Elliptic curves: what they are, why they are called ...group structure on E because it does not determine which point e ∈ E is the identity. Since 3e = 0, e must be a point of inﬂection,

Elliptic curves: what they are, whythey are called elliptic, and why

topologists like them, IWayne State University Mathematics

ColloquiumFebruary 26, 2007

Doug Ravenel

Wayne State Mathematics Colloquium – p. 1/24

Early history of elliptic curvesIn the 18th century it was natural to ask about the arclength of an ellipse.


Early history of elliptic curvesIn the 18th century it was natural to ask about the arclength of an ellipse. This question led to the study ofintegrals involving

√

f(x) wheref(x) is a polynomialof degree 3 or 4.


Early history of elliptic curvesIn the 18th century it was natural to ask about the arclength of an ellipse. This question led to the study ofintegrals involving

√

f(x) wheref(x) is a polynomialof degree 3 or 4.We know now that such integrals cannot be describedin terms of familiar functions that we teach incalculus. They came to be known aselliptic integrals.


Early history of elliptic curvesIf the ellipse is a circle, then one can simplify thingsand replacef(x) by a quadratic polynomial. We teachour calculus students how to handle these “circularintegrals,” for example we know that

∫

dx√1 − x2

= sin−1 x.


Early history of elliptic curvesIf the ellipse is a circle, then one can simplify thingsand replacef(x) by a quadratic polynomial. We teachour calculus students how to handle these “circularintegrals,” for example we know that

∫

dx√1 − x2

= sin−1 x.

Experience has taught us that the inverse of thisfunction, namelysin x, is much easier to deal with.Thus circular integrals lead to trigonometric functionsg(x), many of which are periodic in the sense that

g(x + 2π) = g(x).


Early history of elliptic curvesExperience has taught us that the inverse of thisfunction, namelysin x, is much easier to deal with.Thus circular integrals lead to trigonometric functionsg(x), many of which are periodic in the sense that

g(x + 2π) = g(x).

This means that as a function of a complex variable,g(x) is well defined on the quotientC/2πZ, which istopologically a cylinder.


Early history of elliptic curvesSimilarly, it is convenient to replace certain ellipticintegrals by their inverses, which came to be known aselliptic functions.


Early history of elliptic curvesSimilarly, it is convenient to replace certain ellipticintegrals by their inverses, which came to be known aselliptic functions. This insight was originally due toAbel.


Early history of elliptic curvesSimilarly, it is convenient to replace certain ellipticintegrals by their inverses, which came to be known aselliptic functions. This insight was originally due toAbel.It turns out that an elliptic functiong(x) is doublyperiodic in the following sense. There are nonzerocomplex numbersω1 andω2 (whose ratio is not real)such that

g(x + ω1) = g(x + ω2) = g(x).


Early history of elliptic curvesIt turns out that an elliptic functiong(x) is doublyperiodic in the following sense. There are nonzerocomplex numbersω1 andω2 (whose ratio is not real)such that

g(x + ω1) = g(x + ω2) = g(x).

This means thatg(x) factors through the quotientC/Λ, whereΛ ⊂ C is the lattice (additive subgroup)generated byω1 andω2. Topologically,C/Λ is atorus, and it is by definition anelliptic curve over C.It is also an Abelian group since it is a quotient of theadditive groupC.


Early history of elliptic curvesThere are nonzero complex numbersω1 andω2

(whose ratio is not real) such that

g(x + ω1) = g(x + ω2) = g(x).

One can assume without loss of generality thatτ = ω1/ω2 has positive imaginary part (permutingω1

andω2 if necessary), and by a simple rescaling, wecan replaceω2 by 1. Thus every elliptic curve isisomorphic to one associated with the latticegenerated by 1 andτ , whereτ lies in the upper halfplane.


Elliptic curves as plane cubicsWeierstrass determined the field of meromorphicfunctions that are doubly periodic with respect to agiven lattice. His work led to a description of thecorresponding elliptic curve as a cubic curve in thecomplex projective planeCP 2.


Elliptic curves as plane cubicsWeierstrass determined the field of meromorphicfunctions that are doubly periodic with respect to agiven lattice. His work led to a description of thecorresponding elliptic curve as a cubic curve in thecomplex projective planeCP 2.Recall thatCP 2 is the space of complex lines throughthe origin in the complex vector spaceC3. A nonzeropoint (x, y, z) determines such a line, which wedenote by[x, y, z]. Note that

[λx, λy, λz] = [x, y, z]

for any nonzero scalarλ, and there is no line withcoordinates[0, 0, 0].


Elliptic curves as plane cubicsNow leth(x, y, z) be a homogeneous polynomial ofdegreed. This means that

h(λx, λy, λz) = λdh(x, y, z).

Henceh doesnot give a complex valued function onCP 2, although it can be shown that is corresponds toa section of a certain line bundle overCP 2.


Elliptic curves as plane cubicsNow leth(x, y, z) be a homogeneous polynomial ofdegreed. This means that

h(λx, λy, λz) = λdh(x, y, z).

Henceh doesnot give a complex valued function onCP 2, although it can be shown that is corresponds toa section of a certain line bundle overCP 2.However the set of points inCP 2 whereh vanishes iswell defined and is called aprojective plane curve ofdegree d, which we will denote byVh. For “most”polynomialsh, Vh is an embedded Riemann surface of

genus(

d−1

2

)

.


Elliptic curves as plane cubicsBy “most” I mean the following. The set ofhomogeneous polynomials of degreed is a complex

vector space of dimension(

d+2

2

)

. It is known to have

an open dense subset of polynomialsh for whichVh isas above. It is also clear that this is not true for allh.For example ofh(x, y, z) is a product ofd distinctlinear factors, thenVh will be the union ofd projectivelines instead of a surface of the required genus.


Elliptic curves as plane cubicsWeierstrass showed that every elliptic curveC/Λ isequivalent (in the appropriate sense) to a projectiveplane curveE of degree 3. He gave formulas for thecoefficients of the polynomial in terms of the latticeΛwhich I will not go into here. This geometricdescription leads to the following way to describe theAbelian groups structure induced by complexaddition.

The sum of any three colinear points in E is zero.


Elliptic curves as plane cubicsNote that finding the intersection ofE with aprojective lineL boils down to finding the roots of acubic equation in one variable. The FundamentalTheorem of Algebra tells us that one of three thingswill happen.

(i) There are three distinct roots, which means thatLmeetsE at three distinct points, sayA, B andC.



(ii) There are two distinct roots, which means thatLmeetsE at two distinct pointsA andB. In thiscase the line is tangent to the pointAcorrespsonding to the repeated root, and2A + B = 0.



(iii) There is a single root with multiplicity three. Inthis caseL meetsE tangentially at a single pointof inflectionA, and3A = 0.


Elliptic curves as plane cubicsThe colinear rule is not quite enough to determine thegroup structure onE because it does not determinewhich pointe ∈ E is the identity. Since3e = 0, emust be a point of inflection, and it is known thatthere are 9 of them. This can be seen from the latticepoint of view as follows.


Elliptic curves as plane cubicsSuppose the latticeΛ is generated by1 andτ . Theneach of the 9 points points in the set

{(a + bτ)/3: 0 ≤ a, b < 3}

has order dividing 3 inC/Λ. Similarly, there aren2

points with order dividingn.Once we have chosen one of the 9 points of inflectionas the identity element, the group structure onE isdetermined by the colinear rule


The formal group lawSuppose that

h(x, y, z) = x3 + axy2 + by3 − yz2

for some constantsa andb. This defines an ellipticcurve provided that

4a3 + 27b2 6= 0.


The formal group lawWe can choose[0, 0, 1] to be our identity element.There is an embedding of the affine planeC

2 into theprojective plane given by

(x, y) 7→ [x, y, 1].

Let E ′ denote the intersection of this plane with theelliptic curveE defined by the equationh(x, y, z) = 0, soE ′ is the affine plane curve definedby

y = x3 + axy2 + by3.

The identity element ofE lies inE ′ at the origin.


The formal group lawThe following facts are easily verified.

(i) Near the origin there is an odd power seriesexpansion fory, namely

y = y(x) = x3 + ax7 + bx9 + 2a2x11 + . . . ,

so a point onE ′ near the origin is determined byits first coordinate. This power series hascoefficients in the ringR = Z[a, b].



(ii) (x, y) ∈ E ′ iff (−x,−y) ∈ E ′, so the grouptheoretic inverse of the point(x, y) is (−x,−y).



(ii) (x, y) ∈ E ′ iff (−x,−y) ∈ E ′, so the grouptheoretic inverse of the point(x, y) is (−x,−y).

(iii) Given two points(x1, y1) and(x2, y2) near theorigin, let(x3, y3) denote their group theoreticsum. Then there is a power series expansionF (x1, x2) for x3 in terms ofx1 andx2 withcoefficients inR.


The formal group lawIn addition to having a positive radius of convergence,F must satisfy the following three conditions.

(i) F (u, 0) = F (0, u) = u since(0, 0) is the identityelement.

(ii) F (v, u) = F (u, v) since the group is Abelian.

(iii) F (F (u, v), w) = F (u, F (v, w)) by associativity.

The oddness of the series fory above impliesadditionally thatF (u,−u) = 0.


The formal group lawWe define acommutative 1-dimensional formal grouplaw over a ring R to be a power seriesF ∈ R[[u, v]]satisfying the three conditions above, but without anyconvergence requirement. A commutativen-dimensional formal group law is a collection ofnpower series in2n variable satsifying similarconditions. For a much more thorough discussion[Rav04, Appendix 2].The choice ofh(x, y, z) above is convenient but notessential. The variablex could be replaced by a localcoordinate at the identity, and we would still get aformal group law.


References

[Hus87] Dale Husemoller.Elliptic curves, volume111 ofGraduate Texts in Mathematics.Springer-Verlag, New York, 1987. With anappendix by Ruth Lawrence.

[Kna92] Anthony W. Knapp.Elliptic curves,volume 40 ofMathematical Notes.Princeton University Press, Princeton, NJ,1992.

[Kob84] N. Koblitz. Introduction to Elliptic Curvesand Modular Forms. Graduate Texts inMathematics. Springer-Verlag, New York,1984.


References

[Rav04] D. C. Ravenel.Complex Cobordism andStable Homotopy Groups of Spheres, SecondEdition. American Mathematical Society,Providence, 2004. Available online.

[Sil86] Joseph H. Silverman.The arithmetic ofelliptic curves, volume 106 ofGraduateTexts in Mathematics. Springer-Verlag, NewYork, 1986.

[ST92] Joseph H. Silverman and John Tate.Rational points on elliptic curves.Undergraduate Texts in Mathematics.Springer-Verlag, New York, 1992.


Documents

Elliptic curves: what they are, why they are called ...group structure on E because it does not determine which point e ∈ E is the identity. Since 3e = 0, e must be a point of inﬂection,