Complex tori as elliptic curves

Modular Forms, Elliptic Curves, and Modular Curves

1.4 Complex tori as elliptic curves

Exercises

1.3.1.Prove Lemma 1.3.1.

1.3.2.Prove Corollary 1.3.3. (A hint for this exercise is at the end of the book.)

1.3.3.(a) Show that the Weil pairing is independent of which basis{ω₁, ω₂} is used, providedω₁/ω₂∈ H.

(b) Show that the Weil pairing is bilinear, alternating, and nondegenerate.

(Remember that the groupµN is multiplicative.)

(c) Show that the Weil pairing is compatible withN. This means that for positive integersN andd, the diagram

E[dN]×E[dN] ^edN(·,·) //

d(·,·)

µdN

·^d

E[N]×E[N] ^eN(·,·) // µN

commutes, where the vertical maps are suitable multiplications byd. (A hint for this exercise is at the end of the book.)

(d) Let Λ and Λ be lattices with mΛ =Λ for some m∈ C. Show that the isomorphism of complex elliptic curvesC/Λ−→^∼ C/Λ given byz+Λ→ mz+Λ preserves the Weil pairing.

the terms of the sum from being permuted whenz is translated by a lattice element, so℘doesn’t obviously have periodsΛ. But the derivative

℘(z) =−2

ω∈Λ

1 (z−w)³

clearly does have periodsΛ, and combining this with the fact that℘is an even function quickly shows that in fact℘has periodsΛas well (Exercise 1.4.2). It turns out that℘and℘are the only basic examples we need since the field of meromorphic functions onC/ΛisC(℘, ℘), the rational expressions in these two functions. Since the Weierstrass ℘-function depends on the lattice Λ as well as the variablez we will sometimes write℘Λ(z) and℘_Λ(z); in particular for latticesΛτ we will write℘τ(z) and℘_τ(z).

Eisenstein series generalize to functions of a variable lattice, Gk(Λ) =

ω∈Λ 1

ω^k, k >2 even,

so that G_k(τ) from before is now G_k(Λ_τ). As lattice functions Eisenstein series satisfy thehomogeneity conditionG_k(mΛ) =m^−kG_k(Λ) for all nonzero m∈C. Part (a) of the next result shows that these lattice Eisenstein series appear in the Laurent expansion of the Weierstrass℘-function forΛ. Part (b) relates the functions℘(z) and℘(z) in a cubic equation whose coefficients are also lattice Eisenstein series.

Proposition 1.4.1.Let℘be the Weierstrass function with respect to a lattice Λ. Then

(a)The Laurent expansion of ℘is

℘(z) = 1 z² +

∞ nn=2even

(n+ 1)G_n+2(Λ)zⁿ

for allz such that 0<|z|<inf{|ω|:ω∈Λ− {0}}. (b)The functions ℘and℘ satisfy the relation

(℘(z))²= 4(℘(z))³−g₂(Λ)℘(z)−g₃(Λ) whereg2(Λ) = 60G4(Λ)andg3(Λ) = 140G6(Λ).

(c)LetΛ=ω1Z⊕ω2Zand letω3=ω1+ω2. Then the cubic equation satisfied by ℘and℘,y²= 4x³−g2(Λ)x−g3(Λ), is

y²= 4(x−e1)(x−e2)(x−e3), ei=℘(ωi/2)fori= 1,2,3.

This equation is nonsingular, meaning its right side has distinct roots.

Proof. (Sketch.) For (a), if |z|<|ω|then 1

(z−ω)² − 1 ω² = 1

ω²

1

(1−z/ω)² −1

and the geometric series squares to_∞

n=0(n+ 1)zⁿ/ωⁿ, making the summand 2z/ω³+ 3z²/ω⁴+· · ·. Convergence results from Exercise 1.1.4 allow the re- sulting double sum to be rearranged, and then the inner sum cancels whenn is odd.

For (b), one uses part (a) to show that the nonpositive terms of the Laurent series of both sides are equal. Specifically, since

℘(z) = 1

z² + 3G4(Λ)z²+ 5G6(Λ)z⁴+O(z⁶) (where “O” means “a quantity on the order of”) and

℘(z) =−2

z³ + 6G₄(Λ)z+ 20G₆(Λ)z³+O(z⁵),

a little algebra shows that both (℘(z))² and 4(℘(z))³−g₂(Λ)℘(z)−g₃(Λ) work out to 4/z⁶−24G₄(Λ)/z² −80G₆(Λ) +O(z²). So their difference is holomorphic andΛ-periodic, therefore bounded, therefore constant, therefore zero since it isO(z²) asz→0.

For (c), since ℘ is odd, it has zeros at the order 2 points of C/Λ: if z≡ −z(mod Λ) then ℘(z) =℘(−z) = −℘(z) and thus ℘(z) = 0. Letting Λ = ω1Z⊕ω2Z, the order 2 points are zi = ωi/2 with ℘(zi) = 0 for i = 1,2,3. The relation between℘and℘ from (b) shows that the corresponding valuesxi = ℘(zi) for i = 1,2,3 are roots of the cubic polynomial pΛ(x) = 4x³−g2(Λ)x−g3(Λ), so it factors as claimed. Eachxi is a double value of℘ since℘(zi) = 0, and since℘has degree 2, meaning it takes each value twice counting multiplicity (see Exercise 1.4.1(b)), this makes the threex_i distinct.

That is, the cubic polynomialp_Λ has distinct roots.

Part (b) of the proposition shows that the mapz→(℘Λ(z), ℘_Λ(z)) takes nonlattice points ofCto points (x, y)∈C² satisfying the nonsingular cubic equation of part (c),y²= 4x³−g2(Λ)x−g3(Λ). The map bijects since generally a valuex∈Cis taken by℘_Λtwice onC/Λ, that is,x=℘_Λ(±z+Λ), and then the twoy-values satisfying the cubic equation are℘(±z+Λ) =±℘(z+Λ).

The exceptional x-values where y = 0 occur at the order-2 points of C/Λ, so they are taken once by ℘_Λ as necessary. The map extends to all z ∈ C by mapping lattice points to a suitably defined point at infinity. In sum, for every lattice the associated Weierstrass ℘-function and its derivative give a bijection

(℘, ℘) : complex torus−→elliptic curve.

For example, the value g₃(i) = 0 from Section 1.1 shows that the complex torus C/Λ_i bijects to the elliptic curve with equation y² = 4x³−g₂(i)x.

Similarly the complex torusC/Λ_µ3 (where againµ₃ =e^2πi/3) bijects to the elliptic curve with equation y² = 4x³−g3(µ3). See Exercise 1.4.3 for some values of the functions℘and℘ in connection with these two lattices.

The map (℘, ℘) transfers the group law from the complex torus to the ellip- tic curve. To understand addition on the curve, letz1+Λandz2+Λbe nonzero points of the torus. The image points (℘(z1), ℘(z1)) and (℘(z2), ℘(z2)) on the curve determine a secant or tangent line of the curve inC²,ax+by+c= 0.

Consider the function

f(z) =a℘(z) +b℘(z) +c.

This is meromorphic onC/Λ. Whenb = 0 it has a triple pole at 0 +Λ and zeros atz₁+Λandz₂+Λ, and Exercise 1.4.1(c) shows that its third zero is at the pointz₃+Λsuch thatz₁+z₂+z₃+Λ= 0 +ΛinC/Λ. Whenb= 0,f has a double pole at 0 +Λ and zeros atz₁+Λandz₂+Λ, and Exercise 1.4.1(c) shows thatz₁+z₂+Λ= 0 +ΛinC/Λ. In this case letz₃= 0 +Λso that again z₁+z₂+z₃+Λ= 0 +Λ, and since the line is vertical view it as containing the infinite point (℘(0), ℘(0)) whose second coordinate arises from a pole of higher order than the first. Thus for any value ofbthe elliptic curve points on the lineax+by+c= 0 are the points (xi, yi) = (℘(zi), ℘(zi)) fori= 1,2,3.

Sincez1+z2+z3+Λ= 0 +Λ on the torus in all cases, the resulting group law on the curve is thatcollinear triples sum to zero.

Recall that a holomorphic isomorphism of complex tori takes the form z +Λ → mz + Λ where Λ = mΛ. Since ℘mΛ(mz) = m⁻²℘Λ(z) and

℘_mΛ(mz) = m⁻³℘_Λ(z), the corresponding isomorphism of elliptic curves is (x, y)→(m⁻²x, m⁻³y) or equivalently the substitution (x, y) = (m²x, m³y), changing the cubic equationy²= 4x³−g₂x−g₃associated toΛto the equa- tion y² = 4x³−m⁻⁴g₂x−m⁻⁶g₃ associated to Λ. Suitable choices of m (Exercise 1.4.3 again) normalize the elliptic curves associated toC/mΛ_i and C/mΛ_µ3 to have equations

y²= 4x(x−1)(x+ 1), y²= 4(x−1)(x−µ₃)(x−µ²₃).

We will return to these ideas in Chapter 7.

The appearance of Eisenstein series as coefficients of a nonsingular curve lets us prove as a corollary to Proposition 1.4.1 that the discriminant function from Section 1.1 has no zeros in the upper half plane.

Corollary 1.4.2.The function∆ is nonvanishing on H. That is,∆(τ)= 0 for allτ∈ H.

Proof. For any τ ∈ H, specialize the lattice Λ in the proposition toΛ_τ. By part (c) of the proposition, the cubic polynomialp_τ(x) = 4x³−g₂(τ)x−g₃(τ) has distinct roots. Exercise 1.4.4 shows that∆(τ) is the discriminant ofp_τ up to constant multiple (hence its name), so∆(τ)= 0.

Not only does every complex torusC/Λlead via the Weierstrass℘-function to an elliptic curve

y²= 4x³−a2x−a3, a³₂−27a²₃= 0 (1.8) witha2=g2(Λ) anda3=g3(Λ), but the converse holds as well.

Proposition 1.4.3.Given an elliptic curve(1.8), there exists a latticeΛsuch thata₂=g₂(Λ)anda₃=g₃(Λ).

Proof. The case a₂= 0 and the case a₃ = 0 are Exercise 1.4.5. For the case a₂= 0 and a₃ = 0, sincej : H −→C surjects there exists τ ∈ H such that j(τ) = 1728a³₂/(a³₂−27a²₃). This gives

g2(τ)³

g₂(τ)³−27g₃(τ)² = a³₂ a³₂−27a²₃, or, after taking reciprocals and doing a little algebra,

a³₂

g2(τ)³ = a²₃

g3(τ)². (1.9)

For any nonzero complex numberω₂, letω₁=τ ω₂andΛ=ω₁Z⊕ω₂Z. Then g₂(Λ) =ω₂⁻⁴g₂(τ) and g₃(Λ) =ω⁻₂⁶g₃(τ).

Thus we are done if we can chooseω2 such that

ω₂⁻⁴=a2/g2(τ) and ω⁻₂⁶=a3/g3(τ).

Choose ω2 to satisfy the first condition, so that ω₂⁻¹² = a³₂/g2(τ)³. Then by (1.9)ω₂⁻⁶=±a3/g3(τ), and replacingω2byiω2if necessary completes the

proof.

It follows that any map of elliptic curves (x, y)→(m⁻²x, m⁻³y), changing a cubic equationy²= 4x³−a2x−a3 to the equationy² = 4x³−m⁻⁴a2x− m⁻⁶a3, comes from the holomorphic isomorphism of complex tori z+Λ → mz+Λwherea2=g2(Λ),a3=g3(Λ), andΛ=mΛ. This makes the map of elliptic curves an isomorphism as well.

Thus complex tori (Riemann surfaces, complex analytic objects) and el- liptic curves (solution sets of cubic polynomials, algebraic objects) are inter- changeable. With the connection between them in hand, let the termcomplex elliptic curvebe a synonym for “complex torus” and call meromorphic func- tions with periodsΛelliptic functions with respect toΛ.

The lattice Eisenstein series G_k(Λ) will appear occasionally from now on but generally we will work withG_k(τ).

Exercises

1.4.1.LetE =C/Λbe a complex elliptic curve whereΛ=ω1Z⊕ω2Z, and letf be a nonconstant elliptic function with respect to Λ, viewed either as a meromorphic function onC with periods Λ or as a meromorphic function onE. LetP ={x1ω1+x2ω2:x1, x2∈[0,1]}be the parallelogram representing E when its opposing boundary edges are suitably identified, and let ∂P be the counterclockwise boundary ofP. Sincef has only finitely many zeros and poles inE, some translationt+∂P misses them all. This exercise establishes some necessary properties of f. Showing that these properties are sufficient for an appropriatef to exist requires more work, see for example [JS87].

(a) Compute that 1/(2πi)

t+∂Pf(z)dz = 0. It follows by the Residue Theorem that the sum of the residues of f onE is 0. In particular there is no meromorphic function onE with one simple pole and so the Weierstrass

℘-function, with its double pole at Λ, is the simplest nonconstant elliptic function with respect toΛ.

(b) Compute that 1/(2πi)

t+∂Pf(z)dz/f(z) = 0. It follows by the Ar- gument Principle that f has as many zeros as poles, counting multiplicity.

Replacing f by f −w for any w ∈ C shows that f takes every value the same number of times, counting multiplicity. In particular the Weierstrass

℘-function onE takes every value twice.

(c) Compute that 1/(2πi)

t+∂Pzf(z)dz/f(z)∈Λ. Show that this integral is also

x∈Eνx(f)xwhere νx(f) is the order off atx, meaning thatf(z) = (z−x)^νx(f)g(z) withg(x)= 0. Note thatνx(f) = 0 except at zeros and poles off, so the sum is finite. Thus parts (b) and (c) combine to show that for any nonconstant meromorphic functionf onE,

x∈E

ν_x(f) = 0 inZ and

x∈E

ν_x(f)x= 0 inE.

(A hint for this exercise is at the end of the book.)

1.4.2.LetΛ=ω1Z⊕ω2Zbe a lattice and let℘be its associated Weierstrass

℘-function.

(a) Show that℘is even and that℘ isΛ-periodic.

(b) Fori= 1,2 show that the function℘(z+ω_i)−℘(z) is some constantc_i by taking its derivative. Substitutez=−ω_i/2 to show thatc_i= 0. Conclude that℘isΛ-periodic.

1.4.3.LetΛ=Λ_i. The derivative℘of the corresponding Weierstrass function has a triple pole at 0 and simple zeros at 1/2,i/2, (1 +i)/2. SinceiΛ=Λ it follows that

℘(iz) =℘_iΛ(iz) =i⁻²℘(z) =−℘(z).

Show that in particular ℘((1 +i)/2) = 0. This is a double zero of ℘ since also ℘((1 +i)/2) = 0, making it the only zero of℘ as a function on C/Λ.

SinceΛ=Λ (complex conjugation) it also follows that℘(z) =℘(z), so that

℘(1/2) is real, as is ℘(i/2) = −℘(1/2). Compute some dominant terms of

℘(1/2) and ℘(i/2) to show that ℘(1/2) is the positive value. For what m does the complex torusC/mΛcorrespond to the elliptic curve with equation y²= 4x(x−1)(x+ 1)?

Reason similarly with Λ = Λµ3 to find the zeros of the corresponding Weierstrass function℘and to show that℘(1/2) is real. For whatmdoes the complex torus C/mΛ correspond to the elliptic curve with equation y² = 4(x−1)(x−µ3)(x−µ²₃)? (A hint for this exercise is at the end of the book.) 1.4.4.Forτ ∈ Hletp_τ(x) = 4x³−g₂(τ)x−g₃(τ). Show that the discriminant of p_τ equals ∆(τ) up to constant multiple, where ∆ is the cusp form from Section 1.1.

1.4.5.Show that when a₂ = 0 in Proposition 1.4.3 the desired lattice isΛ= mΛ_µ3 for a suitably chosen m. Prove the case a₃ = 0 in Proposition 1.4.3 similarly.