The J-invariant Of An Elliptic Curve

The j-invariant of an Elliptic CurveDylan Pentland20 May 2018Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20181 / 13

An important questionQuestion. Given a polynomial F (x, y) Q[x, y], for which p Q2 isF (p) 0?It turns out a natural way to attack this problem is to attach a numberg called the genus to F .g 0. This is form conic sections, and these will either have norational points or the rational points will be parameterized byq Q in an easy way.g 1. These are cubic equations, and there can be finitely manyrational points or infinitely many. The points have a nice groupstructure.g 2. There are finitely many rational points (Falting’s theorem).Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20182 / 13

What is an elliptic curve?An elliptic curve E is a curve of the formy 2 x3 ax2 bx c.With substitutions preserving rational points, these can be put inthe Weierstrass form y 2 x3 ax b.E must also be nonsingular. Here, this means there are noself-intersections or cusps. We can check this by lettingF (x, y) x3 ax2 bx c y 2 and checking if F 0,at any point P where F (P ) 0, in which case E is singular.Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20183 / 13

The group structure of EElliptic curves over Q come equipped with a group structure of theset of rational points E(Q).We add P, Q E(Q) to obtain a point R P Q by taking thethird intersection R0 of E and the line (P, Q) through P, Q.Flipping over the x axis, we obtain R.If P Q, (P, Q) is the tangent to E. The identity is given by thepoint at infinity O – we say P Q O if (P, Q) fails intointersect E in R2 .Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20184 / 13

An illustrationFigure 1: Elliptic curve addition (Image from [Sil09])Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20185 / 13

Elliptic curve isogeniesAn isogeny φ : E E 0 is a rational map which satisfiesφ(OE ) OE 0 , which reflects that φ induces a grouphomomorphism. The set of isogenies is denoted Hom(E, E 0 ).When E E 0 , this is End(E).Over a field K, isogenies are maps (x, y) 7 (f (x, y), g(x, y)) wheref, g are in K(x, y).We say E E 0 if φ is an invertible map.Example: The map [n] : E E sending P nP is a member ofEnd(E).Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20186 / 13

An isogeny invariantTake an elliptic curve E/Q and write it in Weierstrass formy 2 x3 ax b. The j-invariant is given byj(E) 17284a3.4a3 27b2TheoremLet E, E 0 be elliptic curves over Q. Then E E 0 over C if and only ifj(E) j(E 0 ). In general, given a field K and elliptic curves E, E 0 overK then E E 0 over K if and only if j(E) j(E 0 ).Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20187 / 13

The functionIn order to motivate j(E), we need to reinterpret what an elliptic curveis. To do this, we look at elliptic functions, or doubly periodicmeromorphic functions. The Weierstrass function describes thesecompletely:TheoremLet Λ C be a lattice, and let Λ (z) 1 z2Xω Λ\{0}11 2.2(z ω)ωThe elliptic function field for C/Λ is given by C( Λ , 0Λ ).Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20188 / 13

Elliptic curves over C are complex toriTheoremGiven a lattice Λ C, there is a corresponding elliptic curve EΛ suchthat C/Λ EΛ (C) as groups. Given an elliptic curve E, there is alattice ΛE such that E C/ΛE as groups.The curve EΛ is given byEΛ : y 2 4x3 g2 (Λ)x g3 (Λ),PPwhere g2 (Λ) 60 ω Λ\{0} ω 4 , g3 (Λ) 140 ω Λ\{0} ω 6 . Theisomorphism is given byz 7 ( Λ (z), 0Λ (z)),when z 6 Λ and z 7 O when z Λ.We can also take any elliptic curve E andR obtain a latticeR dxΛE ω1 Z ω2 Z using integrals ω1 α dxy and ω2 β y toobtain basis elements. Here, α, β generate H1 (E(C), Z).Dylan PentlandThe j-invariant of an Elliptic Curve20 May 20189 / 13

Homothetic LatticesWe say Λ and Λ0 are homothetic if Λ ωΛ0 for ω C . We canequivalently characterize isomorphism classes of elliptic curves asfollows:TheoremThe complex tori C/Λ EΛ and C/Λ0 EΛ0 are isomorphic over C iff0Λ and Λ are homothetic.Now it is very natural to consider the j-invariant from modularforms. This is defined byj(τ ) 1728g23,g23 27g32whereXg2 60(m,n)6 (0,0)Dylan Pentland(m nτ ) 4 , g3 140X(m nτ ) 6 .(m,n)6 (0,0)The j-invariant of an Elliptic Curve20 May 201810 / 13

Why the j-invariant is a perfect fitWe want a homothety invariant j(Λ) such that j(Λ) j(Λ0 ) iff Λ, Λ0are homothetic. Suppose we have such a function:If j is a homothety invariant, j([ω1 , ω2 ]) j([1, ω2 /ω1 ]).Consider τ, τ 0 H. If f (τ ) f (τ 0 ) precisely when the lattices [1, τ ]and [1, τ 0 ] are the same then f should be a modular function as itis invariant under the natural action of SL(2, Z). The space ofsuch functions is C(j), where j j(τ ) is the j-invariant.As a result, we know we should base j(Λ) off of j(τ ). Noticing thatg2 , g3 sum over the lattice [1, τ ], it is natural to definej(EΛ ) j(Λ) 1728g23 (Λ),g23 (Λ) 27g32 (Λ)where g2 (Λ) and g3 (Λ) are the coefficients of EΛ .It remains to check that j(Λ) j(wΛ) – this is not too hard.Dylan PentlandThe j-invariant of an Elliptic Curve20 May 201811 / 13

ConclusionWe can conclude the following about elliptic curves over Q:If j(E) 6 j(E 0 ), then certainly E and E 0 are not isomorphic.If j(E) j(E 0 ), they are isomorphic over C (more specifically, Q)but not necessarily over Q. For example, takeE/Q : y 2 x3 xE 7 /Q : y 2 x3 49x.Here, j(E) j(E 0 ) 1728. However, E(Q) is a finite group butE 7 (Q) is infinite, and hencenot isomorphic to E(Q). These curves are isomorphic over Q( 7).Dylan PentlandThe j-invariant of an Elliptic Curve20 May 201812 / 13

ReferencesJoseph H Silverman, The arithmetic of elliptic curves, vol. 106,Springer Science & Business Media, 2009.Joseph H Silverman and John Torrence Tate, Rational points onelliptic curves, vol. 9, Springer, 1992.Dylan PentlandThe j-invariant of an Elliptic Curve20 May 201813 / 13

rational points or the rational points will be parameterized by q2Q in an easy way. g 1. These are cubic equations, and there can be nitely many rational points or in nitely many. The points have a nice group structure. g 2. There are nitely many rational points (Falting’s theorem). Dylan Pentland The j-invariant of an Elliptic Curve 20 May .