Elliptic curves

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Frey curve

Let be a solution to Fermat's last theorem. Then the corresponding Frey curve is(1)Ribet (1990a) showed that such curves cannot be modular, so if the Taniyama-Shimura conjecture were true, Frey curves couldn't exist and Fermat's last theorem would follow with even and . Frey curves are semistable. Invariants include the elliptic discriminant(2)The minimal discriminant is(3)the j-conductor is(4)and the j-invariant is(5)

Weierstrass form

There are (at least) two mathematical objects known as Weierstrass forms. The first is a general form into which an elliptic curve over any field can be transformed, given bywhere , , , , and are elements of .The second is the definition of the gamma functionaswhere is the Euler-Mascheroni constant (Krantz 1999, p. 157).

Elliptic discriminant

An elliptic curve is the set of solutions to anequation of the form(1)By changing variables, , assuming the field characteristic is not 2, the equation becomes(2)where(3)(4)(5)Define also the quantity(6)then the discriminant is given by(7)The discriminant depends on the choice of equations, and can change after a changeof variables, unlike the j-invariant.If the field characteristic is neither 2or 3, then its equation can be written as(8)in which case, the discriminant is given by(9)Algebraically, the discriminant is nonzero when the right-hand side has three distinct roots. In the classical case of an elliptic curve over the complex numbers, the discriminant has a geometric interpretation. If , then the elliptic curve is nonsingular and has curve genus 1, i.e., it is a torus. If and , then it has a cusp singularity, in which case there is one tangent direction at the singularity. If and , then its singularity is called an ordinary double..

Elliptic curve group law

The group of an ellipticcurve which has been transformed to the formis the set of -rational points, including the single point at infinity. The group law (addition) is defined as follows: Take 2 -rational points and . Now 'draw' a straight line through them and compute the third point of intersection (also a -rational point). Thengives the identity point at infinity. Now find the inverse of , which can be done by setting giving .This remarkable result is only a special case of a more general procedure. Essentially, the reason is that this type of elliptic curve has a single point at infinity which is an inflection point (the line at infinity meets the curve at a single point at infinity, so it must be an intersection of multiplicity three).

Elliptic curve

Informally, an elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus. The Weierstrass elliptic function describes how to get from this torus to the algebraic form of an elliptic curve.Formally, an elliptic curve over a field is a nonsingular cubic curve in two variables, , with a -rational point (which may be a point at infinity). The field is usually taken to be the complex numbers , reals , rationals , algebraic extensions of , p-adic numbers , or a finite field.By an appropriate change of variables, a general elliptic curve over a field with field characteristic , a general cubic curve(1)where , , ..., are elements of , can be written in the form(2)where the right side of (2) has no repeated factors. Any elliptic curve not of characteristic 2 or 3 can also be written in Legendre normal form(3)(Hartshorne 1999).Elliptic curves are illustrated above for various values of and..

Semistable

When a prime divides the elliptic discriminant of a elliptic curve , two or all three roots of become congruent (mod ). An elliptic curve is semistable if, for all such primes , only two roots become congruent mod (with more complicated definitions for or 3).

Hyperelliptic curve

A hyperelliptic curve is an algebraic curve given by an equation of the form , where is a polynomial of degree with distinct roots.If is a cubic or quartic polynomial, then the curve is called an elliptic curve.The genus of a hyperelliptic curve is related to the degree of the polynomial. A polynomial of degree or gives a curve of genus . All curves of genus 2 are hyperelliptic, but for larger genus there are curves that are not hyperelliptic.

Ochoa curve

The Ochoa curve is the elliptic curvegiven in Weierstrass form asThe complete set of 23 integer solutions (where solutions of the form are counted as a single solution) to this equation consists of , (, 4520), (, 13356), (, 14616), (, 10656), (91, 8172), (227, 4228), (247, 3528), (271, 2592), (455, 200), (499, 3276), (523, 4356), (530, 4660), (599, 7576), (751, 14112), (1003, 25956), (1862, 75778), (3511, 204552), (5287, 381528), (23527, 3607272), (64507, 16382772), (100102, 31670478), and (1657891, 2134685628) (OEIS A141144 and A141145; Stroeker and de Weger 1994).

Mordell curve

An elliptic curve of the form for an integer. This equation has a finite number of solutions in integers for all nonzero . If is a solution, it therefore follows that is as well.Uspensky and Heaslet (1939) give elementary solutions for , , and 2, and then give , , , and 1 as exercises. Euler found that the only integer solutions to the particular case (a special case of Catalan's conjecture) are , , and . This can be proved using Skolem's method, using the Thue equation , using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky and Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński and Schinzel (1988, pp. 75-80), and Metsaenkylae (2003).Solutions of the Mordell curve with are summarized in the table below for small .solutions123456none7none8910Values of such that the Mordell curve has no integer solutions are given by 6, 7, 11, 13,..

Burnside curve

The only known classically known algebraic curve of curve genus that has an explicit parametrization in terms of standard special functions (Burnside 1893, Brezhnev 2001). This equation is given by(1)The closed portion of the curve has area(2)(3)where is a gamma function.The closed portion of this curve has a parametrization in terms of the Weierstrasselliptic function given by(4)(5)where(6)the half-periods are given by and ranges over complex values (Brezhnev 2001).

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