a sphere and cylinder intersect? (no)
• Looking for triples of numbers satisfying simultaneous Pell (i.e. quadratic) equations