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 , 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 ).

• 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 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

Let ω1, ω2 be non-zero complex numbers, with ω1/ω2 ∈/ R. Let Λ = ω1Z⊕ω2Z (a in C).

Consider the quotient C/Λ: this is a compact 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 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 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. 2 3 E3 : y = x + 4, torsion ' Z/3Z. 2 3 E4 : y = x + 4x, torsion ' Z/4Z. 2 3 E5 : y − y = x − x, torsion ' Z/5Z. 2 3 E6 : y = x + 1, torsion ' Z/6Z. 2 3 2 E7 : y − xy − 4y = x − x , torsion ' Z/7Z. 2 3 E8 : y + 7xy = x + 16x, torsion ' Z/8Z. 2 3 2 E9 : y + xy + y = x − x − 14x + 29, torsion ' Z/9Z. 2 3 E10 : y + xy = x − 45x + 81, torsion ' Z/10Z. 2 3 2 E11 : y + 43xy − 210y = x − 210x , torsion ' Z/12Z. 2 3 E12 : y = x − 4x, torsion ' Z/2Z × Z/2Z. 2 3 2 E13 : y + xy − 5y = x − 5x , torsion ' Z/4Z × Z/2Z. 2 3 2 E14 : y + 5xy − 6y = x − 3x , torsion ' Z/6Z × Z/2Z. 2 3 2 E15 : y + 17xy − 120y = x − 60x , torsion ' Z/8Z × Z/2Z.

12 Reduction modulo a prime

Let E/Q be an elliptic curve. By change of variable one can assume the equation (1) has integral coefficient. For any prime p, one can reduce modulo p and look at solutions (x, y) in the finite field Z/pZ of 2 3 2 y +a1xy+a3y = x +a2x +a4x+a6 (mod p).

If p - ∆E this is an elliptic curve over Z/pZ.

Theorem (Hasse). The number of projec- tive solutions is Np = p + 1 − ap with |ap| ≤ √ 2 p.

Remark. If a1 = a3 = 0 then 3 2 X x + a2x + a4x + a6  ap = − p x (mod p)   √ with y the Legendre symbol. So size is reason- p p able on probabilistic grounds.

2 3 Example. y = x − x, ∆E = 64. If p ≡ 3 (mod 4), 2 2 ap = 0; if p ≡ 1 (mod 4), write (Fermat) p = a + b with a odd, b even, a + b ≡ 1 (mod 4); then ap = 2a.

13 The (partial) Hasse-Weil L-function

To go from local (modulo primes) to global, define first

Y −s 1−2s −1 `(E, s) = (1 − app + p ) . p-∆E

This product converges absolutely for Re(s) > 3/2 by Hasse’s Theorem.

Hasse conjectured that `(E, s) has analytic continuation to C. This is an imprecise form of the modularity of elliptic curves over Q, proved by Wiles, Taylor-Wiles, Breuil-Conrad- Diamond-Taylor.

14 The conductor

To obtain the “right” L-function, one needs correct factors at the primes p | ∆E. One can have p | ∆E for some equation but not for another (∆ is not an isomorphism-invariant).

One defines the conductor NE ≥ 1 such that p - NE if and only if E has a smooth reduction modulo p, possibly after change of variable. For p | NE, the exponent fp of p in NE is dictated by the geometry of the singular re- duction.

Examples • If the reduction of E modulo p has a node, then fp = 1 (“multiplicative reduction”).

• If p > 3 and the reduction of E modulo p has a cusp, then fp = 2 (“additive reduction”).

2 3 • For y + y = x − x, NE = 11. This is the smallest possible conductor for E/Q.

The cases p = 2, 3 with a cusp are much more intricate. 15 The complete Hasse-Weil L-function

If p | NE, define  0 if fp ≥ 2,  ap = −1 if fp = 1, slopes in Z/pZ,  1 otherwise.

Then Y −s −1 L(E, s) = (1 − app ) p|NE Y −s 1−2s −1 (1 − app + p ) . p-NE

Modularity of E implies that L(E, s) has holo- morphic continuation to an entire function, and that it satisfies 1−s Λ(E, s) = wENE Λ(E, 2 − s) where wE = ±1 (sign of the functional equa- tion) and Λ(E, s) = (2π)−sΓ(s)L(E, s).

Remark. The sign wE factorizes as a prod- uct over p | NE of local signs. It is effectively computable. 16 Modularity explained

In fact the continuation of L(E, s) is proved indirectly.

X −s Write L(E, s) = ann and put n≥1 X 2πinz f(z) = ane for Im(z) > 0. n≥1

Modularity means exactly that f is holomor- phic, satisfies az + b f = (cz + d)2f(z) cz + d for a, b, c, d ∈ Z, ad − bc = 1, NE | c, and Im(z)|f(z)| is bounded (“cusp form of weight 2 for Γ0(NE)”).

Then the formula (Hecke) Z ∞ Λ(E, s) = f(iy)ys−1dy 0 easily gives analytic continuation/functional equation. 17 The Birch and Swinnerton-Dyer Conjecture

Let E be an elliptic curve defined over Q. So L(E, s) is holomorphic, in particular defined at s = 1. The simplest form of the Birch and Swinnerton-Dyer Conjecture is

Conjecture. We have

rank E(Q) = ords=1 L(E, s).

Remark. If wE = −1 (a local condition), then L(E, 1) = 0 so the conjecture implies that rank E ≥ 1 in this case. Find a non- torsion point!

There is a more refined form:

Conjecture. We have L(E, s) ∼ α(s − 1)r as s → 1, where Ω|X(E)|R(E)c = 0 α 2 > . |E(Q)tors|

We will now explain the various factors in the constant α. 18 Explanation I: the “easy” terms

• |E(Q)tors| is the cardinality of the set of rational torsion points on E. This is easy to compute theoretically and algorithmically.

• c is given by the product over primes of cp = |E(Qp)/E0(Qp)|, E0(Qp) being the set of points which have non-singular reduction modulo p. If E has good reduction at p, cp = 1. Otherwise there is an efficient algorithm to compute cp.

19 Explanation II: the regulator

R(E) is the elliptic regulator. Let x1,. . . , xr be a basis for the free part of E(Q). Then

R(E) = det(hxi, xji) where h·, ·i is the canonical height on E(Q) ⊗ R, the bilinear form coming from the quadratic form kpk = lim 4−nh([2n]p) n→+∞ where 1 h(x, y) = log H(x), 2 H(p/q) = max(|p|, |q|) if (p, q) = 1.

There are explicit and efficiently computable formulas for the height.

20 Explanation III: the Tate-Shafarevich group

The group X(E) is the most mysterious term in the Birch and Swinnerton-Dyer conjecture. In contrast with the others, it is not known to be effectively computable, in fact it is not known to be finite. One can sketch how X(E) arises as follows. Let E/Q 2 have equation y = (x − e1)(x − e2)(x − e3) with ei ∈ Q (so E[2] ⊂ E(Q). If (x, y) ∈ E(Q), note that for p - ∆E the smoothness modulo p implies x − ei are pairwise coprime so p occurs with even power in the factorization of x − ei. So there is a computable finite set T of triples b = (b1, b2, b3) of non-zero rationals and rationals z1, z2, z3 such that for some b ∈ T

 2 y = (x − e1)(x − e2)(x − e3)  2 x − e1 = b1z1 2 x − e2 = b2z2  2 x − e3 = b3z3.

For any fixed b, this defines a curve Cb (in affine 5- space). Eliminating some unknowns it is isomorphic to the space curve

 2 2 b1z1 − b2z2 = (e2 − e1) 2 2 b1z1 − b1b2z3 = (e3 − e1).

Finding all Cb which have a rational point allows to compute easily E(Q)/2E(Q) and then E(Q).

21 Problem: There is no algorithm to check whether Cb(Q) 6= ∅. However one can compute easily the subset S ⊂ T of those b for which Cb has “locally” a point at all p, and a real-valued point.

Those elements of S which still do not have a ratio- nal point (they fail the “Hasse principle”) “are” the elements of order 2 in X(E). (It is a finite set here, in fact a group).

General definition as a set: a curve C is a principal homogeneous space for E if one can define p + P for p ∈ C, P ∈ E with p + (P + Q) = (p + P ) + Q and p + P = q has a unique solution q − p for all (p, q). The curves Cb above are examples. Then X(E) is the set of all such C for which C(R) and C(Qp), for all p, are non-empty, modulo iso- morphism (as homogeneous spaces). There is a group structure with E as identity ele- ment. A C ∈ X(E) is trivial if and only if C(Q) 6= ∅ (p 7→ p − p0 gives C ' E).

22 Cohomology definition: ( ) ( ) = ker 1( ) Y 1( ) X E H GQ,E → H GQv,E v

Conjecture. For all E/Q, the Tate-Shafa- revich group X(E) is a finite group.

The refined form of the Birch and Swinnerton- Dyer conjecture does not make sense without this conjecture.

This is known for only very few cases where ords=1 L(E, s) ≤ 1 (Rubin, Kolyvagin...)

23 Example: a Tate-Shafarevich group

Let E be the elliptic curve y2 = x3 − 24300, j = 0. The rank is 0, the regulator is 1, the torsion group is trivial, the Tamagawa number is 1,

L(E, 1) = 4.061375813927 ... Ω = 0.451263979325 ... and so |X(E)| = 9, X(E) ' (Z/3Z)2. In fact, the following are equations for all locally trivial homogeneous spaces under E:

C ' E x3 + y3 + 60x3 = 0 3 3 3 C1 3x + 4y + 5z = 0 3 3 3 C2 12x + y + 5z = 0 3 3 3 C3 15x + 4y + z = 0 3 3 3 C4 3x + 20y + z = 0

(each of the four equations Ci above corre- sponds to two opposite elements of X(E), equivalently to a line in (Z/3Z)2). [ref.: B. Mazur, “On the passage from local to global in number theory”, Bull. A.M.S 29 (1993), 14–50].

24 Enter random matrices...

If the Birch and Swinnerton-Dyer Conjecture holds, the L-function gives an analytic handle on the very mysterious rank of elliptic curves. One may hope to be able in this way to un- derstand (or solve?) some of the outstanding problems, such as:

• Are there elliptic curves over Q of arbitrarily large rank?

• If yes, “how many” are there when one looks at “families” of elliptic curves?

For some of these questions, the link with zeros of L-functions allows the use of mod- els coming from Random Matrix Theory to try and understand those issues. Comparison with insights from algebraic geometry is then very desirable.

25 Quadratic twists

The family of curves y2 = x3 − n2x, indexed by n, are special cases of quadratic twists. More generally if E has equation 2 3 E : y = x + a4x + a6 consider the curves Ed 2 3 Ed : dy = x + a4x + a6. 1/2 Note that E ' Ed over C ((x, y) 7→ (x, d y)).

Question. How does the rank of Ed vary as function of d? How do the other invariants?

Function field analogues and low-lying zeros results suggest that the family (Ed), as d varies, has orthogonal symmetry.

26 The root number is equidistributed in {±1}. One then expects (Goldfeld) 1 |{d ≤ X | rank E = 0}| ∼ X 1 d 2 d≤X 1 |{d ≤ X | rank E = 1}| ∼ X 1. d 2 d≤X

The numerics are ambiguous (suggest “ex- cess rank”, especially for even ranks).

Number of twists with higher rank: Random Matrix Theory has been used to predict, e.g,

|{d ≤ X | w = 1, rank E ≥ 2}| Ed d 3/4 b ∼ cEX (log X) E for some cE > 0, bE ≥ 0 (Conrey, Keating, Rubinstein, Snaith).

Higher order/rank??

27