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Home , Elliptic function, Modular form

The j invariant

Sacha Schweizer

November 22 and 29, 2006

Introduction for the L = Lτ = τZ + Z. This more general expression takes any lattice We pursue the discussion of modular func- ω tions, studying their relationship with ho- L = ω + ω , ω , ω ∈ , 1 ∈/ 1Z 2Z 1 2 C ω R mogeneous lattice functions and the Weier- 2 strass ℘ function, which gives us the oc- as argument and defines a lattice function, casion to define elliptic functions and in- which turns out to be homogeneous of de- troduce some basic results. We explain gree −k: for any λ ∈ C, let λL denote the the importance of the - expansion by λ of L, λω1Z + λω2C, then two of them, in particular - in characteriz- ing ℘. We then introduce a special mod- X 1 Gk(λL) = ular function, the j invariant, in terms ωk ω∈λLr{0} of which any other modular function of weight 0 can be expressed explicitly. j is X 1 = shown to give a bijection between the fun- (λω)k ω∈Lr{0} damental domain and C mapping infinity −k to infinity, which allows us to prove an ex- = λ Gk(L) istence theorem for lattices related to el- a b Note that for τ ∈ and γ = ∈ liptic curves, and Picard’s Little Theorem H c d about the behavior of entire functions. Γ1, Relationship between homogeneous Lτ = γLτ = (aτ + b)Z + (cτ + d)Z lattice functions and modular func- tions For any homogeneous lattice function f of degree −k, Recall The Eisenstein series of order k is a of weight k, defined as f(Lτ ) = f(γLτ )

X 1 = (cτ + d)−kf(L ) G (τ) = γτ k (mτ + n)k (m,n)∈Z×Zr{(0,0)} In other words,

k It can be rewritten as f(Lγτ ) = (cτ + d) f(Lτ ) X 1 1As usual, Γ denotes the full modular G (L) = k ωk SL2(Z). ω∈Lr{0}

1 so f(τ) = f(Lτ ) is modular of weight k. Theorem and Definition The num- ber of zeroes and poles (counted with mul- We have other modular forms tiplicity) of a nonzero f in a period parallelogram coincide; it is g2(τ) = 60G4(τ) called the order of f, and it is always at least 2. g3(τ) = 140G6(τ) Proof Zeroes and poles of a nonzero mero- 3 2 ∆(τ) = g2(τ) − 27g3(τ) morphic function are discrete, hence lo- cally finite. Integrate the logarithmic deriv- of weight 4, 6, and 12, respectively. ative of f along the boundary of a pe- riod parallelogram so chosen that no ze- Elliptic Functions2 roes or poles lie on its boundary (trans- late if needed). By periodicity, opposite Definition A function f : C → C such sides cancel and the integral is 0. The that first part of the claim then follows by the Residue Theorem. The second part fol- f(z + ω) = f(z), ∀z ∈ C, ∀ω ∈ L lows by the Residue Theorem applied to f instead, and considering our previous for some lattice L is called doubly periodic. theorem. A doubly periodic  is called elliptic. An example of elliptic function Elliptic functions form a subfield of the field of meromorphic functions. Definition The Weierstrass ℘ function for a lattice L is defined on as Note that by periodicity, such a function C is fully determined on a period parallelo- 1 X  1 1  ℘(z) = 2 + 2 − 2 gram such as [0, 1)ω1 + [0, 1)ω2 z (z − ω) ω ∼ ω∈Lr{0} Furthermore, L = Z⊕Z as a group under addition acting on C by translation, hence ∼ 2 Theorem ℘ is an elliptic function for C/L = T , and f is actually a function on a . L of order 2, with double poles at lattice points (without proof3). Theorem An elliptic function without poles (or zeroes) is constant. Theorem ℘ possesses the following Lau- Proof If f has no poles, it is entire, and rent expansion around the origin: continuous on the closure of any period ∞ 1 X parallelogram. This is compact, hence f ℘(z) = + (2n + 1)G z2n z2 2n+2 is bounded there. By periodicity, f is n=1 (globally) bounded, and by Liouville’s the- for kzk < min{kωk : ω ∈ L}. orem, it must be constant. If f has no zeroes, consider 1/f.  Proof The proof is based on the Tay- 1 lor expansion of (z−ω)2 and can be found 2This portion has been added to this talk be- in < 1 >, 3. cause, for reasons of time, it had been left out  from the talk < 1 >. 3For a proof, see [1], 1.6

2 Theorem ℘ satisfies the following dif- In particular, any can be ferential equation: brought into Weierstrass form.

0 2 3 [℘ (z)] = 4℘ (z) − g2(L)℘(z) − g3(L). The j invariant

Definition The j invariant7 is defined Proof Knowledge of the above Laurent as expansion of ℘ around the origin allows g3(τ) j(τ) = 1728 2 , τ ∈ . one to conclude that the difference be- ∆(τ) H tween left-hand and right-hand side of the differential equation is a Taylor series with- out constant term. By ellipticity, it must Note j is the ratio of two modular forms be constant, hence zero. For the compu- of weight 12, hence it is a modular func- 4 tational details, see [1], 1.9.  tion of weight 0. Since ∆ has a simple zero at infinity but vanishes nowhere else, Remarks 1) This differential equa- j has a simple pole at infinity and is holo- tion leads us to ask ourselves whether ℘ is morphic on H. fully determined by g2 and g3 alone. This is indeed so, since the Eisenstein series are Let Γd\H denote the closure of the modu- 5 expressible as polynomials in g2 and g3. lar with Γ-equivalent In turn, ℘ suffices for representing all el- sides identified and the point at infinity 6 liptic functions . added. Topologically, Γd\H can be given 2) ∆(L) is the discriminant of the cubic the structure of the one-point compactifi- polynomial in Weierstrass form P (z) = cation of the fundamental domain, which 3 4z − g2(L)z − g3(L); the discriminant is is homeomorphic to the 2-sphere. a constant multiple of

2 Theorem j defines a bijection [(z1 − z2)(z2 − z3)(z1 − z3)] Γd\H −→ C ∪ {∞}. where z1, z2, z3 are the roots of P , so Here the codomain is nothing but the Rie- ∼ 1 ∆(L) 6= 0 ⇐⇒ P has distinct roots mann sphere, C ∪ {∞} = CP .

Any nondegenerate cubic polynomial can Proof As seen above, be transformed by affine change of vari- j(i∞) = ∞ able to the form 4z3 − Az − B. We will see later on that in case the polynomial It remains to show that on the fundamen- has distinct roots, there exists a lattice L tal domain, j attains every complex value such that exactly once. In other words, for each λ ∈ C there exists exactly one z ∈ Γ\H A = g2(L),B = g3(L) such that 4The in square brackets correspond 3 g2(z) − λ∆(z) = 0. to the references given in the outline of the sem- inar, see General Information 7Serre [7] calls it the modular invariant. Apos- 5 g2 and g3 are constant multiples of E4 and tol [1] writes J instead, defines it first in terms of E6, respectively, which are shown to generate all a lattice, leaves out the factor 1728 and calls it modular forms in < 2 >, 4. Klein’s modular function. We adopt the notation 6See [5], §I.5. and definition of Koblitz’s [5].

3 For each λ, the left-hand side of the lat- (i) f is a modular function of weight 0 ter equation defines a modular form fλ of (ii) f is a quotient of modular forms of weight 12. By the valence formula8 with the same weight “k = 12”, we obtain9 (iii) f is a rational function of j. 1 1 v (f ) + v (f ) + v (f ) Proof The implication (ii)⇒(i) is obvi- ∞ λ 2 i λ 3 ω λ ous. X + vp(fλ) = 1. To prove (iii)⇒(ii), suppose p∈(Γ\H)r{i,ω} P (j(z)) f(z) = Because g2(i∞) is a nonzero complex num- Q(j(z)) ber while ∆(i∞) = 0, the term v∞(fλ) vanishes. As a modular form, fλ has no for complex polynomials P and Q (not poles, the remaining terms in the valence necessarily of the same degree), Q 6= 0. formula are thus all nonnegative, so we P(j(z)) and Q(j(z)) are both modular func- get a decomposition tions of weight zero holomorphic on H. They have at most a pole at i∞, which 1 1 a + b + c = 1, a, b, c ∈ N we can kill by multiplication with the 2 3 form ∆, so

It is easy to check by substitution of the k k few possible candidates that the solutions P (j(z))∆ (z),Q(j(z))∆ (z) are are both holomorphic on H ∪ {ı∞} for (a, b, c) ∈ {(2, 0, 0), (0, 3, 0), (0, 0, 1)} some k ∈ N, and modular of weight 12k. So P (j(z))∆k(z) In all three cases, fλ vanishes at exactly f(z) = k one point in the fundamental domain.  Q(j(z))∆ (z) is a ratio of modular forms of the same Remark Γd\H can be given a complex weight. manifold structure arising in a natural way To prove (i)⇒(iii), suppose f is modular from the compactification of the quotient of weight 0. Set by the action of the on the Y −vp(f) complex manifold H. Then the above bi- g(z) = (j(z) − j(p)) f(z) jection is actually an of com- p∈Γ\H:vp(f)<0 plex manifolds, which would not be the in order to kill the poles of f on . g case if we started with an arbitrary home- H is meromorphic at infinity and modular omorphism (such would be fairly easy to of weight 0. In particular, there is some construct). k ∈ such that g(z)∆k(z) is a modular The next proposition shows how any mod- N form of weight 12k, which can therefore ular function of weight 0 can be expressed be written as a linear combination of the in terms of j. form10

Theorem Let f be a meromorphic func- k X n m g(z)∆ (z) = cnmG4 (z)G6 (z) tion on H. Then the following are equiv- 4n+6m=12k alent: 10 The normalized Eisenstein series E∗ consid- 8See < 2 >, 3. √ ered in < 2 > differ only by a numerical factor 9 1 3i As in the previous talks, ω = − 2 + 2 . from the G∗’s here.

4 for complex coefficients cnm. Note that takes on every value including infinity ex- 1 cept on H, since j takes on every com- (60G (z))3 27 j(z) = 1728 4 plex value there, as seen above. On the ∆(z) other hand, the condition A3 6= 27B2 trans- lates into G3(z) B2 1 = 1728 4 , 3 49 2 3 6= . G4(z) − 20 G6(z) A 27 3 2 G4 G6 There exists thus some z ∈ H such that so both ∆ and ∆ have the form aj + b. Furthermore, the above linear combina- g2(z) B2 3 = . tion runs over indices m, n that satisfy 3 3 g2(z) A

12|4n + 6m, Choose λ ∈ C r {0} such that

4 in particular g2(z) = λ A,

3|n, 2|m, then

2 hence 2 B 3 12 2 g3(z) = 3 g2(z) = λ B , 3 2 A X G4(z) n G6(z) m g(z) = cnm( ) 3 ( ) 2 , or ∆(z) ∆(z) 6 4n+6m=12k g3(z) = ±λ A, and g is actually a polynomial in j. Re- In the case of a “−”, replace λ by iλ, so versing our definition of g we get an ex- without loss of generality we may assume pression for f as a ratio of polynomials in the case “+”to hold. The result follows j.  by the homogeneity of g2 and g3 demon- strated at the beginning of the talk.  Remark (contd.) With the appropriate complex manifold structure on Γd\H, this Picard’s Little Theorem theorem shows that the meromorphic func- tions on the are exactly Another application of the j invariant con- the rational functions. sists in the proof of the following well- known theorem from , which We now come to the promised result with however stands quite apart from the top- applications in the theory of elliptic func- ics of this seminar. tions. Theorem Every nonconstant entire func- Theorem For each pair of complex num- tion attains every complex value with at bers A and B such that A3 6= 27B2 there most one exception. exists a lattice L = λLz such that Note Nonconstant polynomials take on A = g2(L),B = g3(L). every complex value (by the Fundamen- tal Theorem of Algebra). The exponen- Proof Note that tial function omits only the value 0. 2 g3 1 1728 3 = (1 − ) g2 27 j

5 Proof (Sketch11) Without loss of gener- ality, let f be an entire function that omits at least the values 0 and 1728 (which turn out to be j(ω) and j(i), respectively). Since H covers the fundamental domain by pro- jection to the quotient by Γ, j maps H to a with branch points at 0, 1728 and ∞ (imagine“folding” H elasti- cally onto itself over and over so as to get Γ-translates of the fundamental domain to lie on top of one another: the “problem points” will be at the “vertices”of each translate). This has a multivalued “in- verse“(similar to the complex logarithm) that can be composed with f, since by assumption f omits the values 0, 1728 and ∞. By the monodromy theorem, the composition has a single-valued analytical continuation on C, but it only takes val- ues in H, so it is constant (the latter step is more elementary, requiring only Liou- ville’s theorem, and is often given as an exercise in complex analysis books), hence so is f by application of j.  References

< 1 > Salvatore Bonaccorso: Modular forms, Eisenstein series and a short in- troduction to elliptic functions.

< 2 > Jonas Jermann: The valence for- mula and the space of modular forms.

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Ver. 1.0

11A complete proof would require knowledge beyond an introductory complex analysis course. A more detailed proof, though still somewhat sketchy, than the one given here is laid out in [1], 2.7.

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