From sum of two squares to arithmetic Siegel–Weil formulas
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 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:
1.3. In how many different ways can one represent as the sum of two squares?
In his book Fundamenta nova theoriae functionum ellipticarum (1829), Jacobi proved the following general formula for the representation numbers.
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 coefficients of the square of Jacobi’s theta series th
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. .
When the series ,Equation 126.96.36.199 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 . of -expansion 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 (see -expansionReference DS05, §4.5)
where is related to a special value of the Dirichlet -function.
When the series ,Equation 188.8.131.52 is not absolutely convergent, but one can still suitably modify it to obtain a modular form
with the same formula Equation 184.108.40.206 for its either using the Weierstrass -expansion, (see -functionReference DS05, §4.8) or using the analytic continuation of
to (see Reference Miy89, §7.2). In particular, when formula ,Equation 220.127.116.11 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.
To summarize, Jacobi’s Theorem 1.3.3 can be proved using the identity of two modular forms Equation 18.104.22.168, 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 22.214.171.124 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.
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.