From sum of two squares to arithmetic Siegel–Weil formulas

By Chao Li

In loving memory of my mother, Xiaoping Mao (1965–2022)


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 can be written as the sum of two squares?

For the first few primes we easily find that

are the sums of two squares, while other primes like are not. The answer seems to depend on the residue class of modulo 4.

Theorem 1.1.1.

A prime is the sum of two squares if and only if .

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 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 Reference Dic66, Ch. VI, pp. 227–231.

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

If () and , then either or , and hence either is also the sum of two squares or (by the quadratic reciprocity). It follows that each with must appear to an even power. On the other hand, the familiar Diophantus identity

shows that a product of integers of the form is also of the same form. Combining with Theorem 1.1.1 we obtain:

Corollary 1.2.1.

A positive integer is of the form if and only if each prime factor of appears to an even power.

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

Definition 1.3.1.

To answer this question, we naturally define the representation number

In particular, is of the form if and only if .

Example 1.3.2.

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

Theorem 1.3.3 (Jacobi).

As a byproduct, Jacobi’s formula shows that

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,

The representation numbers naturally appear as the th coefficients of the square of Jacobi’s theta series

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

which is easily seen to be equivalent to Theorem 1.3.3.

1.5. Another proof using modular forms

An alternative way of evaluating is to view , and as a holomorphic function on the upper half-plane

The function satisfies two transformation rules (see Reference Zag08, Proposition 9):

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 Reference DS05, §4.9). These rules amount to saying that

is a modular form of weight and level . Jacobi’s theta series and its variants (under the general name of theta series) form one of most important classes of modular forms.

It follows that

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

Definition 1.5.1.

Let be the (unique) nontrivial character. We define an Eisenstein series

where is understood to be 0 when .

When , the series Equation is absolutely convergent and is nonzero only when is odd. When is odd, it defines a modular form of weight , level , and character . The constant term of the -expansion of is nonzero, and we let be a scalar multiple of so the constant term is normalized to be 1. This normalized Eisenstein series then has the explicit -expansion (see Reference DS05, §4.5)

where is related to a special value of the Dirichlet -function .

When , the series Equation is not absolutely convergent, but one can still suitably modify it to obtain a modular form

with the same formula Equation for its -expansion, either using the Weierstrass -function (see Reference DS05, §4.8) or using the analytic continuation of

to (see Reference Miy89, §7.2). In particular, when , formula Equation simplifies to

As explained, must be a scalar multiple of . Since both of them have constant coefficient 1, we indeed have the equality

Comparing the coefficient before , we obtain and hence

which proves Theorem 1.3.3.

Remark 1.5.2.

As a byproduct of the proof, we also obtain from . Via the functional equation of , this is equivalent to the famous Leibniz formula for (1676):

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

theta series 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 Reference Sie35 generalizes formula Equation from the binary quadratic form to more general quadratic forms in an arbitrary number of variables. Let be a positive definite quadratic lattice over of rank with quadratic form . Denote by the associated symmetric bilinear form, defined by

(so ). Denote by the set of symmetric matrices whose diagonal entries are in and whose off-diagonal entries are in . Denote by the subset of positive semidefinite matrices.

Definition 2.1.1.

For , define the (generalized) representation number

Define Siegel’s theta series

and a holomorphic function on Siegel’s half-space

Using the Poisson summation formula, Siegel proved that is a Siegel modular form on of weight .

Example 2.1.2.

Notice that when , Siegel’s half-space recovers the usual upper half-plane , and Siegel’s theta series recovers Jacobi’s theta series

In general, the theta series for a lattice 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 :

weighted average of theta series Siegel Eisenstein series.

More precisely, recall that two quadratic lattices are in the same genus, if they are isomorphic over and over for all primes . Denote by the set of isomorphism classes of quadratic lattices in the same genus of . Denote by the automorphism group of as a quadratic lattice.

Theorem 2.1.3 (Siegel).

The following identity holds:

Here is a certain normalized Siegel Eisenstein series on of weight .

Example 2.1.4.

Consider the case , and equipped with the quadratic form . Then

In this case is a singleton and Siegel’s formula recovers Equation

Example 2.1.5 (cf. Reference 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 , defined to be weighted size

as an Euler product of local factors indexed by primes .

For example, consider the simplest case when is

unimodular, i.e., for a -basis of , and

even, i.e., 2 divides for all .

The rank of any unimodular even lattice is necessarily a multiple of 8. Siegel’s mass formula computes the mass of explicitly as

where is the Riemann zeta function, is the volume of the unit -sphere, and is the th Bernoulli number.

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

Let be the root lattice of type , defined by

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

The fact that (which is also the order of the Weyl group of type ) then implies that is the unique unimodular and even lattice of rank 8.

In this case is related to the classical Eisenstein series of weight 4 and level 1 by