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. As for the final one, we easily check that 휃 (휏 + 2) = 휃 (휏 ), and setting 푧 = 0 in the transformation formula gives 1 휃(− ) = √−푖휏 휃 (휏 ). 휏

5 2 Applications to Number Theory

The fact that 휃 is a modular form allows us to effortlessly derive some of the most famous number-theoretic results of the 17th and 18th centuries. Taking 2 ∞ 휋푖푛푧 up the thread from the introduction with 휃 (푧) = ∑푛=1 푟2(푛)푒 , we have the following:

Theorem 2.1 (Fermat). An odd prime 푝 can be written as a sum of two squares if and only if 푝 ≡ 1 (mod 4).

Proof [BvHZ08]. It follows from corollary 1.9 that 휃 2 is a modular form of weight 1 and level 4. It can be shown (cf. Prop. 1.3 in [BvHZ08]) that the vector space of such functions is one-dimensional with basis

∞ 1 휋푖푧푛 퐺 (푧) .= + ∑(∑휒(푑))푒 1,휒 4 푛=1 푑|푛 where 휒(푛) is the Dirichlet character defined by 휒(푛) = 1 if 푛 ≡ 1 (mod 4), 휒(푛) = −1 if 푛 ≡ 3 (mod 4), and 0 otherwise. Comparing constant terms, 휃 2 = 4퐺1,휒, so by equating the coefficients we find that

(푑−1)/2 푟2(푛) = 4 ∑휒(푑) = 4 ∑ (−1) . 푑|푛 푑|푛,푑 odd

(푝−1)/2 In particular, if 푝 ≡ 1 (mod 4), we have that 푟2(푝) = 4(1 + (−1) ) = 8, and if 푝 ≡ 3 (mod 4) we have 푟2(푝) = 0.

2.1 Jacobi’s four-square theorem

Theorem 2.2 ( Jacobi’s four-square theorem). Let 푟4(푛) denote the number of dis- tinct ways in which 푛 can be represented as a sum of four squares of integers. Then

8 ∑ 푚 if 푛 is odd, 푟 (푛) = { 푚|푛 4 24 ∑ 푚 푛 푚|푛,푚 odd if is even.

Proof [Mum83]. Note that 휃 4 satisfies 휃 4(−1/휏 ) = −휏 2휃 4(휏 ) and is a modular form. We claim that the space of functions which transform as 휃 4 and are en- tire in H and finite at 푖∞, denoted by ℳ2,4, is spanned by

∞ ∞ 휃 1 1 . 퐸2 (휏 ) = ∑ ∑ ( 2 − 2 ). 푚=−∞ 푛=−∞ (2푚휏 + (2푛 + 1)) ((2푚 + 1)휏 + 2푛)

Unlike 퐺2, this is in fact absolutely convergent. We leave the details out here, but the standard arguments show that it is entire in H and bounded at 푖∞. Moreover, one easily checks that it transforms as 휃, and that ℳ2,4 is one-dimensional. We

6 휃 can compute the Fourier expansion of 퐸2 in the manner of that of 퐺푘 in [Apo90] – being slightly cavalier with details, which can be found in [Mum83] – to give

2 ∞ ∞ ∞ ∞ 휃 휋 2 푛 2휋푖푛푚휏 2 휋푖푛(2푚+1)휏 퐸2 (휏 ) = − 2휋 ∑ (∑(−1) 푛푒 ) + 2휋 ∑ (∑푛푒 ) 4 푚=1 푛=1 푚=0 푛=1 휋 2 ∞ ∞ = − 2휋 2 ∑ ∑ (−1)푛푛푒2휋푖푚휏 + 2휋 2 ∑ ∑ 푛푒휋푖푚휏 4 푚=1 푛|푚 푚=1 푛|푚 푚/푛 even 푚/푛 odd 휋 2 ∞ = (1 + 24 ∑ ∑ 푛푒2휋푖푛휏 + 8 ∑ ∑푛푒2휋푖푛휏). 4 푚∈ℕ 푛|푚 푚∈ℕ 푛|푚 푚 even 푛 odd 푚 odd

4 휃 2 By comparing constant terms, we see that 휃 = 4퐸2 /휋 , and so we read off the Fourier coefficients

8 ∑ 푚 if 푛 is odd, 푟 (푛) = { 푚|푛 4 24 ∑ 푚 푛 푚|푛,푚 odd if is even.

Since 푟4(푛) > 0 for any 푛 ∈ ℕ, we immediately obtain: Corollary 2.3 (Lagrange’s theorem). Every positive integer can be written as a sum of four squares.

2.2 Quadratic reciprocity The law of quadratic reciprocity was first established by Gauß, who referred to it in his diary as his theorema aureum, the golden theorem, and who published six proofs of it in the course of his career. The fourth proof made use of the following finite sums, which will also be the starting point of our proof:

Definition 2.4. Let 푝 and 푞 be coprime integers. We define the Gauß sum of 푝 and 푞 as 푞−1 2 푆(푝,푞) = ∑푒−휋푖푟 푝/푞. 푟 =0 The theta function lets us compute Gauß sums via the following theorem: 푝 푞 1 푆(푝,푞) = Theorem 2.5 (Landsberg-Schaar). Let and be coprime integers. Then √푞 −휋푖/4 푒 푆(푞,푝) √푝 , where the bar denotes complex conjugation. Explicitly,

푞−1 −휋푖/4 푝−1 1 2 푒 2 ∑푒−휋푖푟 푝/푞 = ∑푒휋푖푟 푞/푝. √푞 푟 =0 √푝 푟 =0 Proof [Bel61]. Set ∞ 푖푡 −푛2푡 푓 (푡) .= 휃( ) = ∑ 푒 . 휋 푛=−∞

7 In short, we will consider the asymptotic behaviours of 푓 before and after apply- ing the transformation formula, and conclude that the asymptotic expressions are equal. Let 푡 .= 휖 + 휋푖푝/푞 for 휖 > 0 small. Then

∞ 푞 ∞ 2 2 2 2 푓 (휖 + 휋푖푝/푞) = 1 + 2 ∑푒−푛 휖푒−휋푖푛 푝/푞 = 1 + 2 ∑푒−휋푖푟 푝/푞(∑푒−(푟 +푠푞) 휖) 푛=1 푟 =1 푠=0

−휋푖푛2푝/푞 ∞ −(푟 +푠푞)2휖 by the periodicity of 푒 . Now as 휖 → 0, ∑푠=0 푒 behaves, up to a constant, like the Riemann sum of

∞ 2 ∞ 2 푑푢 1 ∞ 2 √휋 ∫ 푒−(푟 +푠푞) 휖 = ∫ 푒−푢 휖 ∼ ∫ 푒−푢 푑푢 = . 0 푟 푞 푞√휖 0 2푞√휖 Therefore, noting that 푞−1 푞 2 2 푆(푝,푞) = ∑푒−휋푖푟 푝/푞 = ∑푒−휋푖푟 푝/푞 푟 =0 푟 =1 by periodicity, we obtain 휋푖푝 √휋 푓 (휖 + ) ∼ 푆(푝,푞) 휖 → 0. 푞 푞√휖 as

Next, applying the transformation formula, we have that 휋 휋 2 푓 (푡) = √ 푓 ( ), 푡 푡 and we see that 휋 2 휋 2 휋 2(휖 − 휋푖푝/푞) 휖푞 2 휋푖푞 = = = − + 푂(휖2), 푡 휖 + 휋푖푝/푞 휖2 + 휋 2푝 2/푞 2 푝2 푝 so 휋 2 √휋 푓( ) ∼ 푆(−푞,푝) 휖 → 0. 푡 푞√휖 as As for the transformation factor, we see that √휋 푞 푞 lim = √ = 푒−휋푖/4√ , 휖→0 √휖 + 휋푖푝/푞 푖푝 푝 so that we obtain 휋 휋 2 푞 √휋 푓 (푡) = √ 푓 ( ) ∼ 푒−휋푖/4√ ( )푆(−푞,푝) 휖 → 0. 푡 푡 푝 푞√휖 as

Equating the two asymptotic expressions, we finally get that

푞−1 −휋푖/4 푝−1 1 2 푒 2 ∑푒−휋푖푟 푝/푞 = ∑푒휋푖푟 푞/푝, √푞 푟 =0 √푝 푟 =0 as required.

8 We shall also need the following lemma: Lemma 2.6. Let 퐺(푛,푚) .= 푆(2푛,푚). Then for distinct primes 푝 and 푞, 퐺(1,푝푞) = 퐺(푝,푞)퐺(푞,푝).

Proof. We have that 푞−1 푝−1 푞−1 푝−1 2 2 2 2 퐺(푝,푞)퐺(푞,푝) = ∑푒푖2휋푘 푝/푞 ∑푒2휋푖푙 푞/푝 = ∑∑푒2휋푖푘 푝/푞+2휋푖푙 푞/푝 푘=0 푙=0 푘=0 푙=0 푞−1 푝−1 2 2 2 2 = ∑∑푒2휋푖(푘 푝 +푙 푞 )/푝푞 = 퐺(1,푝푞), 푘=0 푙=0 since 푘2푝2 +푙 2푞 2 ≡ (푘푝 +푙푞)2 (mod 푝푞) and 푘푝 +푙푞 runs through all the values in ℤ/푝푞ℤ exactly once.

Definition 2.7. Let 푝 and 푞 be distinct primes. The Legendre symbol is defined as follows: 푝 1 if 푥2 ≡ 푝 (mod 푞) has a solution for 푥 ∈ ℤ, ( ) .= { 푞 −1 otherwise. Lemma 2.8. Let 푝 and 푞 be distinct primes. Then 푞 퐺(푞,푝) = ( )퐺(1,푝). 푝

Proof. Note that as 푟 runs from 1 to 푝 −1, 푟 2 (mod 푝) goes through the set 푄 of quadratic residues of ℤ/푝ℤ exactly twice since (푟 −푝)2 ≡ 푟 2 (mod 푝). Therefore, 퐺(푞,푝) = 1 + 2 ∑ 푒2휋푖푘푞/푝. 푘∈푄 One easily checks that if 푞 is a quadratic residue, then so is 푘푞 for 푘푞, hence 푞 퐺(푞,푝) = 1 + 2 ∑ 푒2휋푖푙/푝 = 퐺(1,푝) = ( )퐺(1,푝). 푝 푙∈푄 If 푞 is not a quadratic residue of 푝, then 푘푞 runs through the non-quadratic residues 푄′. Noting that 푝−1 1 + ∑ 푒2휋푖푚/푝 + ∑ 푒2휋푖푚/푝 = ∑ 푒2휋푖푚/푝 = 0 푚∈푄 푚∈푄′ 푚=0 by evaluating the geometric series, we find that 푞 퐺(푞,푝) = 1 + 2 ∑ 푒2휋푖푚/푝 = −1 − 2 ∑ 푒2휋푖푛/푝 = ( )퐺(1,푝), 푝 푚∈푄′ 푛∈푄 as required.

9 We are now ready to state and prove our theorem:

Theorem 2.9 (Law of Quadratic Reciprocity). Let 푝 and 푞 be distinct odd primes. Then 푝 푞 (푝−1) (푞−1) ( )( ) = (−1) 2 2 . 푞 푝

Proof. By theorem 2.5 we see that 퐺(1,푝) = 푆(2,푝) = 푒휋푖/4푆(2,푝), so

1 푝 √푝 2 √푝 1 + (−푖) 푆(2,푝) = ∑푒−휋푖푟 푝/2 = (1 + 푒−휋푖푝/2) = √푝푒휋푖/4 . √2 푟 =0 √2 √2

If 푝 ≡ 1 (mod 4), then this equals √푝푒휋푖/4(1 − 푖)/√2 = √푝푒휋푖/4푒−휋푖/4 = √푝, and if 푝 ≡ 3 (mod 4), we obtain √푝푒2휋푖/4 = 푖√푝. It is easily checked that this can be 2 written in a single equation as 퐺(1,푝) = √푝푖 (푝−1) /4. Now, using the two lemmata above, this yields

2 푝 푞 퐺(푝,푞) 퐺(푞,푝) 퐺(1,푝푞) √푝푞푖 (푝푞−1) /4 ( )( ) = = = 2 2 . 푞 푝 퐺(1,푞) 퐺(1,푝) 퐺(1,푝)퐺(1,푞) √푝푞푖 (푝−1) /4+(푞−1) /4

The square roots cancel, and the remaining term is easily computed to yield −1 if 푝 ≡ 푞 ≡ 3 (mod 4) and 1 otherwise, hence is equal to (−1)(푝−1)(푞−1)/4, for which the same holds. But this is precisely what we wanted to show.

We can also use this method for the case where one of the primes is even: Proposition 2.10. Let 푝 be an odd prime. Then

2 푝2−1 1 if 푝 ≡ 1 or 7 (mod 8), ( ) = (−1) 8 = { 푝 −1 if 푝 ≡ 3 or 5 (mod 8).

Proof. By lemma 2.8, the Landsberg-Schaar relation (theorem 2.5) and our pre- vious computation of 퐺(1,푝), we find that

푝 2 √ 휋푖/4 3 휋푖푟 푝/4 휋푖(푝+1)/4 휋푖(9 푝+1)/4 2 퐺(2,푝) 2 푒 ∑푟 =0 푒 푒 + 푒 ( ) = = 2 = 2 , 푝 퐺(1,푝) √푝푖 (푝−1) /4 2푖 (푝−1) /4

푝2−1 and by comparing the various residues in ℤ/8ℤ we see that this equals (−1) 8 , as required.

Our method of computing Gauß sums using theta functions is extended to great generality in a very deep theorem due to Hecke ( [Hec13]), which is far be- yond the scope of this text. In Chap. 8 of the text, he proves quadratic reciprocity for arbitrary number fields, having defined corresponding theta functions.

10 3 Epilogue

In closing we make a few reflections on the significance of these theorems: The proof of quadratic reciprocity shows that like zeta functions, theta functions carry deep information about the structure of the primes in a number field. The proof of the four-square theorem suggests that 휃 also “knows” about the combi- natorial structure of the integers. From a different point of view, Fermat’s theorem is classically proved by look- ing at primes splitting in ℤ[푖], and an analogous proof due to Hurwitz ( [Hur19]) runs a similar argument with the Hurwitz quaternions to prove the four-square theorem. These are both lattices over ℤ, so alternatively we can argue that the properties of 휃 ought to interpreted from, say, a representation-theoretic point of view. Indeed, one of the most fruitful extensions of the classical theory is the theta correspondence, which very roughly realises 휃 as a (still conjectural) corre- spondence between representations of the metaplectic group of order 2푛 and the special orthogonal group of order 2푛 +1. That, however, is another story for an- other essay.

Bibliography

[Apo90] Tom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Graduate Texts in Mathematics. Springer-Verlag, New York, 2 edition, 1990.

[Bel61] Richard Bellman. A Brief Introduction to Theta Functions. Athena Series: Selected Topics in Mathematics. Holt, Rinehart and Winston, New York, 1961.

[BvHZ08] Jan Hendrik Bruinier, Gerard van der Geer, Günter Harder, and Don Zagier. The 1-2-3 of Modular Forms. Universitext. Springer-Verlag, Berlin, 2008.

[GR04] George Gasper and Mizan Rahman. Basic Hypergeometric Series, vol- ume 96 of Encyclopedia of Mathematics and Its Applications. Cam- bridge University Press, Cambridge, second edition, 2004.

[Hec13] E. T. Hecke. Lectures on the Theory of Algebraic Numbers. Springer Science & Business Media, March 2013.

[Hur19] Adolf Hurwitz. Vorlesungen Über die Zahlentheorie der Quaternionen. Springer-Verlag, Berlin Heidelberg, 1919.

[Mum83] David Mumford. Tata Lectures on Theta. I, volume 28 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1983.

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