From sum of two squares to arithmetic Siegel–Weil formulas

Abstract

The main goal of this expository article is to survey recent progress on the arithmetic Siegel–Weil formula and its applications. We begin with the classical sum of two squares problem and put it in the context of the Siegel–Weil formula. We then motivate the geometric and arithmetic Siegel–Weil formula using the classical example of the product of modular curves. After explaining the recent result on the arithmetic Siegel–Weil formula for Shimura varieties of arbitrary dimension, we discuss some aspects of the proof and its application to the arithmetic inner product formula and the Beilinson–Bloch conjecture. Rather than being intended as a complete survey of this vast field, this article focuses more on examples and background to provide easier access to several recent works by the author with W. Zhang and Y. Liu.

1. Sum of two squares

1.1. Which prime $p$ can be written as the sum of two squares?

For the first few primes we easily find that

$$\begin{equation*} 5=1^2+2^2, \quad 13=2^2+3^2,\quad 17=1^2+4^2,\quad 29=2^2+5^2 \end{equation*}$$

are the sums of two squares, while other primes like $3, 7, 11, 19,23$ are not. The answer seems to depend on the residue class of $p$ modulo 4.

Theorem 1.1.1.

A prime $p\ne 2$ is the sum of two squares if and only if $p\equiv 1\ (\mathrm{mod}\ 4)$.

Theorem 1.1.1 is usually attributed to Fermat and appeared in his letter to Mersenne dated Dec 25, 1640 (hence the name Fermat’s Christmas Theorem), although the statement can already be found in the work of Girard in 1625. The “only if” direction is obvious, but the “if” direction is far from trivial. Fermat claimed that he had an irrefutable proof, but nobody was able to find the complete proof among his work—apparently margins were often too narrow for Fermat. The only clue (in his letters to Pascal and to Digby) is that he used a “descent argument”: if such a prime $p$ is not of the required form, then one can construct another smaller prime and so on, until a contradiction occurs when one encounters 5, the smallest such prime. More than 100 years later, Euler (1755) gave the first rigorous proof of Theorem 1.1.1 based on infinite descent. For a detailed history of Theorem 1.1.1, see Dickson Dic66, Ch. VI, pp. 227–231.

1.2. Which positive integer $n$ can be be written as the sum of two squares?

If $n=x^2+y^2$ ($x,y\in \mathbb{Z}$) and $p\mid n$, then either $p\mid \mathrm{gcd}(x,y)$ or $(x/y)^2\equiv -1\ (\mathrm{mod}\ p)$, and hence either $n/p^2$ is also the sum of two squares or $p\not \equiv 3\ (\mathrm{mod}\ 4)$ (by the quadratic reciprocity). It follows that each $p\mid n$ with $p\equiv 3\ (\mathrm{mod}\ 4)$ must appear to an even power. On the other hand, the familiar Diophantus identity

$$\begin{equation*} \left(a^2 + b^2\right)\left(c^2 + d^2\right) =\left(ac-bd\right)^2 + \left(ad+bc\right)^2 \end{equation*}$$

shows that a product of integers of the form $x^2+y^2$ is also of the same form. Combining with Theorem 1.1.1 we obtain:

Corollary 1.2.1.

A positive integer $n$ is of the form $n=x^2+y^2$ if and only if each prime factor $p\equiv 3\ (\mathrm{mod}\ 4)$ of $n$ appears to an even power.

1.3. In how many different ways can one represent $n$ as the sum of two squares?

Definition 1.3.1.

To answer this question, we naturally define the representation number

$$\begin{equation*} r(n)\coloneq \#\{(x,y)\in \mathbb{Z}^2: n=x^2+y^2\}. \end{equation*}$$

In particular, $n$ is of the form $x^2+y^2$ if and only if $r(n)>0$.

Example 1.3.2.

$$\begin{align*} 4&= 0^2+(\pm 2)^2=(\pm 2)^2+0^2, & r(4)=4,\\ 5&=(\pm 1)^2+(\pm 2)^2=(\pm 2)^2+(\pm 1)^2, &r(5)=8,\\ 25&=0^2+(\pm 5)^2=(\pm 3)^2+(\pm 4)^2=(\pm 4)^2+(\pm 3)^2, &r(25)=12. \\ \end{align*}$$

In his book Fundamenta nova theoriae functionum ellipticarum (1829), Jacobi proved the following general formula for the representation numbers.

Theorem 1.3.3 (Jacobi).

$$\begin{equation*} r(n)=4\left(\sum _{d|n\atop d\equiv 1\bmod 4}1-\sum _{d|n\atop d\equiv 3\bmod 4}1\right). \end{equation*}$$

As a byproduct, Jacobi’s formula shows that

$$\begin{align*} p\equiv 1\ (\mathrm{mod}\ 4)&\Rightarrow \ r(p)=4(2-0)=8>0, \\ p\equiv 3\ (\mathrm{mod}\ 4)&\Rightarrow \ r(p)=4(1-1)=0, \end{align*}$$

which gives an immediate (and different) proof of Theorem 1.1.1!

1.4. Jacobi’s proof

Jacobi’s proof of Theorem 1.3.3 involves Jacobi’s theta series,

$$\begin{equation*} \theta \coloneq \sum _{n\in \mathbb{Z}}q^{n^2}=1+2q+2q^4+2q^9+\cdots . \end{equation*}$$

The representation numbers $r(n)$ naturally appear as the $n$th coefficients of the square of Jacobi’s theta series

$$\begin{equation*} \theta ^2=\left(\sum _{n\in \mathbb{Z}}q^{n^2}\right)^2=\sum _{n\ge 0}r(n)q^n=1 + 4 q + 4 q^2 + 4 q^4 + 8 q^5 +4 q^8+\cdots . \end{equation*}$$

Jacobi used his theory of elliptic functions (including his famous Triple Product Identity) to derive the formula (Jac1820, p. 107)

$$\begin{equation*} \theta ^2=1+4\left(\frac{q}{1-q}-\frac{q^3}{1-q^3}+\frac{q^5}{1-q^5}-\frac{q^7}{1-q^7}+\cdots \right), \end{equation*}$$

which is easily seen to be equivalent to Theorem 1.3.3.

1.5. Another proof using modular forms

An alternative way of evaluating $\theta ^2$ is to view $q=e^{2\pi i\tau }$, and $\theta =\theta (\tau )$ as a holomorphic function on the upper half-plane

$$\begin{equation*} \mathcal{H}\coloneq \{\tau =\mathsf{x}+i\mathsf{y}\in \mathbb{C}:\mathsf{y}=\operatorname {Im}(\tau )>0\}. \end{equation*}$$

The function $\theta (\tau )$ satisfies two transformation rules (see Zag08, Proposition 9):

$$\begin{equation*} \theta (\tau +1)=\theta (\tau ),\quad \theta \left(-\frac{1}{4\tau }\right)=(-2i\tau )^{1/2}\theta (\tau ). \end{equation*}$$

The first rule is clear by the periodicity of the exponential function. The second rule can be proved using the Poisson summation formula and also plays a key role in Riemann’s proof of the functional equation of the Riemann zeta function (see DS05, §4.9). These rules amount to saying that

$$\begin{equation*} \theta (\tau )\in M_{1/2}(\Gamma _1(4)) \end{equation*}$$

is a modular form of weight $1/2$ and level $\Gamma _1(4)$. Jacobi’s theta series $\theta (\tau )$ and its variants (under the general name of theta series) form one of most important classes of modular forms.

It follows that

$$\begin{equation*} \theta ^2(\tau )\in M_1(\Gamma _1(4)) \end{equation*}$$

is a modular form of weight 1 and level $\Gamma _1(4)$. The space $M_1(\Gamma _1(4))$ is in fact one dimensional (Zag08, Proposition 3 or DS05, Theorem 3.6.1), so if one can construct another a modular form of weight 1 and level $\Gamma _1(4)$, then it has to be a scalar multiple of $\theta ^2(\tau )$. We next construct such a modular form using Eisenstein series, another of the most important classes of modular forms.

Definition 1.5.1.

Let $\chi : (\mathbb{Z}/4 \mathbb{Z})^\times {\overset{\sim }{\longrightarrow }} \{\pm 1\}$ be the (unique) nontrivial character. We define an Eisenstein series

$$\begin{equation} G_k^{\chi }(\tau )=\sum _{(0,0)\ne (c,d)\in \mathbb{Z}^2, 4|c}\frac{\chi (d)}{(c\tau +d)^k}, {\tag{1.5.1.1}} \end{equation}$$

where $\chi (d)$ is understood to be 0 when $(4,d)\ne 1$.

When $k\ge 3$, the series 1.5.1.1 is absolutely convergent and is nonzero only when $k$ is odd. When $k\ge 3$ is odd, it defines a modular form $G_{k}^\chi (\tau )\in M_k(\Gamma _1(4))$ of weight $k$, level $\Gamma _1(4)$, and character $\chi$. The constant term of the $q$-expansion of $G_k^\chi (\tau )$ is nonzero, and we let $E_k^{\chi }(\tau )$ be a scalar multiple of $G_k^\chi (\tau )$ so the constant term is normalized to be 1. This normalized Eisenstein series $E_k^\chi (\tau )$ then has the explicit $q$-expansion (see DS05, §4.5)

$$\begin{equation} E_k^{\chi }(\tau )=1+c_k^\chi \cdot \sum _{n\ge 1}\left(\sum _{d|n}\chi (d)d^{k-1}\right)q^n, {\tag{1.5.1.2}} \end{equation}$$

where $c_k^\chi =2/L(1-k,\chi )$ is related to a special value of the Dirichlet $L$-function $L(s,\chi )$.

When $k=1$, the series 1.5.1.1 is not absolutely convergent, but one can still suitably modify it to obtain a modular form

$$\begin{equation*} E_1^\chi (\tau )\in M_1(\Gamma _1(4)) \end{equation*}$$

with the same formula 1.5.1.2 for its $q$-expansion, either using the Weierstrass $\sigma$-function (see DS05, §4.8) or using the analytic continuation of

$$\begin{equation} G_1^{\chi }(\tau ,s)\coloneq \sum _{(0,0)\ne (c,d)\in \mathbb{Z}^2, 4|c}\frac{\chi (d)}{(c\tau +d)|c\tau +d|^{2s}},\quad {\mathrm{Re}}(s)>1/2 {\tag{1.5.1.3}} \end{equation}$$

to $s=0$ (see Miy89, §7.2). In particular, when $k=1$, formula 1.5.1.2 simplifies to

$$\begin{equation*} E_1^\chi (\tau )=1+c_1^\chi \cdot \sum _{n\ge 1}\left(\sum _{d|n}\chi (d)\right)q^n. \end{equation*}$$

As explained, $E_1^\chi (\tau )$ must be a scalar multiple of $\theta ^2(\tau )$. Since both of them have constant coefficient 1, we indeed have the equality

$$\begin{equation} \theta ^2(\tau )=E_1^\chi (\tau ). {\tag{1.5.1.4}} \end{equation}$$

Comparing the coefficient before $q$, we obtain $c_1^\chi =4$ and hence

$$\begin{equation*} r(n)=4\sum _{d|n}\chi (d), \end{equation*}$$

which proves Theorem 1.3.3.

Remark 1.5.2.

As a byproduct of the proof, we also obtain $L(0,\chi )=\frac{1}{2}$ from $c_1^\chi =4$. Via the functional equation of $L(s,\chi )$, this is equivalent to the famous Leibniz formula for $\pi$ (1676):

$$\begin{equation*} L(1,\chi )=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots =\frac{\pi }{4}. \end{equation*}$$

To summarize, Jacobi’s Theorem 1.3.3 can be proved using the identity of two modular forms 1.5.1.4, namely using a relation of the form

theta series $\longleftrightarrow$ Eisenstein series.

Notice that the Fourier coefficients of theta series encode representation numbers of quadratic forms, while the Fourier coefficients of Eisenstein series are generalized divisor sums which are more explicit.

2. Siegel–Weil formula

2.1. Siegel’s formula

Siegel Sie35 generalizes formula 1.5.1.4 from the binary quadratic form $x^2+y^2$ to more general quadratic forms in an arbitrary number of variables. Let $\Lambda$ be a positive definite quadratic lattice over $\mathbb{Z}$ of rank $m$ with quadratic form $Q: \Lambda \rightarrow \mathbb{Z}$. Denote by $(\ ,\ )$ the associated symmetric bilinear form, defined by

$$\begin{equation*} (x,y)\coloneq Q(x+y)-Q(x)-Q(y) \end{equation*}$$

(so $Q(x)=\frac{1}{2}(x,x)$). Denote by $\operatorname {Sym}_n(\mathbb{Z})$ the set of symmetric matrices whose diagonal entries are in $\mathbb{Z}$ and whose off-diagonal entries are in $\frac{1}{2}\mathbb{Z}$. Denote by $\operatorname {Sym}_n(\mathbb{Z})_{\ge 0}\subseteq \operatorname {Sym}_n(\mathbb{Z})$ the subset of positive semidefinite matrices.

Definition 2.1.1.

For $T\in \operatorname {Sym}_n(\mathbb{Z})_{\ge 0}$, define the (generalized) representation number

$$\begin{equation*} r_\Lambda (T)\coloneq \#\{(x_1,\ldots ,x_n)\in \Lambda ^n: {\frac{1}{2}}((x_i,x_j))_{1\le i,j\le n}=T\}. \end{equation*}$$

Define Siegel’s theta series

$$\begin{equation} \theta _\Lambda (\tau )\coloneq \sum _{T\in \operatorname {Sym}_n(\mathbb{Z})_{\ge 0}}r_\Lambda (T)q^T, \quad q^T\coloneq e^{2\pi i\operatorname {tr}T\tau }, {\tag{2.1.1.1}} \end{equation}$$

and a holomorphic function on Siegel’s half-space

$$\begin{equation*} \mathcal{H}_n\coloneq \{\tau =\mathsf{x}+i \mathsf{y}: \mathsf{x}\in \operatorname {Sym}_n(\mathbb{R}), \mathsf{y}\in \operatorname {Sym}_n(\mathbb{R})_{>0}\}. \end{equation*}$$

Using the Poisson summation formula, Siegel proved that $\theta _\Lambda (\tau )$ is a Siegel modular form on $\mathcal{H}_n$ of weight $m/2$.

Example 2.1.2.

Notice that when $n=1$, Siegel’s half-space $\mathcal{H}_n$ recovers the usual upper half-plane $\mathcal{H}$, and Siegel’s theta series $\theta _\Lambda (\tau )$ recovers Jacobi’s theta series

$$\begin{equation*} \theta _\Lambda (\tau )\coloneq \sum _{n\ge 0}r_\Lambda (n)q^n,\quad r_\Lambda (n)\coloneq \{x\in \Lambda : Q(x)=n\}. \end{equation*}$$

In general, the theta series $\theta _\Lambda$ for a lattice $\Lambda$ may fail to be an Eisenstein series on the nose, but Siegel’s formula shows that the weighted average of theta series within its genus class is always a Siegel Eisenstein series on $\mathcal{H}_n$:

weighted average of theta series $\longleftrightarrow$ Siegel Eisenstein series.

More precisely, recall that two quadratic lattices $\Lambda ,\Lambda '$ are in the same genus, if they are isomorphic over $\mathbb{R}$ and over $\mathbb{Z}_p$ for all primes $p$. Denote by $\mathrm{Gen}(\Lambda )$ the set of isomorphism classes of quadratic lattices in the same genus of $\Lambda$. Denote by $\operatorname {Aut}(\Lambda )$ the automorphism group of $\Lambda$ as a quadratic lattice.

Theorem 2.1.3 (Siegel).

The following identity holds:

$$\begin{equation} \frac{ \sum _{\Lambda '\in \mathrm{Gen}(\Lambda )}\frac{1}{\#\operatorname {Aut}(\Lambda ')}\cdot \theta _{\Lambda '}(\tau )}{ \sum _{\Lambda '\in \mathrm{Gen}(\Lambda )}\frac{1}{\#\operatorname {Aut}(\Lambda ')}} =E_\Lambda (\tau ). {\tag{2.1.3.1}} \end{equation}$$

Here $E_\Lambda (\tau )$ is a certain normalized Siegel Eisenstein series on $\mathcal{H}_n$ of weight $m/2$.

Example 2.1.4.

Consider the case $m=2$, $n=1$ and $\Lambda =\mathbb{Z}^2$ equipped with the quadratic form $Q=x^2+y^2$. Then

$$\begin{equation*} \theta _\Lambda (\tau )=\theta ^2(\tau ),\quad E_\Lambda (\tau )= E_1^\chi (\tau ). \end{equation*}$$

In this case $\mathrm{Gen}(\Lambda )$ is a singleton and Siegel’s formula recovers 1.5.1.4.

Example 2.1.5 (cf. Ser73, V.2.3).

Siegel’s formula is extremely useful in studying the arithmetic of quadratic forms. For example, one can deduce his famous mass formula (also known as the Smith–Minkowski–Siegel mass formula), which computes the mass of $\mathrm{Gen}(\Lambda )$, defined to be weighted size

$$\begin{equation*} \sum _{\Lambda '\in \mathrm{Gen}(\Lambda )}\frac{1}{\#\operatorname {Aut}(\Lambda ')} \end{equation*}$$

as an Euler product of local factors indexed by primes $p$.

For example, consider the simplest case when $\Lambda$ is

•

unimodular, i.e., $\det (\frac{1}{2}((x_i,x_j)_{i,j=1}^n)\in \{\pm 1\}$ for a $\mathbb{Z}$-basis $\{x_1,\ldots ,x_n\}$ of $\Lambda$, and

•

even, i.e., 2 divides $Q(x)$ for all $x\in \Lambda$.

The rank $m$ of any unimodular even lattice $\Lambda$ is necessarily a multiple of 8. Siegel’s mass formula computes the mass of $\mathrm{Gen}(\Lambda )$ explicitly as

$$\begin{align*} \sum _{\Lambda '\in \mathrm{Gen}(\Lambda )}\frac{1}{\#\operatorname {Aut}(\Lambda ')}&=\frac{2\zeta (2)\zeta (4)\cdots \zeta (m-2)\zeta (m/2)}{\operatorname {vol}(S^0)\operatorname {vol}(S^1)\cdots \operatorname {vol}(S^{m-1})} = {B_{{m/2}} \over m}\prod _{{1\leq j<m/2}}{B_{{2j}} \over 4j}, \end{align*}$$

where $\zeta (s)$ is the Riemann zeta function, $\operatorname {vol}(S^{k-1})=\frac{2\pi ^{k/2}}{\Gamma (k/2)}$ is the volume of the unit $(k-1)$-sphere, and $B_k$ is the $k$th Bernoulli number.

Example 2.1.6 (cf. Ser73, VII.6.6).

Let $\Lambda$ be the root lattice of type $E_8$, defined by

$$\begin{equation*} \Lambda \coloneq \{x=(x_1,\ldots , x_8)\in \mathbb{Z}^8\cup (\mathbb{Z}+1/2)^8: \sum _{i=1}^8x_i\equiv 0\ (\mathrm{mod}\ 2)\},\quad Q=\sum _{i=1}^8 x_i^2. \end{equation*}$$

Then $\Lambda$ is unimodular and even. Siegel’s mass formula computes that

$$\begin{equation*} \sum _{\Lambda '\in \mathrm{Gen}(\Lambda )}\frac{1}{\#\operatorname {Aut}(\Lambda ')}={B_{4} \over 8}{B_{2} \over 4}{B_{4} \over 8}{B_{6} \over 12}={-1/30 \over 8}\;{1/6 \over 4}\;{-1/30 \over 8}\;{1/42 \over 12}={1 \over 696729600}. \end{equation*}$$

The fact that $\#\operatorname {Aut}(\Lambda )=696729600$ (which is also the order of the Weyl group of type $E_8$) then implies that $\Lambda$ is the unique unimodular and even lattice of rank 8.

In this case $E_\Lambda (\tau )$ is related to the classical Eisenstein series $E_4(\tau )\in M_4({\mathrm{SL}}_2(\mathbb{Z}))$ of weight 4 and level 1 by