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13 Elliptic Function 13 Elliptic Function Recall that a function g : C → Cˆ is an elliptic function if it is meromorphic and there exists a lattice L = {mω1 + nω2 | m, n ∈ Z} such that g(z + ω)= g(z) for all z ∈ C and all ω ∈ L where ω1,ω2 are complex numbers that are R-linearly independent. We have shown that an elliptic function cannot be holomorphic, the number of its poles are finite and the sum of their residues is zero. Lemma 13.1 A non-constant elliptic function f always has the same number of zeros mod- ulo its associated lattice L as it does poles, counting multiplicites of zeros and orders of poles. Proof: Consider the function f ′/f, which is also an elliptic function with associated lattice L. We will evaluate the sum of the residues of this function in two different ways as above. Then this sum is zero, and by the argument principle from complex analysis 31, it is precisely the number of zeros of f counting multiplicities minus the number of poles of f counting orders. Corollary 13.2 A non-constant elliptic function f always takes on every value in Cˆ the same number of times modulo L, counting multiplicities. Proof: Given a complex value b, consider the function f −b. This function is also an elliptic function, and one with the same poles as f. By Theorem above, it therefore has the same number of zeros as f. Thus, f must take on the value b as many times as it does 0. If f − b in the previous proof has a multiple root, then b is called a ramification point of f. The order of an elliptic function is the number of poles counting orders, modulo its lattice. By Lemma 12.1, Lemma 16.2 and 16.3, the order is the number of zeros counting multi- plicities, and also the number of times any other value is taken on, accounting for ramification points. An elliptic function has order 0 if and only if it is a constant function, according to Lemma 12.1. Next, there are no elliptic functions of order 1, since that would entail having a residue that is both zero because of Lemma 12.2, and non-zero because it’s at a pole of order one. 31See Theorem 36.1, in my notes last semester. 80 Some idea to construct periodical meromorphic functions Weierstrass P function is the simplest non-trivial elliptic function. To illustrate the idea how to define this function, we consider some examples on periodic functions first. [Example] Consider ∞ 1 f (z) := 1 (z − n)2 nX= −∞ which is meromorphic on C, with poles in Z, and satisfies f1(z +1) = f1(z) for any z ∈ C. We claim: ∞ 1 π2 = . (45) (z − n)2 sin2(πz) nX= −∞ π2 In fact, note that both sides have the same poles, and their difference g1(z) := f1(z)− sin2(πz) is an entire function, satisfying g1(z +1) = g1(z) for any z ∈ C. Also it is seen that, for z = x + iy with x in a fixed small closed interval, both sides of (45) tend to 0 as |y|→∞. By Liouville’s theorem, this forces g1 ≡ 0. [Example] Consider 1 1 1 f (z) := + + 2 z z − n n Xn=0 6 which is meromorphic on C, with poles in Z, and satisfies f2(z +1) = f2(z) for any z ∈ C. We claim: 1 1 1 + + = π cot(πz). (46) z z − n n Xn=0 6 1 1 In fact, note that both sides have the same poles, and their difference g2(z) := z + z n + n=0 − 6 1 C P n − π cot(πz) is an entire function, satisfying g2(z +1) = g2(z) for any z ∈ . Applying d −dz on the both sides of (46), we obtain the identity (45). Then g2 ≡ constant. 1 2z Looking at the last term in f2(z) = z + n∞=1 z2 n2 , we see that the left side of (46) is − odd in z; so is the right side; hence g2 ≡ 0. P Weierstrass ℘ function Let L denote the lattice Zω1 + Zω2 of the periods generated two two R-linearly independent complex numbers ω1 and ω2 over the integers. We define the Weierstrass ℘-function for the lattice by 1 1 1 ℘(w)= + − . w2 (w − ℓ)2 ℓ2 ℓXL 0 ∈ − 81 1 This series is motivated by using a double pole w2 at w = 0 an then use the nonzero periods 1 1 ℓ ∈ L to translate w2 to (w ℓ)2 . Each point in L is a pole of the meromorphic function ℘ with order 2. − By the way, to recognize Weierstrass’ contribition, in latex, we use “\wp” to denote ℘. We show several properties of this function ℘ as follows. 1 • We claim that ℘(z) is convergent and hence meromorphic . Subtraction of (w ℓ)2 by 1 − ℓ2 is used to guarantee the convergence of the series. In fact, we note that 1 1 |w||w − ℓ| C − ≤ ≤ (w − ℓ)2 ℓ2 |ℓ|2|w − ℓ|2 |ℓ|3 for some constant provided |ℓ| > 2|w| so that the series defining ℘ converges absolutely and uniformly on compact subsets of C−L. This proves that ℘ has isolated singularities at points of L by consideration of Laurent series. • The function ℘ is an even function : ℘(−z)= ℘(z), ∀z ∈ C, which follows from its definition. Also ℘′(z) is an odd function . • We claim the double periodical property: ℘(w + ℓ)= ℘(w), ∀ℓ ∈ L, ∀w ∈ C. In fact, note that −2 ℘′(w)= (47) (w − ℓ)3 Xℓ L ∈ is periodic relative to the lattice L. Then for any ℓ ∈ L, we have −2 ℘′(w + ℓ)= = ℘′(w) (w + ℓ − k)3 Xk L ∈ which implies ℘′(w + ℓ) − ℘′(w)= 0 for any w so that ℘(w + ℓ) − ℘(w)= C(ℓ), ∀w ∈ C 82 1 for some constant C(ℓ). We want to prove C(ℓ) = 0. Taking w = − 2 ℓ, we find ℓ ℓ ℘(− + ℓ) − ℘(− )= C(ℓ), 2 2 i.e., ℓ ℓ ℘( ) − ℘(− )= C(ℓ). 2 2 Since ℘(w) is an even function, we conclude that the constant C(ℓ) is zero. • Since ℘(w + ω1)= ℘(w), we take differentiation to get ℘′(w + ω1)= ℘′(w), and hence ω1 ω1 ω1 ω1 ℘′(− 2 + ω1) = ℘′(− 2 ), i.e., ℘′( 2 ) = ℘′(− 2 ). Since ℘′(w) is odd function by (47), ω1 ω2 ω1+ω2 it implies 2 is a zero of ℘′(w) . Similarly, 2 and 2 are zeros of ℘′(w) . Since ℘′(w) has a single pole of order 3 in a period parallelogram, these must be all the zeros. While it is quite easy to determine zeros of ℘′, it is very difficult to determine the zeros of ℘ (see the appendix below). Zeros of the Weierstrass elliptic functions Let us assume that L is a lattice Z + τZ where τ ∈ H. Since ℘ assumes every value in C∪{∞} exactly twice in C/(Z+τZ), it follows that ℘ has two zeros there which, ℘ being even, can be written in the form ±z0. Almost a century after Weierstrass’ lectures on elliptic functions were published, Eichler and Zagier 32 found the first explicit formula for z0 in terms of some integrals. In 2008 W. Duke and O.¨ Imamoglu found another formula 33: 1 4 1 2 3 5 1+ τ c2x F ( 3 , 3 , 1; 4 , 4 | x) z0 = + 1 5 2 F ( 12 , 12 ;1 | 1 − x) i√6 where c2 = − 3π , F is generalized hypergeometric series defined for |x| < 1 by n ∞ (a1)n...(am)n x F (x)= F (a1, ..., am; b1, ..., bm 1 | x)= · − (b1)n...(bm 1)n n! Xn=0 − Γ(a+n) 1728 where (a)n = Γ(a) and no (bk)n = 0, and x =1 − j and 1 j = + 744 + 196884q + ... q 32M. Eichler and D. Zagier, On the zeros of the Weierstrass ℘-function, Math Ann. 258(1981/82), 399-407. 33W. Duke and O.¨ Imamoglu, The zeros of the Weierstrass ℘ - function and hypergeometric series, Kath Ann, (2008), 897-905. 83 where q = e2πiτ . Weierstrass’ fundamental work At around 1860, Weierstrass, impressed by Abel’s and Jacob’s work, began his own investigations into the theory of elliptic functions. In his theory, the fundamental elliptic function was z = ℘(w), defined as the inverse function of z dt w = F (z)= , 3 Zz0 4t − g2t − g3 p 3 2 where g2 and g3 are constants such that g2 − 27g3 =6 0. He also showed that every elliptic function could be expressed in terms of z = ℘(w) and its derivative ℘′(w). According to Weierstrass 34 , every function with an algebraic addition theorem is an elliptic function or a limiting case of one (i.e. rational, trigonometric, and exponential functions). Thus, with Weierstrass’s contributions, the intimate connection between algebraic addition theorems and elliptic functions was established. Lemma 13.3 Let ℘ be the Weierstrass elliptic function associated the lattice L = Zω1 +Zω2 as in last section. Then ℘ satisfies the following differential equation: 2 3 ℘′(w) =4 ℘(w) − g2℘(w) − g3, ∀w ∈ C with constants gj = gj(ω1,ω2), j =1, 2. Proof: Let us consider the Laurent series near 0 as follows. By the geometrical series, 1 1 1 w w 2 w 3 w w 2 w 3 2 = 2 w 2 = 2 1+ + + +...... 1+ + + +...... (w − ℓ) ℓ (1 − ℓ ) ℓ ℓ ℓ ℓ ℓ ℓ ℓ 1 w w2 = +2 +3 + ..., for |w| < |ℓ|. ℓ2 ℓ3 ℓ4 1 1 1 Recall ℘(w)= w2 + ℓ L 0 (w ℓ)2 − ℓ2 . It follows that ∈ − P − 2 1 w w 2 2 4 ℘(w)= + 2 +3 + ...... = w− +3s w +5s w + ..., w2 ℓ3 ℓ4 4 6 ℓXL 0 ∈ − 34H.
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