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Theta Functions and Their Applications

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Theta Functions and Their Applications Theta functions and their applications Håvard Damm-Johnsen 2019 Contents 1 Certain Theta Functions 1 1.1 Introduction ............................. 1 1.2 The Jacobi theta functions ..................... 3 1.3 The transformation formula .................... 4 2 Applications to Number Theory 6 2.1 Jacobi’s four-square theorem .................... 6 2.2 Quadratic reciprocity ........................ 7 3 Epilogue 11 Bibliography 11 1 Certain Theta Functions 1.1 Introduction The classical theta function, ∞ ∞ 휋푖푛2휏 휋푖푛2휏 휃 (휏 ) .= ∑ 푒 = 1 + 2 ∑푒 푛=−∞ 푛=1 was introduced by Euler in a letter to Goldbach, as a tool in the study of repre- sentation of integers as sums of squares. Observe that ∞ ∞ ∞ 2 휋푖푚2휏 휋푖푛2휏 휋푖푘휏 휃 (휏 ) = ( ∑ 푒 )( ∑ 푒 ) = ∑푟2(푘)푒 , 푚=−∞ 푛=−∞ 푘=1 where 푟2(푘) is the number of ways in which 푘 can be written as a sum of two squares, since we get one contribution from each ordered pair (푚,푛) such that 2 2 푒휋푖푚 휏푒휋푖푛 휏 = 푒휋푖푘휏, or equivalently, 푘 = 푛2+푚2. This line of inquiry was taken up by Jacobi, who first proved the fundamental transformation formula for 휃, and the four-square theorem of section 2.1. Our point of departure will be slightly more general. Definition 1.1. For 푧 ∈ ℂ and 휏 ∈ H .= {휏 ∈ ℂ∶ Im휏 > 0}, we define the theta function to be ∞ 2 휃 (푧,휏 ) = ∑ 푒2휋푖푛푧+휋푖푛 휏. 푛=−∞ 1 Remark. Our notation follows Riemann and Mumford [Mum83], but there is no shortage of alternative conventions. We see that 휃 (휏 ) = 휃 (0,휏 ). The condition that 휏 ∈ H ensures that by the ratio test, 휃 (푧,휏 ) converges absolutely, and moreover uniformly on compact sets in the first variable, hence is an entire function in 푧. A fundamental property of the theta function is the following: Proposition 1.2. The theta function satisfies the transformation equations 휃 (푧 + 1,휏 ) = 휃 (푧,휏 ) and 휃 (푧 + 휏 ,휏 ) = 푒−2휋푖푧푒−휋푖휏휃 (푧,휏 ). Proof. The first is immediate from the definition, the second follows from the quick computation ∞ ∞ 2 2 휃 (푧 + 휏 ,휏 ) = ∑ 푒2휋푖푛푧+2휋푖푛휏 +휋푖푛 휏 = 푒−2휋푖푧푒−휋푖휏 ∑ 푒2휋푖(푛+1)푧+2휋푖(푛+1) 휏 푛=−∞ 푛=−∞ = 푒−2휋푖푧푒−휋푖휏휃 (푧,휏 ). Recall that by Liouville’s theorem, every entire doubly periodic function is constant. One might ask what happens if we weaken the periodicity condition slightly. The above proposition can be interpreted as saying that when viewed as a function of 푧, 휃 has period 1, and a quasi-period 휏 with periodicity factor 푒−2휋푖푧푒−휋휏. In other words, it is a quasi-periodic function with respect to the lattice 훬 .= ℤ+휏 ℤ. Conversely, considering quasi-periodic functions of this sort naturally gives rise to the theta function: Theorem 1.3 (Functional equation for 휃). Suppose 푓∶ ℂ → ℂ is entire and non- constant, 푓 (푧) = 푓 (푧 +1) and 푓 (푧 +휏 ) = 푒푎푧+푏푓 (푧) for some fixed 푎,푏 ∈ ℂ and any 푧 ∈ ℂ. If 푎 = 0, then 푓 (푧) = 푒2휋푖푧, and if 푎 = −2휋푖, then 1 푏 푓 (푧) = 푎 휃(−푧 − 휏 − ,휏), 0 2 2휋푖 for some 푎0 ∈ ℂ. Proof [Mum83]. Expanding 푓 in terms of its Fourier series, we have that ∞ 2휋푖푛푧 푓 (푧) = ∑ 푎푛푒 for some 푎푛 ∈ ℂ. 푛=−∞ Using the periodicity relations, we compute 푓 (푧 + 휏 + 1) in two ways, namely 푓 (푧 + 휏 + 1) = 푓 (푧 + 휏 ) = 푒푎푧+푏푓 (푧) and 푓 (푧 + 휏 + 1) = 푒푎(푧+1)+푏푓 (푧 + 1) = 푒푎(푧+1)+푏푓 (푧) = 푒푎푒푎푧+푏푓 (푧), 2 so 푎 = 2휋푖푘 for some 푘 ∈ ℤ. Substituting in the Fourier series of 푓, we find that ∞ ∞ 2휋푖푘푧+푏 2휋푖푘푧+푏+2휋푖푛푧 2휋푖푛푧+푏 푓 (푧 + 휏 ) = 푒 푓 (푧) = ∑ 푎푛푒 = ∑ 푎푛−푘푒 , 푛=−∞ 푛=−∞ 푏−2휋푛푖휏 which gives a recurrence relation for the Fourier coefficients, 푎푛 = 푎푛−푘푒 . 2휋푖푧 If 푘 = 0, then 푎푛 = 0 for all 푛 but possibly 푛 = 0, so 푓 (푧) = 푒 . Taking 푘 = −1, we readily see that −푛푏+휋푖푛(푛−1)휏 푎푛 = 푎0푒 for all 푛 ∈ ℤ, and so ∞ −푛푏+휋푖푛(푛−1)휏 +2휋푖푛푧 1 푏 푓 (푧) = 푎0 ∑ 푒 = 푎0휃(−푧 − 휏 − ,휏). 푛=−∞ 2 2휋푖 Suppose we take 푘 ≥ 1 in the proof above; then the Fourier coefficients grow rapidly, and so the corresponding function 푓 is not entire. On the other hand, for 푘 < −1, one can show that the solutions of the functional equations form a vector space of dimension |푘|. 1.2 The Jacobi theta functions Returning to the lattice 훬 = ℤ + 휏 ℤ, we define the following: Definition 1.4. Let 휏 ∈ H and 푧 ∈ ℂ. The Jacobi theta functions are . 휋푖휏 /4+휋푖푧 휏 휃00(푧,휏 ) .= 휃 (푧,휏 ), 휃10(푧,휏 ) .= 푒 휃 (푧 + /2,휏 ), . 1 . 휋푖휏 /4+휋푖(푧+1/2) (1+휏 ) 휃01(푧,휏 ) .= 휃 (푧 + /2,휏 ), 휃11(푧,휏 ) .= 푒 휃 (푧 + /2,휏 ). If 휏 ∈ H is fixed, we write 휃푎푏(푧) .= 휃푎푏(푧,휏 ). The value 휃푎푏(0) is traditionally called the theta-nullwerte of 휃푎푏(푧,휏 ). These functions were studied in great detail by Jacobi. Note that the Jacobi theta functions are simply 휃 shifted by the half-periods of 훬, modulo a certain scaling factor. This is explained in the more general theory of theta functions with characteristic, which is unfortunately beyond the scope of this essay. The Jacobi theta functions are also quasi-elliptic with respect to the same lattice 훬. An astonishing number of formulae involving these functions exist, but we will limit our attention to the following identity: Theorem 1.5 ( Jacobi’s identity). For 휏 ∈ H fixed, we have 4 4 4 휃00(0) = 휃01(0) + 휃10(0). It is easily seen that for any 푧 ∈ ℂ, 휃11(−푧) = −휃11(푧), so that 휃11(0) = 0, which explains why 휃11 is absent from the identity. The identity is particularly interest- ing from an algebro-geometric point of view, since it gives a parameterisation of the Fermat quartic 푋 4 + 푌 4 = 푍4 in terms of theta functions. 3 휋푖휏 Let 푞 .= 푒 . Using Jacobi’s triple product identity,1 one can prove that ∞ 1/4 2푛 2푛 4푛 휃11(푧,푞) = −2푞 sin휋푧 ∏(1 − 푞 )(1 − 2푞 cos2휋푧 + 푞 ), 푛=1 ∞ 1/4 2푛 2푛 4푛 휃10(푧,푞) = 2푞 cos휋푧 ∏(1 − 푞 )(1 + 2푞 cos2휋푧 + 푞 ), 푛=1 and from these we readily compute 휃11(푧,푞) 휃10(푧,푞) lim = sin휋푧 and lim = cos휋푧. 푞→0 −2푞 1/4 푞→0 2푞 1/4 Therefore, Jacobi’s theta functions can be considered as natural one-parameter deformations of sin and cos. For more details, see [GR04], section 1.6. 1.3 The transformation formula The transformation formula for the theta function is arguably its most useful property in applications. Without further ado: Theorem 1.6. The theta function satisfies the following identity: 1 1 2 휃( ,− ) = √−푖휏 푒휋푖푧 /휏휃 (푧,휏 ). 푧 휏 The theorem is not “optimal”, since there exists a more general transforma- tion formula, cf. Thm. 7.1 in [Mum83]. However, that one follows without too much effort from the ours, which is quite sufficient for our purposes. We start off with two lemmata: Lemma 1.7. We have that ∞ 2 ∫ 푒−푥 푑푥 = √휋. −∞ Proof (Poisson). Let 퐼 denote the integral on the left. We can consider 퐼 2 as an integral in two variables 푥 and 푦, and introduce polar coordinates 푟 and 휙 in the (푥,푦)-plane to yield −푢 ∞ ∞ 2 2 2휋 ∞ 2 ∞ 푒 2휋 퐼 2 = ∫ ∫ 푒−푥 −푦 푑푥푑푦 = ∫ ∫ 푒−푟 푟 푑푟 푑휙 = 2휋 ∫ 푑푢 = = 휋, −∞ −∞ 0 0 0 2 2 so 퐼 = √휋. Lemma 1.8 (Poisson summation formula). Suppose 푓∶ ℝ → ℂ is a Schwarz func- tion. Then ∞ ∞ ∑ 푓 (푥 + 푛) = ∑ 푒2휋푖푛푥푓̂ (푛), 푛=−∞ 푛=−∞ where 푓̂ denotes the Fourier transform of 푓. 1Not to be confused with the above identity. 4 Proof (sketch). Recognise the left-hand side as a periodic function, write out its Fourier series and re-index after a change of variables in the integral. Remark. The condition that 푓 be Schwarz is very far from optimal, but sufficient for our purposes. The classical statement of the Poisson formula is obtained by taking 푥 = 0. Proof of theorem 1.6. Fixing 푡 ∈ ℂ with Re푡 > 0, for 푥 ∈ ℝ we define 푓 (푥) .= 2 푒−휋푡푥 and compute −휋푦 2/푡 −휋푦 2/푡 ∞ 2 푒 ∞ 2 푒 푓̂ (푦) = ∫ 푒휋푖푥푦−휋푡푥 푑푥 = ∫ 푒−푢 푑푢 = −∞ √푡√휋 −∞ √푡 by the substitution 푢 = 휋√푡(푥−푖푦/푡) and lemma 1.7. Now by Poisson summation (lemma 1.8), we obtain ∞ ∞ ∞ ∞ 2 1 2 ∑ 푓 (푥 + 푛) = ∑ 푒−휋푡(푥+푛) = ∑ 푒2휋푖푛푥푓̂ (푛) = ∑ 푒2휋푖푛푥−휋푛 /푡. (1) 푛=−∞ 푛=−∞ 푛=−∞ √푡 푛=−∞ Taking 휏 = 푖푡, we have that 휏 ∈ H, and we see that ∞ ∞ 2 2 2 2 ∑ 푒휋푖휏 (푥+푛) = 푒휋푖휏 푥 ∑ 푒2휋푖휏 푥푛+휋푖푛 휏 = 푒휋푖휏 푥 휃 (휏 푥,휏 ). 푛=−∞ 푛=−∞ But now, by eq. (1), ∞ 2 1 2 1 1 푒휋푖휏 푥 휃 (휏 푥,휏 ) = ∑ 푒2휋푖푛푥−휋푖푛 /휏 = 휃(푥,− ), √−푖휏 푛=−∞ √−푖휏 휏 and by analytic continuation, this is valid for any 푥 ∈ ℂ. Setting 푧 = 휏 푥 and rearranging, we obtain 1 1 2 휃( ,− ) = √−푖휏 푒휋푖푧 /휏휃 (푧,휏 ), 푧 휏 as required. Corollary 1.9. Euler’s theta function is a modular form of weight 1/2 and level 2. Proof. In addition to the regular growth and analyticity conditions, this means that 휃 transforms like a modular form under 휏 ↦ 휏 +2 and 휏 ↦ −1/휏. The first two conditions hold because the series defining 휃 converges absolutely and uniformly on compact sets.
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