12 Complex Tori II

Matrix representation of the linear map F We knew that a map f : T = C/Γ 1 → T = C/Γ′ if and only if There exists F (z)= az with a = 0 such that F (Γ) = Γ′. 2 6 2 2 Let Γ = g(z) = z + m ω , m Z , Γ′ = g(z)= z + m′ ω′ , m′ Z . For { j=1 j j j ∈ } { j=1 j j j ∈ } F (z) = az, since it maps the equivalent classes w.r.t Γ1 to equivalent classes w.r.t Γ2, we have P P

F (ω )= aω = a ω′ + a ω′ , F (ω )= aω = a ω′ + a ω′ , with a Z (37) 1 1 11 1 12 2 2 2 21 1 22 2 ij ∈ or ω a a ω ω F 1 = 11 12 1′ = A 1′ , ω a a ω′ ω′  2   21 22   2   2  a a where A = 11 12 GL(2, Z), i.e., the group of 2 2 non-singular matrices with a21 a22 ∈ × integer entries.  1 Consider F − we obtain

1 ω1′ b11 b12 ω1 ω1 F − = = B , ω′ b b ω ω  2   21 22   2   2  b b where B = 11 12 GL(2, Z). b21 b22 ∈   Then

ω1 1 ω1 1 ω1′ 1 a11ω1′ + a12ω2′ = F − F = F − A = F − ω2 ◦ ω2 ◦ ω′ a21ω′ + a22ω′      2   1 2  1 1 a11F − (ω1′ )+ a12F − (ω2′ ) 1 ω1′ ω1 = 1 1 = AF − = AB . a F − (ω′ )+ a F − (ω′ ) ω′ ω  21 1 22 2   2   2  Hence AB = Id so that det(B) det(A) = 1. Since A and B GL(2, Z), it concludes det(A)= 1 so that A SL(2, Z).· ∈ ± ∈ Functions defined by In order to understand complex tori, we review the history of elliptic functions. x In general, a defined by integral f(x)= a h(t)dt is more complicated than the function h(t). For examples, R x x 1 1 1 dt = log x and dt = sin− x. 2 1 t 0 √1 t Z Z − 71 Let us look back some history.24 In 1702, John Bernoulli conjectured that the integral of any is expressible in terms of other rational functions, trigonometric functions, and logarithmic functions. It turns out that this statement is true. 25 x2 y2 There are more interesting and important functions defined by integral. Let a2 + b2 =1 be the equation of a general . While the area of an ellipse is easy to calculated, the length of an ellipse is very difficult to compute. In fact, its arc-length is given by

x dy 2 x a4 (a2 b2)x2 1+ dx = −2 2 − 2 dx 0 s dx 0 a (a x ) Z   Z s − x dx a2 b2 x x2dx = a3 − . 4 2 2 2 2 2 4 2 2 2 2 2 0 (a (a b )x )(a x ) − a 0 (a (a b )x )(a x ) Z − − − Z − − − 2 p 2 p 4 2 2 2 2x 2y dy 2 x 2 dy b x b x Here we used a2 + b2 dx = 0 and y = 1 a2 b = so that dx = a4y2 = a2(a2 x2) . We − − call such integral elliptic integral.     Elliptic can also be obtained from physics: from motion of a simple . The above examples suggest to study integral of the form x dt

a P (t) Z where P (t) is a of degree 4. Or morep generally, the integrals of the form x R(t, P (t))dx. a Z p where R(t, s) is a rational function of t and s and P (t) is a polynomial of degree 3 or 4. That such integrals cannot be evaluated in terms of the elementary functions was finally proved by Liouville in 1833. 26

Earlier works up to Euler In 1718, Fagnano made a discovery related to the rectification of the arc of the lemniscate.27 He considered, in particular, the lemniscate whose rectangular

24Jose Barrios, A brief history of elliptic integral addition theorems, www.rose- hulman.edu/mathjournal/archives/2009/vol10-n2/paper2/v10n2-2pd.pdf 25Goursat, Edouard and Hedrick, E.R., A course in Mathematical Analysis, vol. 1, Ginn and Company, Boston, 1904, p.210. 26J. Liouville, Memoire sur les transcendantes ellipques de premiere et de seconde espece considerees comme fonctions de leur amplitude, Journal de l’Ecole Polytechnique 23, 37-83, 1833. 27C.L. Siegel, Topics in Complex Function Theory, voll. 1, Wiley-Interscience: New York, 1969, p.1-10.

72 equation is (x2 + y2)2 =(x2 y2). The graph has the shape of the infinity symbol “ ” and intersects the x-axis at x =0−, 1. The length of the arc in the first quadrant beginning∞ at ± the origin and terminating at the point a distance r from the origin is r dx s(r)= 4 0 √1 x Z − where r [0, 1]. While the lemniscatic integral itself remained unevaluated, Fagnano did discover a∈ formula which doubles the arc of the lemniscate: u dy r dx 2 = 4 4 0 1 y 0 √1 x Z − Z − where p 2u√1 u4 r = T (u)= − . 1+ u4 Expanding on Fagnano’s work, Euler obtained the addition theorem for lemniscatic integrals. u dx v dx T (u,v) dx + = , 4 4 4 0 √1 x 0 √1 x 0 √1 x Z − Z − Z − where u√1 v4 + v√1 u4 T (u, v)= − − . 1+ u2v2 Euler next obtained the addition theorem for elliptic integrals of the first kind, which were the chief result in the theory of elliptic integrals at the time: u dx v dx T (u,v) dx + = , 0 P (x) 0 P (x) 0 P (x) Z Z Z where P (x)=(1 x2)(1 pk2x2), and p p − − u P (v)+ v P (u) T (u, v)= . 1 k2u2v2 p − p Legendre’s reduction The next mathematician who investigated elliptic integrals was Legendre. In his Traite des fonctions elliptiques 28, he showed that integration of the elliptic integral R(t) dt P (t) Z 28A.M. Legendre, Traite des fonctions elliptiquesp et des integrals Eulerienne, 3 vols., Huzard-Courier: Paris, 1825, 1826, 1828.

73 where R(t) is a rational function of t and P (t) is a polynomial of fourth degree, can be reduced to the integration of the three integrals dx x2dx dx , , √1 x2√1 l2x2 √1 x2√1 l2x2 (x a)√1 l2x2 Z − − Z − − Z − − which he called the elliptic integrals of the first, second, and third kinds, respectively. By applying substitutions, he reduced these integrals to the following three integrals : φ dφ φ φ dφ , 1 k2sin2φdφ, 2 2 2 2 2 0 1 k sin φ 0 − 0 (1 + nsin φ) 1 k sin φ Z − Z p Z − where 0

x dt y log x = 1 t e log(x1 + x2)= log x1 + log x2 R sin 1x = x dt sin y sin(y + y )= sin y cos y + cos y sin y − 0 √1 t2 1 2 1 2 1 2 − R F (z)= z dt ? g(w) ?? z0 √P (t) R where P (t) is a polynomial of degree 4. ≤ Notice that the function ez has periodic property: ez+2iπn = ez for any integer n. In fact, the complex valued function defined by z 1 dw w Z1 has multiple valued log z +2iπn, depending on the curve γ from 1 to z in C 0 . −{ } Namely, from a local commutative diagram ez C C∗ −→ (38) ||log z || C C∗ ←− 74 where C∗ = C 0 , we can obtain −{ } ez C/ X ∼ −→ (39) ||log z || C/ X ∼ ←− where X is the maximum domain of definition of log z, a , and C/ is the ∼ quotient space defined by the equivalent relation: z1 z2 z1 = z2 +2iπm for some integer m. ∼ ⇐⇒ Abel and Jacobi discovered in 1827 that the inverse functions are doubly periodic. In fact, we’ll see that the maximum domain X of definition of the elliptic integral x 1 is a torus . z0 √P (t) x dt Therefore, by choosing curves from z0 to z in X, we find the function R has multiple z0 √P (t) values: x R dt dt dt dt = + m + n z0 P (t) γ0 P (t) γ1 P (t) γ2 P (t) Z Z Z Z for integers m, n wherepγ0 is a fixedp curve from z0 topz in X, and γp1 and γ2 are two “base” curves from z0 to z in the torus X. To visualize the problem clearly, we may assume a = a = a = a . Then the elliptic integral above becomes 1 2 6 3 4 z dζ z 1 w = F (z)= = dζ. 2 2 z0 (ζ a ) (ζ a ) z0 (ζ a1)(ζ a3) Z − 1 − 3 Z − − It turns out that f(z) is a multi-valuedp function z dζ w = f(z) := + mω1 + nω2, m, n Z. (40) z0 P (ζ) ∀ ∈ Z R where ω1,ω2 are two complex numbersp and they are linearly independent. This chain of events caused these inverse functions to be called elliptic functions. Namely, from a local commutative diagram

g(w) C C a , a , a , a −→ −{ 1 2 3 4} (41) ||F (z) || C C a , a , a , a ←− −{ 1 2 3 4} we can obtain g(w) C/Λ X −→ (42) ||F (z) || C/Λ X ←− 75 where X is the maximum domain of definition of F (z), a Riemann surface, and C/Λ is the torus defined by the equivalent relation: z z z = z +2iπn +2iπm for some 1 ∼ 2 ⇐⇒ 1 2 integers n and m. We will explain how X and C/Λ are defined. Further, we’ll show that for a fixed periodicity lattice, the elliptic functions for that lattice satisfy algebraic relations. The basic relation is of the form (found by Weierstrass) y2 =4x3 ax b. − − Then such curves were called elliptic curves, because they relate the elliptic functions that invert the elliptic integrals that generalize the integral computing the arc length of an el- lipse. That story continues with abelian integrals, meaning more complicated multi-valued algebraic integrals, whose inverse functions are multiply periodic, and called abelian func- tions. These abelian functions satisfy algebraic relations which define abelian varieties as generalizations of elliptic curves. Jacobian varieties are a special case. These are so-named exactly because Abel and Jacobi studied them. Elliptic integrals and ellipitic functions Let us consider an indefinite integral of the form, called elliptic integrals, dx . P (x) Z where p4 P (x)= (x a ) − j j=1 Y is a polynomial of degree four. Let us assume that a1, a2, a3, a4 are distinct complex numbers. We claim that we can assume a , a , a , a are 1, 1, 1 , 1 for some k C . In fact, 1 2 3 4 − − k k ∈ ∪{∞} four distinct points a1, a2, a3, a4 in Cˆ can be mapped to four distinct points w1,w2,w3,w4 by a linear transformation if and only if the cross-ratio

(z1, z2, z3, z4)=(w1,w2,w3,w4). αt+β Also by linear transformation x = γt+δ , we have (αδ γβ)dt dx = − (γt + δ)2 and dx dt =(αδ γβ) (43) f(x) − q(t) Z Z p 76 p where q(t)= 4 [(αt + β) a (γt + δ)]. j=1 − j Fixing a point z C a , a , a , a , for any z C a , a , a , a , we consider a Q 0 ∈ −{ 1 2 3 4} ∈ −{ 1 2 3 4} holomorphic function z dζ f(z) := z0 P (ζ) Z C where the integral is along a curve C from a fixedp point z0 to z in a1, a2, a3, a4 . The value of f(z) clearly depends of choice of the curve C. −{ } If a meromorphic function defined on C satisfying

g(w)= g(w + mω + nω ), m, n Z, w C, (44) 1 2 ∀ ∈ ∀ ∈ it is called an elliptic functions. Just as the exponential function ez can be restricted on the quotient space C/ , we ∼ want to show that can be restricted on the quotient space C/L, a torus. More precisely, a subset L C is a lattice if there exist R-linearly independent ω and ω ⊂ 1 2 such that L can be written as mω1 + nω2 m, n Z . The torus C/L corresponding to the lattice L is formally the quotient{ of C by| the additive∈ } subgroup L. 29 Then we consider the equivalence relation: z z z z L to define a quotient space C/L so that g 1 ∼ 2 ↔ 1 − 2 ∈ induces a well defined function on C/L. A function f : C Cˆ is an elliptic function if it is meromorphic and there exists a lattice L such that f(z + mω→ + nω )= f(z) for all z C and all m, n Z. The domain of such a 1 2 ∈ ∈ function can also be considered C/L. The construction of the space X The function P (z) is not single-valued. In fact, except at the roots of P (z) the function P (z) has two distinct values, one being 1 times p the another. The idea of a Riemann surface is to replace a point of C by two points− (except p at the roots of P (z) where no replacement is made) to form a new set M to replace C so that the P (z) can be regarded a single-valued function defined on M. 4 To preciselyp describe the set M, we assume that P (z) = (z aj) with all four aj j=1 − distinct (1 j 4). We take a slit from a1 to a2 and another slit from a3 to a4 and take two identical≤ copies≤ of C with the two slits. Without loss of generality,Q we can assume that the four roots aj (j =1, 2, 3, 4) are all real and arranged in ascending order. We replace the

29The terminology “torus” comes from considering a parallelogram and identifying the two edges in each pair of opposite edges with one another. A common visualization aid is forming a piece of paper into a cylinder, and then connecting the two boundary circles together.

77 second copy C by its conjugation. Then for each slit, we identify (or “glue”) one side of the slit on the same side of another copy of C with another copy of C. Instead of considering C and its two copies we consider the C and two copies of it. We can do the same four identifications by using the two slits on∪ {∞} each of the two copies of the Riemann sphere C . Instead of getting the new set M we get ∪ {∞} from the four identifications of the two copies of the Riemann sphere C a compact topological space X. 30 ∪ {∞} As we open up the slits into circles we see that the final topological space can be described by a torus. Since X is a torus and X is the domain of definition for 1 , we see that (44) √P (z) 1 is true for f(z)= dζ where C is a curve in a torus X with some point z0 as initial C √P (ζ) point and z as terminalR point. We know that we can study f(x) = x dt = sin 1x by studying its inverse function 0 √1 t2 − − sin y, that we can study f(z) = z dζ = log z by studying its inverse function ew, and 1 ζ R that we can study elliptic integral F (z)= z dζ by studying its inverse function: elliptic R z0 √P (ζ) function g. Recall that a function g : C RCˆ is an elliptic function if it is meromorphic and there exists a lattice L = mω + nω →m, n Z such that g(z + ω) = g(z) for all z C { 1 2 | ∈ } ∈ and all ω L where ω1,ω2 are complex numbers that are R-linearly independent. If elliptic function exists,∈ it can be defined on the induced torus C/L as a meromorphic function. Basic properties for ellipitic functions Before we construct explicit elliptic function, we show that such function cannot be holomorphic on C.

Lemma 12.1 If an elliptic function has no poles, then it is constant.

Proof: By the classical Liouville theorem. 

Lemma 12.2 An elliptic function f always has finitely many poles modulo its associated lattice L, and the sum of their residues is zero.

Proof: For the first half, simply note that the intersection of a discrete set with a compact set must be finite. 30X is a Riemann Surface for the Square Root of a Quartic Polynomial. We may define this Riemann surface in another way: (z, w) w2 = P (z) C2, which is an in C2. { | }⊂

78 For the second half, consider the parallelogram Q = z + ω t + ω t t , t [0, 1) , { 0 1 1 2 2 | 1 2 ∈ } where z0 is chosen so that there are no poles of f on the boundary of Q. By evaluating

f(z)dz I∂Q where we travel along the boundary with respect to a choice of positive orientation. Here this integral is the sum of the residues of the poles inside Q due to the residue theorem. But as we travel in opposite directions on parallel edges of the boundary, the value of the function is identical on parallel edges, so their integrals along each pair of edges cancel, giving 0. 

79