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 integral In order to understand complex tori, we review the history of elliptic functions. x In general, a function 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 rational function 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 ellipse. 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 integrals can also be obtained from physics: from motion of a simple pendulum. The above examples suggest to study integral of the form x dt a P (t) Z where P (t) is a polynomial 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 <k<p 1 and n is an arbitrary constant. p Abel and Jacobi’s idea Elliptic really does have a connection with literal geometric ellipses, removed by a few steps. Integrals expressing arc length of ellipses are less elementary than integrals for inverse trigonometric functions, and were named elliptic because of this. Function Its inverse function Addition formula 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 Riemann surface, 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.