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Elliptic Curves Elliptic curves Main ref.: J. Silverman, “The arithmetic of elliptic curves”, G.T.M 106. Elliptic curves as algebraic curves A plane affine cubic curve over C is the set of complex solutions (x, y) of an equation 2 3 2 y + a1xy + a3y = x + a2x + a4x + a6 (1) where the ai are complex numbers. If the ai are rational, the curve is said to be defined over Q, and one can consider the basic dio- phantine question: find all solutions (x, y) in Q. It is more convenient to look at the equation in homogeneous form and solutions in the projective plane 2 2 Y Z + a1XYZ + a3YZ = 3 2 2 3 X + a2X Z + a4XZ + a6Z where (X : Y : Z) = (αX : αY : αZ) if α 6= 0. If Z 6= 0, x = X/Z, y = Y/Z gives back the first equation. If Z = 0 one adds one point at infinity ∞ = (0 : 1 : 0) (always rational). 1 To “be” an elliptic curve, a plane cubic must be smooth, which means that the partial deri- vatives 2 2y + a1x + a3 and a1y − 3x − 2a2x − a4 do not have a common zero on the cubic. If a1 = a3 = 0, this means the cubic polyno- 3 2 mial x + a2x + a4x + a6 has simple roots, 3 2 equivalently that ∆E = −16(4a4+27a6) 6= 0. Examples • y2 = x3 is not smooth (it has a cusp). • y2 = x3 + x2 is not smooth (it has a node). • y2 = x3 − x is an elliptic curve (the “congruent num- ber” curve), of CM type. • y2 + y = x3 − x2 is an elliptic curve (the modular curve X1(11)). • If ` > 2 is a prime, (a, b, c) are non-zero and satisfy a` + b` = c` then y2 = x(x − a`)(x + b`) is a remarkable elliptic curve. It can not exist if (a, b, c) are rationals. 2 An elliptic curve as a complex torus Let ω1, ω2 be non-zero complex numbers, with ω1/ω2 ∈/ R. Let Λ = ω1Z⊕ω2Z (a lattice in C). Consider the quotient C/Λ: this is a compact Riemann surface and it “is” an elliptic curve; topologically this is a torus. In terms of functions, consider meromorphic functions f : C → C which are ω1 and ω2-periodic: f(z + ω1) = f(z) and f(z + ω2) = f(z). Those f are called elliptic functions because the arc-length on an ellipse can be expressed in terms of (inverses of) such functions. 3 Correspondance between the two viewpoints For a given Λ, there is an elliptic function ℘ which has a pole of order 2 at points of Λ and satisfies the algebraic differential equation 02 3 ℘ = 4℘ − g2℘ − g3 for some g2, g3 ∈ C. In fact 1 1 1 ℘(z) = + X − , z2 (z − ω)2 ω2 ω∈Λ ω6=0 X 1 X 1 g2 = 60 , g3 = 140 . ω4 ω6 ω∈Λ ω∈Λ ω6=0 ω6=0 √ Sending z 7→ (2℘(z), 2℘0(z)) gives (bijec- tively) the points on the plane cubic 2 3 y = x − g2x − 2g3 with 0 7→ ∞ since ℘ has a pole at z = 0. One shows this is smooth. All elliptic curves with a1 = a3 = a2 = 0 arise in this manner. 4 The group law The quotient C/Λ has a natural abelian group structure. So there must be group structure on elliptic curves. Geometrically, if P , Q, R are distinct points on the elliptic curve (as plane cubic curve), we have P + Q + R = 0 for this group law if and only if P , Q and R are colinear. [Sketch of “if”: The equation F (x, y, z) = 0 of the line joining P , Q and R gives an elliptic function f such that the divisor of f is (p) + (q) + (r) − 3(0). By integrating zf 0/f along the boundary of a fundamental parallelogram, one gets p + q + r ∈ Λ]. This group law is algebraic: for instance one 2 3 finds for y = x + a4x + a6 that −(x, y) = (x, −y), and (x1, y1) + (x2, y2) = (x3, y3) with 2 y2 − y1 x3 = − x1 − x2, x2 − x1 y2 − y1 y3 = (x3 − x1) − y1 x2 − x1 if x1 6= x2. 5 Maps between elliptic curves Some polynomial or rational change of co- ordinates transform an elliptic curve into an- other. If they are “well-defined” everywhere, these are morphisms of elliptic curves. Examples. • For fixed x0 ∈ E(C), f(x) = x+x0 (with the group law above) is a morphism E → E. • For any integer n ∈ Z,[n]: x 7→ nx (for the group law) is a morphism E → E (defined over Q if E is). • If a2 6= 4b, {y2 = x3 + ax2 + bx} → {w2 = v3 − 2av2 + (a2 − 4b)v} 2 ( ) (y2 y(b−x) ) x, y 7→ x2 , x2 is a morphism with (0, 0) 7→ ∞. • Let E : y2 = x3 − x. Then [i]:(x, y) 7→ (−x, iy) is a morphism defined over Q(i). Any morphism is of the form f(x) = g(x)+x0 where g is a morphism preserving the group law (isogeny). For a complex torus C/Λ, translations corre- spond to translations and isogenies C/Λ1 → C/Λ2 correspond to multiplication by α ∈ C such that Λ1 ⊂ Λ2. 6 Isogenies, torsion points Isogenies E → E form a ring, End(E). One may have EndC(E) 6= EndQ(E). Usually one has End(E) ' Z. Otherwise, E is a CM curve. A non-zero isogeny f : E1 → E2 is surjective. The kernel ker f = {x ∈ E1 | f(x) = 0} is a finite abelian group. For f = [n], n 6= 0, one has ker f = E[n] ' (Z/nZ)2 by the complex description E(C) = C/Λ. If E is defined over Q, the points of E[n] have algebraic coordinates. They are analogues of the classical roots of unity (torsion points in C×) and are very important in the arithmetic of E. 2 3 Example. If E has equation y = x + a4x + a6 then E[2] = {∞, (e1, 0), (e2, 0), (e3, 0)} 3 where ei are the (distinct) roots of x +a4x+ a6 = 0. 7 Classification of elliptic curves An isomorphism of elliptic curve is an isogeny which is one-to-one. One can try to classify elliptic curves up to isomorphism. By simple changes of variable, any Weier- strass equation over C can be brought to 2 3 the form y = x + c4x + c6 for some c4, c6. Such curves are elliptic curves if ∆ = 3 2 −16(4c4 + 27c4) 6= 0. They are classified up to C-isomorphism by the j-invariant j = 3 1728(4c4) /∆. Elliptic curves defined over Q might be iso- morphic but only over a bigger field (twists). For complex tori, one checks that there is a unique τ ∈ SL(2, Z)\H such that C/Λ ' C/(Z ⊕ τZ). The j-invariant is then a holo- morphic map H → C which is SL(2, Z)-in- variant. 8 Enter arithmetic (Mordell’s theorem) If the coefficients ai defining the elliptic curve are rational, one checks immediately that the set of rational solutions is a subgroup of E(C). The main structure theorem is due to Mordell in this case. Theorem. The group E(Q) is finitely gen- erated. Thus one can write E(Q) ' Zr ⊕ F where r ≥ 0 is the rank of E (over Q) and F = E(Q)tors is the finite torsion subgroup of E. The proof of the theorem is ineffective; more precisely, it yields an upper bound on r, but no effective way of testing whether a finite family x1, ..., xk ∈ E(Q) generates the whole of E(Q) up to torsion. However the proof shows it suffices to com- pute E(Q)/mE(Q) for some m ≥ 1 to com- pute effectively E(Q). 9 Example: the congruent number problem Here is a beautiful instance of the intrusion of elliptic curves in a very classical problem: what are the rationals (so-called congruent numbers) r such that there is a right-triangle with rational lengths a, b, c and area r. Proposition. A squarefree integer n ≥ 1 is a congruent number if and only if the elliptic 2 3 2 curve En : y = x − n x has rank rn ≥ 1. This has led to an algorithm to check whether a given squarefree integer n is a congruent number. Theorem (Tunnell). If the Birch and Swin- nerton-Dyer Conjecture holds, then (for odd squarefree n), n is a congruent number if and only if the number of triples of integers (x, y, z) such that 2x2 + y2 + 8z2 = n is twice the number of triples such that 2x2 + y2 + 32z2 = n. 10 Example: n = 41 is congruent; 4 8 z }| { z }| { 41 = 2(±4)2 + (±3)2 = 2(±4)2 + (±1)2 + 8(±1)2 8 4 z }| { z }| { = 2(±2)2 + (±5)2 + 8(±1)2 = (±3)2 + 8(±2)2 8 z }| { = 2(±2)2 + (±1)2 + 8(±2)2 4 8 z }| { z }| { 41 = 2(±4)2 + (±3)2 = 2(±2)2 + (±1)2 + 32(±1)2 4 z }| { = (±3)2 + 32(±1)2. [ref.: N. Koblitz, “Introduction to elliptic curves and modular forms”, G.T.M 97.] 11 Digression on torsion The finite torsion group E(Q)tors is easily computable (the prime-to-p part of it injects in the finite group of points modulo a prime p of good reduction). There is a finite list of possible E(Q)tors for E/Q (a theorem of Mazur). 2 3 E1 : y = x − 2, torsion = {0}. 2 3 E2 : y = x + 8, torsion ' Z/2Z.
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