Lattices and Modular Forms

I am not a number theorist but this topic has fascinated me and I’ve decided to write about it as one of the first blog posts. The classification of definite forms is difficult but also appears in the study of 4-manifolds since Freedman proved that closed, oriented, simply-connected, topological 4-manifolds are classified by their intersection form. This post isn’t about that so much. Much of what I write here can be found in A First Course in Modular Forms by Fred Diamond and Jerry Shurman.

Without loss of generality, let’s assume that $Q$ is positive definite. If $Q$ is even, then its rank and signature are both divisible by 8. Under the identification of $Q$ with the Euclidean inner product, the domain of $Q$ is identified with some lattice $\Gamma \subset \mathbb{R}^n$. We can write a basis for the lattice and the absolute value of the determinant of the basis is called the covolume. In our case, the covolume is 1.

Since $Q$ is unimodular, $\Gamma$ is isomorphic to its dual. Associated to $\Gamma$ is a theta function: $\theta_\Gamma(q) := \sum_{x\in \Gamma} q^{(x \cdot x)/2}$.

Example: Let $\Gamma = 2 \mathbb{Z}$ and so $\theta(q) = 1+ \sum_{n=1}^\infty 2q^{\frac{1}{2}(2n)^2} = \sum^\infty_{-\infty} q^{2n^2}$. If we square this function $\theta(q)^2 = (1+2q^2+2q^8+2q^{32}+…)(1+2q^2+2q^8+2q^{32}+…)$ and write it as $\sum c_{2m}q^{2m}$, the coefficient $c_{2m}$ counts ordered pairs $(a,b) \in \mathbb{Z}^2$ such that $2m = \frac{1}{2}((2a)^2+(2b)^2)$; in other words, it tells us the number of ordered pairs $(a,b) \in \mathbb{Z}^2$ such that $a^2+b^2 = m$. Note that sometimes, we’re counting $(\pm a,\pm b)$ and also $(\pm b,\pm a)$ as distinct pairs. And if we have $(0,b)$, then the $\pm$ doesn’t affect 0. Here are the first few coefficients: $\theta(q)^2 = 1 + 4q^2+4q^4+4q^8+8q^{10}+…$ The number of ways to write $1$ as a sum of squares is 4. We have $(\pm 1)^2 + 0^2$ and $0^2 + (\pm 1)^2$. We can write 5 as $(\pm 1)^2+(\pm 2)^2$ or $(\pm 2)^2 + (\pm 1)^2$. Hence, 8 ways. Note that there aren’t any ways to write 3 as a sum of two squares.

Being even, $(x\cdot x)/2$ is an integer and so this is an ordinary Taylor series in $q$. These exponents are also nonnegative (that’s how the Euclidean dot product is). Also, if we have a basis for the lattice ${e_1,…,e_n}$ and $x = \sum a_i e_i$, then $x\cdot x = \sum^n a^2_i e^2_i + 2\sum_{i,j} a_i a_j e_i\cdot e_j$.

The first term is at least $\sum^n a^2_i$ since $e^2_i \geq 1$. As the $a_i$ grow, $x\cdot x$ grows like a polynomial of degree $n/2$. If we make the substitution $q = e^{2\pi i t}$, then $\theta_\Gamma(t) = \sum_{x \in \Gamma} e^{\pi it (x\cdot x)}$ and $\theta_\Gamma(it ) = \sum_{x \in \Gamma} e^{-\pi t (x\cdot x)}$.

The Poisson summation formula for a function $f$ on $\mathbb{R}^n$ decaying sufficiently fast at infinity, says that for any lattice $\Lambda \subset \mathbb{R}^n$, the sum of values of $f$ on $\Lambda$ equals the sum of values of its Fourier transform on the dual lattice $\Lambda$. Here, $\Gamma \cong \Gamma^*$ because of unimodularity. Moreover, $e^{-\pi |x|^2}$ is its own Fourier transform (here, I’m just writing $|x|^2$ to mean $x \cdot x$). I’ll prove it below for $n=1$.

Lemma: If $\varphi(x) = e^{-\pi |x|^2}$, then the Fourier transform $\hat{\varphi}(\xi) = e^{-\pi |\xi|^2}$.

Proof: Let’s just work on $\mathbb{R}$ though the same proof works on $\mathbb{R}^n$. $\hat{\varphi}(\xi):= \int^\infty_{-\infty} \exp(-\pi x^2-2\pi ix \xi) \, dx$. We can multiply by $1 = e^{-\pi \xi^2} e^{\pi \xi^2}$. Moving the second term into the exponent since it has no $x$ dependence, we have that the exponential becomes: $\exp(-\pi(x^2+2ix \xi -\xi^2)) = \exp(-\pi x+i\xi^2)$. Substituting $u = x+i\xi, du = dx$, we have that $\hat{\varphi}(\xi) = e^{-\pi \xi^2} \int^\infty_{-\infty} e^{-\pi u^2}\, du$. The integral is the Gaussian integral and it equals 1. So $\hat{\varphi}(\xi) = e^{-\pi \xi^2}$. For $\mathbb{R}^n$, we would basically do the same thing by studying a product: $\prod^n_j \int^\infty_{-\infty} \exp(-\pi x^2_j - 2\pi i x_j \xi_j)$. $\square$

So, each term in the theta function is its own Fourier transform and $\Gamma \cong \Gamma^*$. The Fourier transform of $e^{-\pi t x^2}$ is almost the same but is $t^{-n/2}e^{-\pi \xi^2/t}$. So, by the Poisson summation formula, $\theta_\Gamma(it) = \sum_{x \in \Gamma} e^{-\pi t x^2} = \sum_{x \in \Gamma} t^{-n/2} e^{-\pi \xi^2/t} = t^{-n/2} \sum_{x \in \Gamma} e^{-\pi \xi^2/t}$. On the other hand, $\theta_\Gamma((i/\sqrt{t})^2) =\theta_\Gamma(-1/t) = \sum_{x \in \Gamma} e^{- \pi i x^2/t}$. Hence, $\theta_\Gamma(-1/t) = t^{n/2}\theta_\Gamma(it)$. Lastly, $\theta_\Gamma(-1/t) = (it)^{n/2}\theta_\Gamma(t)$.

Recall that the transformation $t \mapsto -1/t$ is given by one of the generators of $SL(2,\mathbb{Z})$.

Also, $\theta_\Gamma(t + 1) = \sum e^{\pi i(t+1)x^2}$. Since $x\cdot x$ is even, $e^{\pi i x^2} = 1$. Hence, $\theta_\Gamma(t+1) = \theta_\Gamma(t)$. This transformation $t \mapsto t+1$ is the other generator of $SL(2,\mathbb{Z})$. Hence, $\theta_\Gamma$ is a modular form of weight $n/2$. This is an integer since $n$ is a multiple of 8.

Brief Aside on Modular Forms

Let $\Gamma \subset SL_2(\mathbb{Z})$ be a finite index subgroup; this is not the same $\Gamma$ as above but since this is the standard notation in the literature, I’ll use it here. Recall that a modular form of weight $k$ and level $\Gamma$ is a holomorphic function $f:\mathbb{H} \to \mathbb{C}$ on the upper-half plane satisfying:

1. $f(\frac{a\tau + b}{c \tau +d})= (c\tau+d)^{k} f(\tau)$ for \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\Gamma\)

2. $f$ is bounded as $\tau \to i \infty$.

Let $\pi_N:SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/N\mathbb{Z})$ be reduction mod $N$ on matrices. Typically, we want $\Gamma$ to contain $\Gamma(N) = \ker \pi_N$; we call such $\Gamma$ congruence subgroups. If we’re studying $\Gamma = SL_2(\mathbb{Z})$, then we only need to check that $f$ satisfies property (1) for the generators \(T=\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\)

This is what we did above with the theta function $\theta_\Gamma$. One can sort of see why modular forms are related to rank 2 lattices in $\mathbb{C}$ since $SL_2(\mathbb{Z})$ just changes the basis vectors and we only need to check the upper-half plane since we can take $\tau, 1$ as a basis where $\tau \in \mathbb{H}$.

The boundary of $\mathbb{H}$ is $\mathbb{RP}^1$ and we can also consider $\mathbb{QP}^1 = \mathbb{Q} \cup \infty$. Then, an element \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{Z})\)

sends $x \in \mathbb{Q}$ to $(ax+b)/(cx+d)$ and sends $\infty$ to $a/c$. If $cx+d = 0$, then $x \mapsto \infty$. The cusps of a subgroup $\Gamma$ are the $\Gamma$-orbits. For example, if $\Gamma = SL_2(\mathbb{Z})$ we see that $0 \mapsto b/d$. Given any $b,d$ with $\gcd(b,d) = 1$ (so the fraction is fully reduced), then we just need to find $a,c$ so that $ad-bc=1$ to form an element of $\Gamma$. This is guaranteed by the Euclidean algorithm. And also, any matrix with $d=0$ sends $0 \mapsto \infty$. Hence, there is one orbit and thus, $SL_2(\mathbb{Z})$ itself has one cusp.

As we saw above, it turns out that any even, unimodular rank $8m$ lattice produces a modular form of weight $4m$ for $SL_2(\mathbb{Z})$ because we have a Poisson summation formula in the setting of unimodular lattices. The definitions of modular forms makes it clear that the modular forms of weight $k$ and level $\Gamma$ form a vector space and in fact, can be alternatively described in the following way.

Let $M_k(\Gamma)$ denote the space of modular forms of weight $k$, level $\Gamma$. Define the modular curve to be $X_\Gamma = ( \mathbb{H} \cup \mathbb{QP}^1)/\Gamma$. That is, we take the upper-half plane; it has $\mathbb{RP}^1$ as the boundary at infinity; we add in the rational points. We then quotient by the action of $\Gamma$. Let $\omega$ be the canonical line bundle on this curve; then the holomorphic sections of $\omega^{\otimes k}$ is denoted $H^0(X_\Gamma,\omega^{\otimes k})$ and it’s not hard to see that this is isomorphic to $M_k(\Gamma)$. Hence, the dimensions of these spaces of modular forms can be computed using the Riemann-Roch formula. One of the interesting things is that $M_k(SL_2(\mathbb{Z})) = \mathbb{C}\cdot G_{2k} \oplus S_k$ decomposes into the direct sum of a 1-dim subspace generated by the $2k$-Eisenstein series and the so-called cusp forms of weight $k$. These are modular forms that, when we extend them to $\overline{\mathbb{H}} = \mathbb{H} \cup \mathbb{QP}^1$, they vanish on the cusps of $\Gamma$.

The definition of the Eisenstein series is straightforward: $G_{2k}(t):= \sum_{(m,n) \in \mathbb{Z}^2 \setminus (0,0)} (m+n \tau)^{-2k}$ where $k \geq 2$ (this last condition is to get absolute convergence). We note that a proper congruence subgroup $\Gamma$ may be such that the vector space of Eisenstein series is not just 1-dim. Also, note that if we replaced $-2k$ with a negative odd integer, terms would cancel in pairs which is why we only work with even powers. We can use $G_4,G_6$ to define the $j$-invariant for elliptic curves. It turns out that for $\Gamma = SL_2(\mathbb{Z})$, there are no cusp forms of weight $k < 12$ and hence, $\dim_\mathbb{C} M_k = 1$ for $k<12$. Since constant functions are modular forms of weight 0, it turns out that constant functions are the only modular forms of weight 0 by this dimension consideration.

Example: The $E_8$ lattice is rank 8 and hence, its theta function gives a modular form of weight 4 which means it must be a multiple of the Eisenstein series $G_8$. Since the theta function of any lattice has terms of the form $q^{(x \cdot x)/2}$, we always have $x=0$ in the lattice and hence, we have some constant terms. When the lattice is unimodular and definite, then the constant term is 1. Thus, $\theta_{E_8}(q) = \frac{1}{c_0}G_8$ where $c_0$ is the constant term of $G_8$. Amazingly, $c_0 = 2\zeta(4) = 2\sum^\infty_{n=1} \frac{1}{n^4} = \frac{\pi^4}{45}$ where $\zeta$ is the Riemann-Zeta function. In fact, $\theta_{E_8}(q) = 1+ 240\sum^\infty_{n=1} \sigma_3(n)q^n$ where $\sigma_3(n)$ is computed by taking all the positive divisors of $n$, cubing them, and then summing. For example, $\sigma_3(2) = 1^3+2^3 = 9$.

Example: In the example above with $\Lambda = 2\mathbb{Z}$, we saw that the square of the theta function has coefficients that tell us the number of ways to write an integer $m$ as a sum of two squares; more generally, the coefficients of $q^{2m}$ in $\theta^k_{2\mathbb{Z}}(q)$ tell us when $m$ can be written as a sum of $k$ squares. Let $\Gamma_1(4) \subset SL_2(\mathbb{Z})$ be the subgroup of matrices congruent to \(\begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix}\)

after taking mod 4. It turns out that $\theta^{2j}_{2\mathbb{Z}}$ is itself a modular form of weight $j$, level $\Gamma_1(4)$. By Riemann-Roch, one finds that for $j=1$, $M_1(\Gamma_1(4))$ is 1-dimensional so this modular form is a multiple of some Eisenstein series. In fact, $\theta^2_{2\mathbb{Z}}(q)=1+4\sum^\infty_{m=1}(\sum_{d|m} \chi_4(d))q^{2m}$ where $\chi_4(d)=0$ when $d \equiv 0,2 \pmod{4}$, $\chi_4(d)=+1$ when $d \equiv 1 \pmod{4}$, and $\chi_4(d)=-1$ when $d \equiv 3 \pmod{4}$. In other words, the number of ordered pairs $(a,b)$ such that $a^2+b^2 = m$ is given by $4(d_1(m)-d_3(m))$ where $d_1(m)$ is the number of divisors $d$ of $m$ congruent to 1 mod 4 and $d_3(m)$ is similar but congruent to 3 mod 4. This recovers a theorem of Jacobi. For example, $d_1(3)=d_3(3)$ and hence, 3 cannot be written as a sum of two squares. On the other hand, $d_1(5) = 2,d_3(5)=0$ and so there are 8 ordered pairs for 5.

Corollary (originally due to Euler, after 1640): If $p$ is an odd prime, then it can be written as a sum of two squares if and only if $p \equiv 1 \pmod{4}$.

Corollary: An integer $m$ can be written as the sum of two squares if and only if when writing out its prime factorization, all the primes $p \equiv 3 \pmod{4}$ have even powers.

If we let $k=2j = 4$, then we obtain a modular form where for every $m$, the coefficient for $q^{2m}$ is positive. This recovers a theorem of Lagrange:

Theorem (Lagrange’s 4 Squares Theorem, 1770): Every positive integer can be written as a sum of four squares.

But more than that, we get a theorem of Jacobi, proved in 1834, which gives the exact number of 4-tuples $(w,x,y,z) \in \mathbb{Z}^4$ such that $m=w^2+x^2+y^2+z^2$. See this masters thesis for more on this topic of modular forms and sums of squares.