Difference between revisions of "Elliptic curve"
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− | A non-singular complete | + | A non-singular complete [[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic curves is the source of a large part of contemporary algebraic geometry. But historically the theory of elliptic curves arose as a part of analysis, as the theory of elliptic integrals and elliptic functions (cf. [[Elliptic integral|Elliptic integral]]; [[Elliptic function|Elliptic function]]). |
− | [[Algebraic curve|algebraic curve]] of genus 1. The theory of elliptic | ||
− | curves is the source of a large part of contemporary algebraic | ||
− | geometry. But historically the theory of elliptic curves arose as a | ||
− | part of analysis, as the theory of elliptic integrals and elliptic | ||
− | functions (cf. | ||
− | [[Elliptic integral|Elliptic integral]]; | ||
− | [[Elliptic function|Elliptic function]]). | ||
− | Examples. A non-singular plane projective cubic curve; the | + | Examples. A non-singular plane projective cubic curve; the intersection of two non-singular quadrics in three-dimensional projective space; a two-sheeted covering of the projective line ramified at exactly four points; and also a one-dimensional Abelian variety are elliptic curves. |
− | intersection of two non-singular quadrics in three-dimensional | ||
− | projective space; a two-sheeted covering of the projective line | ||
− | ramified at exactly four points; and also a one-dimensional Abelian | ||
− | variety are elliptic curves. | ||
==The geometry of an elliptic curve.== | ==The geometry of an elliptic curve.== | ||
− | Let | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354501.png" /> be an elliptic curve over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354502.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354503.png" /> is biregularly isomorphic to a plane cubic curve (see [[#References|[1]]], [[#References|[9]]], [[#References|[13]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354504.png" />, then in the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354505.png" /> there is an affine coordinate system in which the equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354506.png" /> is in normal Weierstrass form: |
− | over an algebraically closed field | + | |
− | isomorphic to a plane cubic curve (see | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table> |
− | [[#References|[1]]], | + | |
− | [[#References|[9]]], | + | The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354508.png" /> is non-singular if and only if the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e0354509.png" /> does not have multiple zeros, that is, if the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545010.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545011.png" /> the curve (1) has a unique point at infinity, which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545013.png" /> is a point of inflection of (1), and the tangent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545014.png" /> is the line at infinity. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545016.png" />-invariant of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545017.png" />, |
− | [[#References|[13]]]). If | + | |
− | is an affine coordinate system in which the equation of | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545018.png" /></td> </tr></table> |
− | normal Weierstrass form: | + | |
− | + | does not depend on the choice of the coordinate system. Two elliptic curves have the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545019.png" />-invariant if and only if they are biregularly isomorphic. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545020.png" /> there is an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545021.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545022.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545023.png" />. | |
− | |||
− | only if the polynomial | ||
− | the discriminant | ||
− | infinity, which is denoted by | ||
− | (1), and the tangent at | ||
− | of an elliptic curve | ||
− | |||
− | |||
− | coordinate system. Two elliptic curves have the same | ||
− | and only if they are biregularly isomorphic. For any | ||
− | elliptic curve | ||
==The group structure on an elliptic curve.== | ==The group structure on an elliptic curve.== | ||
− | Let | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545024.png" /> be a fixed point on an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545025.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545026.png" /> assigning to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545027.png" /> the [[Divisor|divisor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545028.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545029.png" /> establishes a one-to-one correspondence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545030.png" /> and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545031.png" /> of divisor classes of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545033.png" />, that is, the [[Picard variety|Picard variety]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545034.png" />. This correspondence endows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545035.png" /> with the structure of an Abelian group that is compatible with the structure of an algebraic variety and that turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545036.png" /> into a one-dimensional Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545037.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545038.png" /> is the trivial element of the group. This group structure has the following geometric description. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545039.png" /> be a smooth plane cubic curve. Then the sum of two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545041.png" /> is defined by the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545043.png" /> is the third point of intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545044.png" /> with the line passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545046.png" />. In other words, the sum of three points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545047.png" /> vanishes if and only if the points are collinear. |
− | on an elliptic curve | ||
− | [[Divisor|divisor]] | ||
− | between | ||
− | that is, the | ||
− | [[Picard variety|Picard variety]] of | ||
− | |||
− | structure of an algebraic variety and that turns | ||
− | one-dimensional Abelian variety | ||
− | of the group. This group structure has the following geometric | ||
− | description. Let | ||
− | two points | ||
− | third point of intersection of | ||
− | and | ||
− | and only if the points are collinear. | ||
==An elliptic curve as a one-dimensional Abelian variety.== | ==An elliptic curve as a one-dimensional Abelian variety.== | ||
− | Let | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545048.png" /> denote the endomorphism of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545049.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545050.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545051.png" /> is an elliptic curve with distinguished point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545052.png" />, then any rational mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545053.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545056.png" /> is a homomorphism of Abelian varieties. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545057.png" /> is either a constant mapping at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545058.png" /> or is an [[Isogeny|isogeny]], that is, there is a homomorphism of Abelian varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545059.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545061.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545062.png" /> (see [[#References|[1]]], [[#References|[6]]]). |
− | denote the endomorphism of multiplication by | ||
− | elliptic curve with distinguished point | ||
− | |||
− | varieties. Here | ||
− | [[Isogeny|isogeny]], that is, there is a homomorphism of Abelian | ||
− | varieties | ||
− | [[#References|[1]]], | ||
− | [[#References|[6]]]). | ||
− | The automorphism group of an elliptic curve | + | The automorphism group of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545063.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545064.png" />, and its subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545065.png" /> of automorphisms leaving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545066.png" /> fixed is non-trivial and finite. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545067.png" /> is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545068.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545069.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545070.png" /> is neither 0 nor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545071.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545072.png" /> consists of the two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545074.png" />. The order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545075.png" /> is 4 when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545076.png" /> and 6 when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545077.png" /> (see [[#References|[1]]], [[#References|[6]]], [[#References|[13]]]). |
− | |||
− | non-trivial and finite. Suppose that | ||
− | is neither 0 nor | ||
− | |||
− | [[#References|[1]]], | ||
− | [[#References|[6]]], | ||
− | [[#References|[13]]]). | ||
− | An important invariant of an elliptic curve is the endomorphism ring | + | An important invariant of an elliptic curve is the endomorphism ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545078.png" /> of the Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545079.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545080.png" /> defines an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545081.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545082.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545083.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545084.png" /> is an elliptic curve with complex multiplication. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545085.png" /> can be of one of the following types (see [[#References|[1]]], [[#References|[9]]], [[#References|[13]]]): I) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545086.png" />; II) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545087.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545088.png" /> is the ring of algebraic integers of an imaginary quadratic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545089.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545090.png" />; or III) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545091.png" /> is a non-commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545092.png" />-algebra of rank 4 without divisors of zero. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545094.png" /> is a maximal order in the quaternion algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545095.png" /> ramified only at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545097.png" />. Such elliptic curves exist for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545098.png" /> and are called supersingular; elliptic curves in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e03545099.png" /> that are not supersingular are said to be ordinary. |
− | |||
− | of | ||
− | complex multiplication. The ring | ||
− | types (see | ||
− | [[#References|[1]]], | ||
− | [[#References|[9]]], | ||
− | [[#References|[13]]]): I) | ||
− | algebraic integers of an imaginary quadratic field | ||
− | III) | ||
− | of zero. In this case | ||
− | algebra over | ||
− | exist for all | ||
− | characteristic | ||
− | The group | + | The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450100.png" /> of points of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450101.png" /> with orders that divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450102.png" /> has the following structure: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450103.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450104.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450105.png" /> and ordinary elliptic curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450106.png" />, while for supersingular elliptic curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450107.png" />. For a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450108.png" /> the [[Tate module|Tate module]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450109.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450110.png" />. |
− | divide | ||
− | ordinary elliptic curves | ||
− | |||
− | [[Tate module|Tate module]] | ||
==Elliptic curves over non-closed fields.== | ==Elliptic curves over non-closed fields.== | ||
− | Let | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450111.png" /> be an elliptic curve over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450112.png" />. If the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450113.png" />-rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450114.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450115.png" /> is not empty, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450116.png" /> is biregularly isomorphic to a plane cubic curve (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450117.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450118.png" />). The point at infinity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450119.png" /> of (1) is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450120.png" />. As above, one can introduce a group structure on (1), turning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450121.png" /> into a one-dimensional Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450122.png" /> and turning the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450123.png" /> into an Abelian group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450124.png" /> as trivial element. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450125.png" /> is finitely generated over its prime subfield, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450126.png" /> is a finitely-generated group (the Mordell–Weil theorem). |
− | curve over an arbitrary field | ||
− | |||
− | cubic curve (1) with | ||
− | defined over | ||
− | (1), turning | ||
− | turning the set | ||
− | element. If | ||
− | |||
− | For any elliptic curve | + | For any elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450127.png" /> there is defined the [[Jacobi variety|Jacobi variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450128.png" />, which is a one-dimensional Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450129.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450130.png" /> is a [[Principal homogeneous space|principal homogeneous space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450131.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450132.png" /> is not empty, then the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450133.png" /> specifies an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450134.png" /> under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450135.png" /> becomes the trivial element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450136.png" />. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450137.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450138.png" /> are isomorphic over a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450139.png" /> (see [[#References|[1]]], [[#References|[4]]], [[#References|[13]]]). |
− | [[Jacobi variety|Jacobi variety]] | ||
− | Abelian variety over | ||
− | [[Principal homogeneous space|principal homogeneous space]] over | ||
− | |||
− | isomorphism | ||
− | general, | ||
− | (see | ||
− | [[#References|[1]]], | ||
− | [[#References|[4]]], | ||
− | [[#References|[13]]]). | ||
==Elliptic curves over the field of complex numbers.== | ==Elliptic curves over the field of complex numbers.== | ||
− | An elliptic | + | An elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450140.png" /> is a compact [[Riemann surface|Riemann surface]] of genus 1, and vice versa. The group structure turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450141.png" /> into a complex Lie group, which is a one-dimensional complex torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450142.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450143.png" /> is a lattice in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450144.png" />. Conversely, any one-dimensional complex torus is an elliptic curve (see [[#References|[3]]]). From the topological point of view, an elliptic curve is a two-dimensional torus. |
− | curve over | + | |
− | [[Riemann surface|Riemann surface]] of genus 1, and vice versa. The | + | The theory of elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450145.png" /> is in essence equivalent to the theory of elliptic functions. An identification of a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450146.png" /> with an elliptic curve can be effected as follows. The elliptic functions with a given period lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450147.png" /> form a field generated by the Weierstrass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450148.png" />-function (see [[Weierstrass elliptic functions|Weierstrass elliptic functions]]) and its derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450149.png" />, which are connected by the relation |
− | group structure turns | + | |
− | one-dimensional complex torus | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450150.png" /></td> </tr></table> |
− | complex plane | ||
− | elliptic curve (see | ||
− | [[#References|[3]]]). From the topological point of view, an elliptic | ||
− | curve is a two-dimensional torus. | ||
− | The | + | The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450151.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450152.png" />) induces an isomorphism between the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450153.png" /> and the elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450154.png" /> with equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450155.png" />. The identification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450156.png" /> given by (1) with the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450157.png" /> is effected by curvilinear integrals of the holomorphic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450158.png" /> and gives an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450159.png" />. |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | elliptic curve | ||
− | by (1) with the torus | ||
− | holomorphic form | ||
− | The description of the set of all elliptic curves as tori | + | The description of the set of all elliptic curves as tori <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450160.png" /> leads to the [[Modular function|modular function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450161.png" />. Two lattices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450162.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450163.png" /> determine isomorphic tori if and only if they are similar, that is, if one is obtained from the other by multiplication by a complex number. Therefore it may be assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450164.png" /> is generated by the numbers 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450165.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450166.png" />. Two lattices with bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450167.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450168.png" /> are similar if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450169.png" /> for an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450170.png" /> of the [[Modular group|modular group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450171.png" />. The modular function |
− | the | ||
− | [[Modular function|modular function]] | ||
− | determine isomorphic tori if and only if they are similar, that is, if | ||
− | one is obtained from the other by multiplication by a complex | ||
− | number. Therefore it may be assumed that | ||
− | numbers 1 and | ||
− | similar if and only if | ||
− | [[Modular group|modular group]] | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | An elliptic curve | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450172.png" /></td> </tr></table> |
− | an imaginary | + | |
− | [[Quadratic irrationality|quadratic irrationality]]. In this case | + | is also called the absolute invariant; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450173.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450174.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450175.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450176.png" /> produces a one-to-one correspondence between the classes of isomorphic elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450177.png" /> and the complex numbers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450178.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450179.png" />. |
− | is a subring of finite index in the ring of algebraic integers of the | + | |
− | imaginary quadratic field | + | An elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450180.png" /> has complex multiplication if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450181.png" /> is an imaginary [[Quadratic irrationality|quadratic irrationality]]. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450182.png" /> is a subring of finite index in the ring of algebraic integers of the imaginary quadratic field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450183.png" />. Elliptic curves with complex multiplication are closely connected with the [[Class field theory|class field theory]] for imaginary quadratic fields (see [[#References|[4]]], [[#References|[8]]]). |
− | multiplication are closely connected with the | ||
− | [[Class field theory|class field theory]] for imaginary quadratic | ||
− | fields (see | ||
− | [[#References|[4]]], | ||
− | [[#References|[8]]]). | ||
==Arithmetic of elliptic curves.== | ==Arithmetic of elliptic curves.== | ||
− | Let | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450184.png" /> be an elliptic curve over the finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450185.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450186.png" /> elements. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450187.png" /> is always non-empty and finite. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450188.png" /> is endowed with the structure of a one-dimensional Abelian variety over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450189.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450190.png" /> with that of a finite Abelian group. The order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450191.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450192.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450193.png" />. The characteristic polynomial of the [[Frobenius endomorphism|Frobenius endomorphism]] acting on the Tate module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450194.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450195.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450196.png" />. Its roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450197.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450198.png" /> are complex-conjugate algebraic integers of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450199.png" />. For any finite extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450200.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450201.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450202.png" />, the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450203.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450204.png" />. The [[Zeta-function|zeta-function]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450205.png" /> is |
− | the finite field | + | |
− | non-empty and finite. Hence | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450206.png" /></td> </tr></table> |
− | one-dimensional Abelian variety over | + | |
− | finite Abelian group. The order | + | For any algebraic integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450207.png" /> of modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450208.png" /> in some imaginary quadratic field (or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450209.png" />) one can find an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450210.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450211.png" /> such that the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450212.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450213.png" />. |
− | characteristic polynomial of the | + | |
− | [[Frobenius endomorphism|Frobenius endomorphism]] acting on the Tate | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450214.png" /> be the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450215.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450216.png" />-adic numbers or a finite algebraic extension of it, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450217.png" /> be the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450218.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450219.png" /> be an elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450220.png" />, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450221.png" /> is non-empty. The group structure turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450222.png" /> into a commutative compact one-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450223.png" />-adic Lie group (cf. [[Lie-group-adic|Lie group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450224.png" />-adic]]). The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450225.png" /> is Pontryagin-dual to the [[Weil–Châtelet group|Weil–Châtelet group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450226.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450227.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450228.png" /> is a Tate curve (see [[#References|[1]]], [[#References|[5]]]) and there exists a canonical uniformization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450229.png" /> analogous to the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450230.png" />. |
− | module | + | |
− | algebraic integers of modulus | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450231.png" /> be an elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450232.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450233.png" /> is not empty. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450234.png" /> is biregularly isomorphic to the curve (1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450235.png" />. Of all curves of the form (1) that are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450236.png" /> with integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450237.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450238.png" />, one chooses the one for which the absolute value of the discriminant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450239.png" /> is minimal. The conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450240.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450242.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450243.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450244.png" /> are defined as formal products of local factors: |
− | of degree | + | |
− | [[Zeta-function|zeta-function]] of | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450245.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table> |
− | + | ||
− | + | over all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450246.png" /> (see [[#References|[1]]], [[#References|[5]]], [[#References|[13]]]). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450247.png" /> is some power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450248.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450249.png" /> is a meromorphic function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450250.png" /> that has neither a zero nor a pole at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450251.png" />. To determine the local factors one considers the reduction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450252.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450253.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450254.png" />), which is a plane projective curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450255.png" /> over the residue class field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450256.png" /> and is given in an affine coordinate system by the equation | |
− | integer | + | |
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450257.png" /></td> </tr></table> | |
− | of | + | |
+ | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450258.png" /> be the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450259.png" />-points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450260.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450261.png" /> does not divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450262.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450263.png" /> is an elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450264.png" />, and one puts | ||
+ | |||
+ | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450265.png" /></td> </tr></table> | ||
+ | |||
+ | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450266.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450267.png" />, then the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450268.png" /> has a multiple root, and one puts | ||
+ | |||
+ | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450269.png" /></td> </tr></table> | ||
− | + | (depending on whether it is a triple or a double root). The product (2) converges in the right half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450270.png" />. It has been conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450271.png" /> has a meromorphic extension to the whole complex plane and that the function | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450272.png" /></td> </tr></table> | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450273.png" /> is the [[Gamma-function|gamma-function]]) satisfies the functional equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450275.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450276.png" /> (see [[#References|[5]]], [[#References|[3]]]). This conjecture has been proved for elliptic curves with complex multiplication. | |
− | |||
− | [[Gamma-function|gamma-function]]) satisfies the functional equation | ||
− | |||
− | [[#References|[5]]], | ||
− | [[#References|[3]]]). This conjecture has been proved for elliptic | ||
− | curves with complex multiplication. | ||
− | The group | + | The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450277.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450278.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450279.png" /> is a finite Abelian group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450280.png" /> is a free Abelian group of a certain finite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450281.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450282.png" /> is isomorphic to one of the following 15 groups (see [[#References|[11]]]): <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450283.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450284.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450285.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450286.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450287.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450288.png" /> is called the rank of the elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450289.png" />, or its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450291.png" />-rank. Examples are known of elliptic curves over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450292.png" /> of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450293.png" />. There is a conjecture (see [[#References|[1]]], [[#References|[13]]]) that over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450294.png" /> there exist elliptic curves of arbitrary large rank. |
− | group and | ||
− | |||
− | [[#References|[11]]]): | ||
− | is called the rank of the elliptic curve over | ||
− | |||
− | |||
− | [[#References|[1]]], | ||
− | [[#References|[13]]]) that over | ||
− | arbitrary large rank. | ||
− | In the study of | + | In the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450295.png" /> one uses the Tate height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450296.png" />, which is a non-negative definite quadratic form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450297.png" /> (see [[#References|[1]]], [[#References|[3]]], [[#References|[8]]], and also [[Height, in Diophantine geometry|Height, in Diophantine geometry]]). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450298.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450299.png" /> is finite. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450300.png" /> vanishes precisely on the torsion subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450301.png" />. |
− | non-negative definite quadratic form on | ||
− | [[#References|[1]]], | ||
− | [[#References|[3]]], | ||
− | [[#References|[8]]], and also | ||
− | [[Height, in Diophantine geometry|Height, in Diophantine | ||
− | geometry]]). For any | ||
− | vanishes precisely on the torsion subgroup of | ||
− | An important invariant of an elliptic curve is its Tate–Shafarevich | + | An important invariant of an elliptic curve is its Tate–Shafarevich group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450302.png" /> (see [[Weil–Châtelet group|Weil–Châtelet group]]). The non-trivial elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450303.png" />, an elliptic curve without <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450304.png" />-points, provide examples of elliptic curves for which the [[Hasse principle|Hasse principle]] fails to hold. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450305.png" /> is periodic and for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450306.png" /> the subgroup of its elements of order dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450307.png" /> is finite. For a large number of elliptic curves it has been verified that the 2- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450308.png" />-components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450309.png" /> are finite (see [[#References|[1]]], [[#References|[4]]], [[#References|[5]]]). There is a conjecture that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450310.png" /> is finite. |
− | group | ||
− | [[Weil–Châtelet group|Weil–Châtelet group]]). The non-trivial elements | ||
− | of | ||
− | elliptic curves for which the | ||
− | [[Hasse principle|Hasse principle]] fails to hold. The group | ||
− | periodic and for every | ||
− | dividing | ||
− | been verified that the 2- and | ||
− | [[#References|[1]]], | ||
− | [[#References|[4]]], | ||
− | [[#References|[5]]]). There is a conjecture that | ||
− | A conjecture of Birch and Swinnerton-Dyer asserts (see | + | A conjecture of Birch and Swinnerton-Dyer asserts (see [[#References|[5]]], [[#References|[13]]]) that the order of the zero of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450311.png" />-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450312.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450313.png" /> is equal to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450314.png" />-rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450315.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450316.png" /> has a zero at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450317.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450318.png" /> is infinite. So far (1984) the conjecture has not been proved for a single elliptic curve, but for elliptic curves with complex multiplication (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450319.png" />) it has been established that when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450320.png" /> is infinite, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450321.png" />-function has a zero at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450322.png" /> (see [[#References|[14]]]). The conjecture of Birch and Swinnerton-Dyer gives the principal term of the asymptotic expansion of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450323.png" />-function as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450324.png" />; in it there occur the orders of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450325.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450326.png" /> and the determinant of the Tate height [[#References|[1]]]. It can be restated in terms of the Tamagawa numbers (cf. [[Tamagawa number|Tamagawa number]], see [[#References|[7]]]). |
− | [[#References|[5]]], | ||
− | [[#References|[13]]]) that the order of the zero of the | ||
− | |||
− | zero at | ||
− | conjecture has not been proved for a single elliptic curve, but for | ||
− | elliptic curves with complex multiplication (and | ||
− | established that when | ||
− | zero at | ||
− | [[#References|[14]]]). The conjecture of Birch and Swinnerton-Dyer | ||
− | gives the principal term of the asymptotic expansion of the | ||
− | |||
− | and | ||
− | [[#References|[1]]]. It can be restated in terms of the Tamagawa | ||
− | numbers (cf. | ||
− | [[Tamagawa number|Tamagawa number]], see | ||
− | [[#References|[7]]]). | ||
− | There is a conjecture of Weil that an elliptic curve | + | There is a conjecture of Weil that an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450327.png" /> has a uniformization by modular functions relative to the congruence subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450328.png" /> of the modular group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450329.png" /> (see [[#References|[5]]] and also [[Zeta-function|Zeta-function]] in algebraic geometry). This conjecture has been proved for elliptic functions with complex multiplication. It is known (see [[#References|[15]]]) that every algebraic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450330.png" /> can be uniformized (cf. [[Uniformization|Uniformization]]) by modular functions relative to some subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450331.png" /> of finite index. |
− | uniformization by modular functions relative to the congruence | ||
− | subgroup | ||
− | [[#References|[5]]] and also | ||
− | [[Zeta-function|Zeta-function]] in algebraic geometry). This | ||
− | conjecture has been proved for elliptic functions with complex | ||
− | multiplication. It is known (see | ||
− | [[#References|[15]]]) that every algebraic curve over | ||
− | uniformized (cf. | ||
− | [[Uniformization|Uniformization]]) by modular functions relative to | ||
− | some subgroup of | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" ''J. London Math. Soc.'' , '''41''' (1966) pp. 193–291</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Yu.I. Manin, "Cyclotomic fields and modular curves" ''Russian Math. Surveys'' , '''26''' : 6 (1971) pp. 6–78 ''Uspekhi Mat. Nauk'' , '''26''' : 6 (1971) pp. 7–71</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S. Bloch, "A note on height pairings, Tamagawa numbers, and the Birch and Swinnnerton-Dyer conjecture" ''Invent. Math.'' , '''58''' (1980) pp. 65–76</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Lang, "Elliptic curves; Diophantine analysis" , Springer (1978)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> S. Lang, "Elliptic functions" , Addison-Wesley (1973)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> B. Mazur, "Rational isogenies of prime degree" ''Invent. Math.'' , '''44''' (1978) pp. 129–162</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , ''Modular functions of one variable. 4'' , ''Lect. notes in math.'' , '''476''' , Springer (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J.F. Mestre, "Construction d'une courbe elliptique de rang <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035450/e035450332.png" />" ''C.R. Acad. Sci. Paris Sér. 1'' , '''295''' (1982) pp. 643–644</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> J. Tate, "The arithmetic of elliptic curves" ''Invent. Math.'' , '''23''' (1974) pp. 197–206</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> J. Coates, A. Wiles, "On the conjecture of Birch and Swinnerton-Dyer" ''Invent. Math.'' , '''39''' (1977) pp. 223–251</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> G.V. Belyi, "On Galois extensions of a maximal cyclotomic field" ''Math. USSR Izv.'' , '''14''' : 2 (1980) pp. 247–256 ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''43''' (1979) pp. 267–276</TD></TR></table> |
− | valign="top"> J.W.S. Cassels, "Diophantine equations with special | ||
− | reference to elliptic curves" ''J. London Math. Soc.'' , '''41''' | ||
− | (1966) pp. 193–291</TD></TR><TR><TD valign="top">[2]</TD> <TD | ||
− | valign="top"> A. Hurwitz, R. Courant, "Vorlesungen über allgemeine | ||
− | Funktionentheorie und elliptische Funktionen" , Springer | ||
− | (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> | ||
− | D. Mumford, "Abelian varieties" , Oxford Univ. Press | ||
− | (1974)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> | ||
− | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' | ||
− | , Acad. Press (1967)</TD></TR><TR><TD valign="top">[5]</TD> <TD | ||
− | valign="top"> Yu.I. Manin, "Cyclotomic fields and modular curves" | ||
− | ''Russian Math. Surveys'' , '''26''' : 6 (1971) pp. 6–78 ''Uspekhi | ||
− | Mat. Nauk'' , '''26''' : 6 (1971) pp. 7–71</TD></TR><TR><TD | ||
− | valign="top">[6]</TD> <TD valign="top"> R. Hartshorne, "Algebraic | ||
− | geometry" , Springer (1977) pp. 91</TD></TR><TR><TD | ||
− | valign="top">[7]</TD> <TD valign="top"> S. Bloch, "A note on height | ||
− | pairings, Tamagawa numbers, and the Birch and Swinnnerton-Dyer | ||
− | conjecture" ''Invent. Math.'' , '''58''' (1980) | ||
− | pp. 65–76</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> | ||
− | S. Lang, "Elliptic curves; Diophantine analysis" , Springer | ||
− | (1978)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> | ||
− | S. Lang, "Elliptic functions" , Addison-Wesley (1973)</TD></TR><TR><TD | ||
− | valign="top">[10]</TD> <TD valign="top"> B. Mazur, "Rational isogenies | ||
− | of prime degree" ''Invent. Math.'' , '''44''' (1978) | ||
− | pp. 129–162</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> | ||
− | J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , ''Modular | ||
− | functions of one variable. 4'' , ''Lect. notes in math.'' , '''476''' | ||
− | , Springer (1975)</TD></TR><TR><TD valign="top">[12]</TD> <TD | ||
− | valign="top"> J.F. Mestre, "Construction d'une courbe elliptique de | ||
− | rang | ||
− | pp. 643–644</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> | ||
− | J. Tate, "The arithmetic of elliptic curves" ''Invent. Math.'' , | ||
− | '''23''' (1974) pp. 197–206</TD></TR><TR><TD valign="top">[14]</TD> | ||
− | <TD valign="top"> J. Coates, A. Wiles, "On the conjecture of Birch and | ||
− | Swinnerton-Dyer" ''Invent. Math.'' , '''39''' (1977) | ||
− | pp. 223–251</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> | ||
− | G.V. Belyi, "On Galois extensions of a maximal cyclotomic field" | ||
− | ''Math. USSR Izv.'' , '''14''' : 2 (1980) pp. 247–256 | ||
− | ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''43''' (1979) | ||
− | pp. 267–276</TD></TR></table> | ||
Line 350: | Line 98: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B. Mazur, "Modular curves and the Eisenstein ideal" ''Publ. Math. IHES'' , '''47''' (1978) pp. 33–186</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986)</TD></TR></table> |
− | valign="top"> B. Mazur, "Modular curves and the Eisenstein ideal" | ||
− | ''Publ. Math. IHES'' , '''47''' (1978) pp. 33–186</TD></TR><TR><TD | ||
− | valign="top">[a2]</TD> <TD valign="top"> J.H. Silverman, "The | ||
− | arithmetic of elliptic curves" , Springer (1986)</TD></TR></table> |
Revision as of 14:33, 12 September 2011
A non-singular complete algebraic curve of genus 1. The theory of elliptic curves is the source of a large part of contemporary algebraic geometry. But historically the theory of elliptic curves arose as a part of analysis, as the theory of elliptic integrals and elliptic functions (cf. Elliptic integral; Elliptic function).
Examples. A non-singular plane projective cubic curve; the intersection of two non-singular quadrics in three-dimensional projective space; a two-sheeted covering of the projective line ramified at exactly four points; and also a one-dimensional Abelian variety are elliptic curves.
The geometry of an elliptic curve.
Let be an elliptic curve over an algebraically closed field
. Then
is biregularly isomorphic to a plane cubic curve (see [1], [9], [13]). If
, then in the projective plane
there is an affine coordinate system in which the equation of
is in normal Weierstrass form:
![]() | (1) |
The curve is non-singular if and only if the polynomial
does not have multiple zeros, that is, if the discriminant
. In
the curve (1) has a unique point at infinity, which is denoted by
;
is a point of inflection of (1), and the tangent at
is the line at infinity. The
-invariant of an elliptic curve
,
![]() |
does not depend on the choice of the coordinate system. Two elliptic curves have the same -invariant if and only if they are biregularly isomorphic. For any
there is an elliptic curve
over
with
.
The group structure on an elliptic curve.
Let be a fixed point on an elliptic curve
. The mapping
assigning to a point
the divisor
on
establishes a one-to-one correspondence between
and the group
of divisor classes of degree
on
, that is, the Picard variety of
. This correspondence endows
with the structure of an Abelian group that is compatible with the structure of an algebraic variety and that turns
into a one-dimensional Abelian variety
; here
is the trivial element of the group. This group structure has the following geometric description. Let
be a smooth plane cubic curve. Then the sum of two points
and
is defined by the rule
, where
is the third point of intersection of
with the line passing through
and
. In other words, the sum of three points on
vanishes if and only if the points are collinear.
An elliptic curve as a one-dimensional Abelian variety.
Let denote the endomorphism of multiplication by
in
. If
is an elliptic curve with distinguished point
, then any rational mapping
has the form
, where
and
is a homomorphism of Abelian varieties. Here
is either a constant mapping at
or is an isogeny, that is, there is a homomorphism of Abelian varieties
such that
and
for some
(see [1], [6]).
The automorphism group of an elliptic curve acts transitively on
, and its subgroup
of automorphisms leaving
fixed is non-trivial and finite. Suppose that
is not
or
. When
is neither 0 nor
, then
consists of the two elements
and
. The order of
is 4 when
and 6 when
(see [1], [6], [13]).
An important invariant of an elliptic curve is the endomorphism ring of the Abelian variety
. The mapping
defines an imbedding of
in
. If
, one says that
is an elliptic curve with complex multiplication. The ring
can be of one of the following types (see [1], [9], [13]): I)
; II)
, where
is the ring of algebraic integers of an imaginary quadratic field
and
; or III)
is a non-commutative
-algebra of rank 4 without divisors of zero. In this case
and
is a maximal order in the quaternion algebra over
ramified only at
and
. Such elliptic curves exist for all
and are called supersingular; elliptic curves in characteristic
that are not supersingular are said to be ordinary.
The group of points of an elliptic curve
with orders that divide
has the following structure:
when
. For
and ordinary elliptic curves
, while for supersingular elliptic curves
. For a prime number
the Tate module
is isomorphic to
.
Elliptic curves over non-closed fields.
Let be an elliptic curve over an arbitrary field
. If the set of
-rational points
of
is not empty, then
is biregularly isomorphic to a plane cubic curve (1) with
(
). The point at infinity
of (1) is defined over
. As above, one can introduce a group structure on (1), turning
into a one-dimensional Abelian variety over
and turning the set
into an Abelian group with
as trivial element. If
is finitely generated over its prime subfield, then
is a finitely-generated group (the Mordell–Weil theorem).
For any elliptic curve there is defined the Jacobi variety
, which is a one-dimensional Abelian variety over
, and
is a principal homogeneous space over
. If
is not empty, then the choice of
specifies an isomorphism
under which
becomes the trivial element of
. In general,
and
are isomorphic over a finite extension of
(see [1], [4], [13]).
Elliptic curves over the field of complex numbers.
An elliptic curve over is a compact Riemann surface of genus 1, and vice versa. The group structure turns
into a complex Lie group, which is a one-dimensional complex torus
, where
is a lattice in the complex plane
. Conversely, any one-dimensional complex torus is an elliptic curve (see [3]). From the topological point of view, an elliptic curve is a two-dimensional torus.
The theory of elliptic curves over is in essence equivalent to the theory of elliptic functions. An identification of a torus
with an elliptic curve can be effected as follows. The elliptic functions with a given period lattice
form a field generated by the Weierstrass
-function (see Weierstrass elliptic functions) and its derivative
, which are connected by the relation
![]() |
The mapping (
) induces an isomorphism between the torus
and the elliptic curve
with equation
. The identification of
given by (1) with the torus
is effected by curvilinear integrals of the holomorphic form
and gives an isomorphism
.
The description of the set of all elliptic curves as tori leads to the modular function
. Two lattices
and
determine isomorphic tori if and only if they are similar, that is, if one is obtained from the other by multiplication by a complex number. Therefore it may be assumed that
is generated by the numbers 1 and
in
. Two lattices with bases
and
are similar if and only if
for an element
of the modular group
. The modular function
![]() |
is also called the absolute invariant; if and only if
for some
, and the function
produces a one-to-one correspondence between the classes of isomorphic elliptic curves over
and the complex numbers. If
, then
.
An elliptic curve has complex multiplication if and only if
is an imaginary quadratic irrationality. In this case
is a subring of finite index in the ring of algebraic integers of the imaginary quadratic field
. Elliptic curves with complex multiplication are closely connected with the class field theory for imaginary quadratic fields (see [4], [8]).
Arithmetic of elliptic curves.
Let be an elliptic curve over the finite field
with
elements. The set
is always non-empty and finite. Hence
is endowed with the structure of a one-dimensional Abelian variety over
, and
with that of a finite Abelian group. The order
of
satisfies
. The characteristic polynomial of the Frobenius endomorphism acting on the Tate module
,
, is
. Its roots
and
are complex-conjugate algebraic integers of modulus
. For any finite extension
of
of degree
, the order of
is
. The zeta-function of
is
![]() |
For any algebraic integer of modulus
in some imaginary quadratic field (or in
) one can find an elliptic curve
over
such that the order of
is
.
Let be the field
of
-adic numbers or a finite algebraic extension of it, let
be the ring of integers of
, let
be an elliptic curve over
, and suppose that
is non-empty. The group structure turns
into a commutative compact one-dimensional
-adic Lie group (cf. Lie group,
-adic). The group
is Pontryagin-dual to the Weil–Châtelet group
. If
, then
is a Tate curve (see [1], [5]) and there exists a canonical uniformization of
analogous to the case of
.
Let be an elliptic curve over
for which
is not empty. Then
is biregularly isomorphic to the curve (1) with
. Of all curves of the form (1) that are isomorphic to
with integers
and
, one chooses the one for which the absolute value of the discriminant
is minimal. The conductor
and the
-function
of
are defined as formal products of local factors:
![]() | (2) |
over all prime numbers (see [1], [5], [13]). Here
is some power of
, and
is a meromorphic function of the complex variable
that has neither a zero nor a pole at
. To determine the local factors one considers the reduction of
modulo
(
), which is a plane projective curve
over the residue class field
and is given in an affine coordinate system by the equation
![]() |
Let be the number of
-points on
. If
does not divide
, then
is an elliptic curve over
, and one puts
![]() |
If divides
, then the polynomial
has a multiple root, and one puts
![]() |
(depending on whether it is a triple or a double root). The product (2) converges in the right half-plane . It has been conjectured that
has a meromorphic extension to the whole complex plane and that the function
![]() |
(where is the gamma-function) satisfies the functional equation
with
(see [5], [3]). This conjecture has been proved for elliptic curves with complex multiplication.
The group is isomorphic to
, where
is a finite Abelian group and
is a free Abelian group of a certain finite rank
.
is isomorphic to one of the following 15 groups (see [11]):
,
or
, and
,
. The number
is called the rank of the elliptic curve over
, or its
-rank. Examples are known of elliptic curves over
of rank
. There is a conjecture (see [1], [13]) that over
there exist elliptic curves of arbitrary large rank.
In the study of one uses the Tate height
, which is a non-negative definite quadratic form on
(see [1], [3], [8], and also Height, in Diophantine geometry). For any
the set
is finite. In particular,
vanishes precisely on the torsion subgroup of
.
An important invariant of an elliptic curve is its Tate–Shafarevich group (see Weil–Châtelet group). The non-trivial elements of
, an elliptic curve without
-points, provide examples of elliptic curves for which the Hasse principle fails to hold. The group
is periodic and for every
the subgroup of its elements of order dividing
is finite. For a large number of elliptic curves it has been verified that the 2- and
-components of
are finite (see [1], [4], [5]). There is a conjecture that
is finite.
A conjecture of Birch and Swinnerton-Dyer asserts (see [5], [13]) that the order of the zero of the -function
at
is equal to the
-rank of
. In particular,
has a zero at
if and only if
is infinite. So far (1984) the conjecture has not been proved for a single elliptic curve, but for elliptic curves with complex multiplication (and
) it has been established that when
is infinite, then the
-function has a zero at
(see [14]). The conjecture of Birch and Swinnerton-Dyer gives the principal term of the asymptotic expansion of the
-function as
; in it there occur the orders of the groups
and
and the determinant of the Tate height [1]. It can be restated in terms of the Tamagawa numbers (cf. Tamagawa number, see [7]).
There is a conjecture of Weil that an elliptic curve has a uniformization by modular functions relative to the congruence subgroup
of the modular group
(see [5] and also Zeta-function in algebraic geometry). This conjecture has been proved for elliptic functions with complex multiplication. It is known (see [15]) that every algebraic curve over
can be uniformized (cf. Uniformization) by modular functions relative to some subgroup of
of finite index.
References
[1] | J.W.S. Cassels, "Diophantine equations with special reference to elliptic curves" J. London Math. Soc. , 41 (1966) pp. 193–291 |
[2] | A. Hurwitz, R. Courant, "Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen" , Springer (1968) |
[3] | D. Mumford, "Abelian varieties" , Oxford Univ. Press (1974) |
[4] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) |
[5] | Yu.I. Manin, "Cyclotomic fields and modular curves" Russian Math. Surveys , 26 : 6 (1971) pp. 6–78 Uspekhi Mat. Nauk , 26 : 6 (1971) pp. 7–71 |
[6] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 |
[7] | S. Bloch, "A note on height pairings, Tamagawa numbers, and the Birch and Swinnnerton-Dyer conjecture" Invent. Math. , 58 (1980) pp. 65–76 |
[8] | S. Lang, "Elliptic curves; Diophantine analysis" , Springer (1978) |
[9] | S. Lang, "Elliptic functions" , Addison-Wesley (1973) |
[10] | B. Mazur, "Rational isogenies of prime degree" Invent. Math. , 44 (1978) pp. 129–162 |
[11] | J.-P. Serre (ed.) P. Deligne (ed.) W. Kuyk (ed.) , Modular functions of one variable. 4 , Lect. notes in math. , 476 , Springer (1975) |
[12] | J.F. Mestre, "Construction d'une courbe elliptique de rang ![]() |
[13] | J. Tate, "The arithmetic of elliptic curves" Invent. Math. , 23 (1974) pp. 197–206 |
[14] | J. Coates, A. Wiles, "On the conjecture of Birch and Swinnerton-Dyer" Invent. Math. , 39 (1977) pp. 223–251 |
[15] | G.V. Belyi, "On Galois extensions of a maximal cyclotomic field" Math. USSR Izv. , 14 : 2 (1980) pp. 247–256 Izv. Akad. Nauk SSSR Ser. Mat. , 43 (1979) pp. 267–276 |
Comments
References
[a1] | B. Mazur, "Modular curves and the Eisenstein ideal" Publ. Math. IHES , 47 (1978) pp. 33–186 |
[a2] | J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986) |
Elliptic curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Elliptic_curve&oldid=19580