Editing Elliptic curve (section)

==Elliptic curves over a general field==
Elliptic curves can be defined over any [[field (mathematics)|field]] ''K''; the formal definition of an elliptic curve is a non-singular projective algebraic curve over ''K'' with [[genus (mathematics)|genus]] 1 and endowed with a distinguished point defined over ''K''.

If the [[characteristic (algebra)|characteristic]] of ''K'' is neither 2 nor 3, then every elliptic curve over ''K'' can be written in the form
:<math>y^2 = x^3 - px - q</math>

after a linear change of variables. Here ''p'' and ''q'' are elements of ''K'' such that the right hand side polynomial ''x''<sup>3</sup> − ''px'' − ''q'' does not have any double roots. If the characteristic is 2 or 3, then more terms need to be kept: in characteristic 3, the most general equation is of the form
:<math>y^2 = 4x^3 + b_2 x^2 + 2b_4 x + b_6</math>

for arbitrary constants ''b''<sub>2</sub>, ''b''<sub>4</sub>, ''b''<sub>6</sub> such that the polynomial on the right-hand side has distinct roots (the notation is chosen for historical reasons).  In characteristic 2, even this much is not possible, and the most general equation is

:<math>y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6</math>

provided that the variety it defines is non-singular. If characteristic were not an obstruction, each equation would reduce to the previous ones by a suitable linear change of variables.

One typically takes the curve to be the set of all points (''x'',''y'') which satisfy the above equation and such that both ''x'' and ''y'' are elements of the [[algebraic closure]] of ''K''. Points of the curve whose coordinates both belong to ''K'' are called '''''K''-rational points'''.

Many of the preceding results remain valid when the field of definition of ''E'' is a [[number field]] ''K'', that is to say, a finite [[field extension]] of '''Q'''. In particular, the group ''E(K)'' of ''K''-rational points of an elliptic curve ''E'' defined over ''K'' is finitely generated, which generalizes the Mordell–Weil theorem above. A theorem due to [[Loïc Merel]] shows that for a given integer ''d'', there are ([[up to]] isomorphism) only finitely many groups that can occur as the torsion groups of ''E''(''K'') for an elliptic curve defined over a number field ''K'' of [[degree of a field extension|degree]] ''d''. More precisely,<ref>{{cite journal |first=L. |last=Merel | author-link=Loïc Merel | title=Bornes pour la torsion des courbes elliptiques sur les corps de nombres | language=fr | journal=[[Inventiones Mathematicae]] |volume=124 |year=1996 |issue=1–3 |pages=437–449 | zbl=0936.11037 | doi=10.1007/s002220050059 |bibcode=1996InMat.124..437M |s2cid=3590991 }}</ref> there is a number ''B''(''d'') such that for any elliptic curve ''E'' defined over a number field ''K'' of degree ''d'', any torsion point of ''E''(''K'') is of [[order (group theory)|order]] less than ''B''(''d''). The theorem is effective: for ''d'' > 1, if a torsion point is of order ''p'', with ''p'' prime, then
:<math>p < d^{3d^2}</math>

As for the integral points, Siegel's theorem generalizes to the following: Let ''E'' be an elliptic curve defined over a number field ''K'', ''x'' and ''y'' the Weierstrass coordinates. Then there are only finitely many points of ''E(K)'' whose ''x''-coordinate is in the [[ring of integers]] ''O''<sub>''K''</sub>.

The properties of the Hasse–Weil zeta function and the Birch and Swinnerton-Dyer conjecture can also be extended to this more general situation.