From sum of two squares to arithmetic Siegel–Weil formulas

By Chao Li

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

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 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 1.5.1.1 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 1.5.1.1 is not absolutely convergent, but one can still suitably modify it to obtain a modular form

with the same formula Equation 1.5.1.2 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 1.5.1.2 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 1.5.1.4, 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 1.5.1.4 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 1.5.1.4.

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

Here, the factor agrees with the constant for the weight . Siegel’s formula then implies that

In particular, we recover that there are roots in the root system (see Figure 1).

2.2. Siegel–Weil formula

In a series of works (Reference Sie36Reference Sie37Reference Sie51Reference Sie52), Siegel further generalized his formula from definite to indefinite quadratic forms and from the base field to totally real fields. In the indefinite case the formula is more difficult to state, as the theta series is divergent, and it is necessary to introduce extra weight functions in the definition to ensure convergence. The situation was greatly clarified by Weil Reference Wei65 using powerful tools from representation theory, especially due to his use of the Weil representation.

Let be a quadratic space over of dimension with bilinear form . For simplicity we assume that is even (so the weight of the relevant Siegel modular forms is integral; see Remark 2.2.12). Consider the reductive dual pair , where is the symplectic group of the standard -dimensional symplectic space over and is the orthogonal group of . Let be the standard Siegel parabolic subgroup, so that under the standard basis we have

Definition 2.2.1.

Let be the ring of adèles of . We fix the standard additive character whose archimedean component is given by . The (Schrödinger model of the) Weil representation is the representation of on the space of Schwartz functions such that for any and ,

Here

is the quadratic character that corresponds to the quadratic extension , and is the discriminant of defined to be

for any -basis of .

is the normalized absolute value,

for and ,

is the Fourier transform of using the self-dual Haar measure on with respect to ,

Remark 2.2.2.

There are many nice expository articles on the Weil representation; see e.g., Reference Kud96Reference Pra93Reference Pra98. See also a more general formula of the Weil representation in Reference Kud96, Proposition 4.3).

Example 2.2.3.

When we have . The standard Siegel parabolic is the standard Borel subgroup of consisting of upper triangle matrices , and , are the diagonal and upper unipotent matrices, respectively. In this case the first two formulas in Equation 2.2.1.1 simplify to

for any , , and .

Our next goal is to use the Weil representation to construct theta series and Siegel Eisenstein series, starting from any common choice of a Schwartz function .

Definition 2.2.4.

Associated to , define the (two-variable) theta function

Then is invariant under (automorphic on both and in a broad sense).

Remark 2.2.5.

Using as an integral kernel allows one to lift automorphic forms on to automorphic forms on (and vice versa): for a cuspidal automorphic representation of and , define the theta lift of to by the Petersson inner product on ,

Then is an automorphic form on . This may be viewed as the starting point of the modern theory of theta correspondence, which is indispensable in the study of automorphic forms and the Langlands correspondence. We refer to Gan Reference Gan14 for an excellent recent survey on theta correspondence.

Example 2.2.6.

Assume that is positive definite, then the theta integral

(or in other words, the theta lift of the constant function on to ) is closely related to Siegel’s theta series. More precisely, for a lattice over , we take the Schwartz function such that

is the characteristic function of ,

is the standard Gaussian function .

For , we consider , where such that . Then . By Equation 2.2.1.1 we have

Define the classical theta integral

Then it recovers the weighted average of theta series in Equation 2.1.3.1 (see e.g., Reference Han13, §4.6, Reference KR14, §7).

In fact, let be the stabilizer of , then we have a bijection

Let be a complete set of representatives of and let be the corresponding representatives of under this bijection. Then

Using , each summand evaluates to

Unfolding the definition, the second integral equals

which by our choice of evaluates to

Thus, combining with Equation 2.2.6.2, we know that the classical theta integral Equation 2.2.6.3 evaluates to

Finally notice that

thus if we normalize the Haar measure such that , then the classical theta integral Equation 2.2.6.3 recovers the weighted average of theta series in Equation 2.1.3.1.

Definition 2.2.7.

Associated to , also define the Siegel Eisenstein series

where

is the standard Siegel–Weil section of the degenerate principal series representation of and

Here we write under the Iwasawa decomposition for the standard maximal open compact subgroup of , and the quantity is well defined.

Example 2.2.8.

Similarly, define the classical Siegel Eisenstein series

When is positive definite and is chosen as in Example 2.2.6, the special value at essentially recovers in Equation 2.1.3.1.

For example, consider the case , and equipped with the quadratic form . We have and the quadratic character corresponds to the quadratic extension , and hence corresponds to the Dirichlet character . For , we have

For , one can compute that the Siegel–Weil section evaluates to

using Equation 2.2.6.2 together with Equation 2.2.3.1 (or the more general Reference Kud96, Proposition 4.3). Comparing with Equation 1.5.1.3, we see that the classical Eisenstein series

at essentially recovers the Eisenstein series of weight 1 (up to a nonzero constant).

The Siegel Eisenstein series converges absolutely when . It has a meromorphic continuation to and satisfies a functional equation relating (i.e., centered at ). The Siegel–Weil formula gives a precise identity of the form

theta integral value of Siegel Eisenstein series at .

Notice that is the unique point such that the map in Equation 2.2.7.1 is -equivariant, so that both sides of the identity at least have the same transformation behavior with respect to the Weil representation.

Theorem 2.2.9 (Siegel–Weil formula Reference Wei65Reference KR88aReference KR88b).

Let be the dimension of a maximal isotropic subspace of . If (i.e., is anisotropic) or and , then is holomorphic at and

where if or otherwise. Here the Haar measure is normalized so that .

Example 2.2.10.

When is positive definite (so ), the Siegel–Weil formula recovers Siegel’s formula (see Examples 2.2.8 and 2.2.6), and in particular recovers Jacobi’s formula as a special case of , , (see Example 2.1.4).

Remark 2.2.11.

The condition in Theorem 2.2.9 is known as Weil’s convergence condition, which ensures the convergence of the theta integral. It has been a long effort, starting with the work of Kudla–Rallis Reference KR94, to generalize the Siegel–Weil formula outside the convergence range and for all reductive dual pairs of classical groups. We refer to Gan–Qiu–Takeda Reference GQT14 for the most general Siegel–Weil formula and a nice summary of its literature and history.

Remark 2.2.12.

When is odd, the theta series and Eisenstein series have half integral weights , and are automorphic forms on the metaplectic cover of . In this case the Weil representation needs to be modified to be a representation of and the Siegel–Weil formula still holds after modification.

3. Geometric Siegel–Weil formula

In this section we discuss an example of the Siegel–Weil formula in the indefinite case originating from the classical work of Hurwitz, and we use it to motivate the more general geometric Siegel–Weil formula.

3.1. Hurwitz class number relation

Definition 3.1.1.

For any positive integer , the Hurwitz class number is defined to be the weighted size of -equivalence classes of positive definite binary quadratic forms

Here the forms equivalent to () and () are counted with multiplicities and , respectively, due to extra symmetry.

Example 3.1.2.
Example 3.1.3.

Table 1 lists the first few Hurwitz class numbers.

Example 3.1.4.

When is a fundamental discriminant and , the Hurwitz class number is equal to the class number of the imaginary quadratic field .

Understanding these class numbers remains a central subject in algebraic number theory. The following remarkable formula, which we call the Hurwitz class number relation or the Hurwitz formula, gives an elementary expression for a certain sum of Hurwitz class numbers.

Theorem 3.1.5 (Kronecker Reference Kro60, Gierster Reference Gie83, Hurwitz Reference Hur85).

If is not a perfect square, then

Example 3.1.6.

When , the Hurwitz class number relation says

When , the Hurwitz class number relation says

A quite nontrivial way to decompose the integers 6 and 10, respectively!

Hurwitz Reference Hur85 proved this formula using the modular -invariant and the modular polynomial of level (which defines the modular curve over ). He recognized the left-hand side (LHS) of Equation 3.1.5.1 as and computed this degree in a different way involving Hurwitz class numbers to arrive at formula Equation 3.1.5.1.

3.2. A geometric proof

(cf. Reference GK93). From the modern point of view, we have a nice geometric interpretation of Hurwitz’s proof, in terms of the geometry of the modular curve

The modular curve is the moduli space of elliptic curves:

which allows one to define a canonical model of as an algebraic curve over . Each elliptic curve has a Weierstrass equation

The -invariant

only depends on the isomorphism class of and gives rise to an isomorphism

Definition 3.2.1.

Define the surface to be the product of two modular curves,

which is the moduli space of pairs of elliptic curves . For each positive integer , we define the modular correspondence over the surface by

parameterizing a pair of elliptic curves together with a degree isogeny .

The isogeny imposes one nontrivial condition, and thus defines a divisor on . For example, when the modular correspondence is nothing but the diagonally embedded modular curve

Given two divisors and on the surface , one expects that the intersection should be zero dimensional. When this is the case (i.e., when and intersect properly), we obtain a geometric intersection number by counting the number of intersection points weighted by intersection multiplicities (see Figure 2).

A curious observation comes: the geometric intersection number is equal to the LHS of Equation 3.1.5.1,

In fact, under the isomorphism Equation 3.2.0.1 we know that has a natural compactification

Since , the well-known cohomological equivalence on ,

then implies that

where the last equality comes from counting the number of degree isogenies with a fixed source (resp., target) elliptic curve. The desired identity Equation 3.2.1.1 then follows from subtracting the contribution at .

On the other hand, via the moduli interpretation we have

When is a perfect square, we know that contains (by considering the multiplication-by- isogeny ), and thus and do not intersect properly. However, when is not a perfect square (as assumed in Theorem 3.1.5), has to be an elliptic curve with complex multiplication by an imaginary quadratic order , where , and thus and do intersect properly. Using the theory of complex multiplication, counting the weighted number of such elliptic curves with complex multiplication exactly gives the sum of Hurwitz class numbers as the right-hand side (RHS) of Equation 3.1.5.1,

Combining Equation 3.2.1.1 and Equation 3.2.1.2 completes our sketch of the geometric proof of the Hurwitz formula (see Gross–Keating Reference GK93 for complete details).

3.3. Hurwitz formula as a geometric Siegel–Weil formula

The Hurwitz class numbers appearing in the Hurwitz formula Equation 3.1.5.1 also naturally appear as Fourier coefficients of Siegel Eisenstein series. More precisely, consider the Siegel Eisenstein series on of weight 2,

Then we have (up to a normalizing constant)

Notice that the condition neatly translates to the condition .

We summarize our discussion with the following diagram.

In this way the Hurwitz formula can be viewed as a geometric Siegel–Weil formula for the pair , where one replaces the theta integral on by a geometric theta series, i.e., the generating series of geometric intersection numbers of modular correspondences for the surface ,

geometric theta series on value of Siegel Eisenstein series on at .

Notice here the natural appearance of the product of modular curves due to the exceptional isomorphism . This geometric Siegel–Weil formula further computes a more general geometric intersection number as the sum of with .

Remark 3.3.1.

The remarkable discovery that generating series involving intersection numbers of cycles are modular originates from the work of Hirzebruch–Zagier Reference HZ76 on Hilbert modular surfaces. Historically Reference HZ76 was the primary motivation in Kudla’s work discussed below (cf. the introduction of Reference KM90Reference Kud97a) and also in the work of Gross–Kohnen–Zagier Reference GKZ87. See Example 3.5.1 and Remark 3.5.5(i), (ii).

3.4. Orthogonal Shimura varieties

Kudla proved a more general geometric Siegel–Weil formula by replacing the surface by an orthogonal Shimura variety of arbitrary dimension. Our next goal is to discuss Kudla’s formula.

Let be a totally real number field. Pick a real place of . Let be a quadratic space over of dimension such that for any place of ,

Let , which sits in an exact sequence

Associated to any open compact subgroup , we have a GSpin Shimura variety , which has a smooth canonical model of dimension over the reflex field (viewed as a subfield of via the embedding induced by the place ) and admits complex uniformization

Here is the hermitian symmetric domain of oriented negative 2-planes in . Unfolding the definition we may rewrite as a disjoint union of quotients of by congruence subgroups of (see Reference Kud04, §1). The Shimura variety is quasi-projective, and it is projective when is anisotropic (e.g., when , by the signature condition).

Remark 3.4.1.

One technical reason that one would prefer to work with the Shimura variety associated to (instead of ) is that it is of Hodge type (instead of abelian type), and it admits an embedding into a Siegel modular variety (the moduli space of polarized abelian varieties) of larger dimension.

Example 3.4.2 (cf. Reference Kud04Reference HP14Reference FH00).

Consider . Via accidental isomorphisms between and classical groups of symplectic type in low ranks, the Shimura varieties recover many classical modular varieties in low dimensions (see Table 2, where is a quaternion algebra over and is a real quadratic field). When , is also closely related to the moduli space of polarized K3 surfaces and has proved to be useful for studying the arithmetic of K3 surfaces.

3.5. Kudla’s generating series of special cycles and the modularity conjecture

The Shimura variety is equipped with special divisors generalizing in the case . Via an embedding into a Siegel modular variety, parameterizes certain polarized abelian varieties together with a special endomorphism of degree (see Reference MP16, §5).

Example 3.5.1.

The special divisor recovers

Heegner points on modular curves and Shimura curves when (cf. Reference Kud04, Appendix),

modular correspondences on considered in §3.2 when and is split,

Hirzebruch–Zagier cycles Reference HZ76 on Hilbert modular surfaces when and is nonsplit.

More generally, for any with , its orthogonal complement has rank . The embedding defines a Shimura subvariety of codimension 1,

For any with , there exists and such that . Define the special divisor

to be the -translate of . For any with , define the special cycle (of codimension )

Here denotes the fiber product over . More generally, for a -invariant Schwartz function and , define the weighted special cycle

Here is the Chow group of algebraic cycles of codimension on (up to rational equivalence). With extra care, we can also define for any (see Reference Kud04Reference YZZ09).

Definition 3.5.2.

Define Kudla’s generating series of special cycles,

as a formal sum valued in , where

Remark 3.5.3.

Analogous constructions of special divisors and Kudla’s generating series also apply to Shimura varieties of unitary type associated to hermitian spaces with signature at one archimedean place and signature at all other archimedean places (see Reference Liu11a). These Shimura varieties of orthogonal/unitary type can be naturally viewed as Shimura varieties associated to totally definite incoherent quadratic/hermitian spaces (Reference Z19Reference Gro20); see §6.5. In the unitary case, we obtain a generating series of the form

where we replace positive semidefinite symmetric matrices by positive semidefinite hermitian matrices , and we replace Siegel’s half-space by the hermitian half-space

We may view as a geometric theta series, now valued in Chow groups for cycles of arbitrary codimension . The analogy to Siegel’s theta series Equation 2.1.1.1 and the theta integral Equation 2.2.6.1 leads to Kudla’s modularity conjecture:

Conjecture 3.5.4 (Kudla’s modularity).

The formal generating series converges absolutely and defines a modular form on of weight valued in .

Remark 3.5.5.
(i)

The analogous modularity in Betti cohomology, i.e., the modularity of the generating series valued in defined by the image of under the cycle class map

is known by the classical work of Kudla–Millson Reference KM90. The special case of special divisors on Hilbert modular surfaces dates back to Hirzebruch–Zagier Reference HZ76 (see also Funke–Millson Reference FM14).

(ii)

Kudla’s modularity conjecture was originally formulated for orthogonal Shimura varieties over (Reference Kud97aReference Kud04). In this case, Borcherds Reference Bor99 proved the conjecture for the divisor case (the special case of Heegner points on modular curves dates back to the classical work of Gross–Kohnen–Zagier Reference GKZ87). Zhang Reference Zha09 proved the modularity for general assuming the absolute convergence of the series. Bruinier–Westerholt-Raum Reference BWR15 proved the desired convergence and hence established Kudla’s modularity conjecture for orthogonal Shimura varieties over . More recently, Bruinier–Zemel Reference BZ22 have extended the modularity to toroidal compactifications of orthogonal Shimura varieties when .

(iii)

For orthogonal Shimura varieties over totally real fields, Yuan–Zhang–Zhang Reference YZZ09 proved the modularity for (see also Bruinier Reference Bru12 for a different proof) and reduced the case to the convergence.

(iv)

Conjecture 3.5.4 in the unitary case was formulated by Liu Reference Liu11a, who also proved the case and reduced the case to the convergence. Recently Xia Reference Xia21 proved the desired convergence when for (in the notation of §6.5), and thus established Conjecture 3.5.4 in these cases.

(v)

Kudla Reference Kud04, Problem 4 also proposed the modularity problem in the arithmetic Chow group of a suitable (compactified) integral model of (see Reference GS90Reference BGKK07 and also Reference Sou92). The problem seeks to define canonically an explicit arithmetic generating series valued in which lifts under the restriction map

and such that is modular. When , this arithmetic modularity was proved by Howard–Madapusi Pera Reference HMP20 (orthogonal groups over ) and Bruinier–Howard–Kudla–Rapoport–Yang Reference BHK20 (unitary groups over ). Several low dimensional cases were also proved:

Shimura/modular curves (Kudla–Rapoport–Yang Reference KRY06, Sankaran Reference San14, and Du–Yang Reference DY19),

Hilbert modular surfaces (Bruinier–Burgos Gil–Kühn Reference BBGK07),

Product of modular curves (Berndt–Kühn Reference BK12Reference BK12a).

We also mention the arithmetic modularity of the difference of two arithmetic theta series by Ehlen–Sankaran Reference ES18 for (unitary groups over ), the almost arithmetic modularity by Mihatsch–Zhang Reference MZ21, Theorem 4.3 for (unitary groups over totally real fields ), the arithmetic modularity of Fourier–Jacobi coefficients for general by Sankaran Reference San20 (anisotropic orthogonal groups), and several striking recent works involving applications of arithmetic modularity Reference AGHM18Reference Zha21Reference SSTT22.

(vi)

Finally, we mention several recent works on the modularity conjecture for more general classes of orthogonal and unitary Shimura varieties (indefinite at more than one archimedean place) by Rosu–Yott Reference RY20, Kudla Reference Kud21, and Maeda Reference Mae21Reference Mae22.

3.6. Kudla’s geometric Siegel–Weil formula

In the special case , the generating series in Equation 3.5.2.1 is valued in (i.e., the Chow group of 0-cycles). When is projective, composing with the degree map

we obtain a generating series valued in . Its terms encode geometric intersection numbers between special divisors on generalizing the case considered in §3.3. Kudla proved the following remarkable geometric version of the Siegel–Weil formula (by analogy with Theorem 2.2.9 specialized to the case and so ).

Theorem 3.6.1 (Kudla’s geometric Siegel–Weil formula Reference Kud97a, Corollary 10.5).

Assume that is projective (i.e., is anisotropic). Take . Then for any the following identity holds (up to a nonzero constant depending only on choices of measures)

Here is a certain Schwartz function constructed from the Kudla–Millson Schwartz form (Reference KM86).

Thus Kudla’s geometric Siegel–Weil formula is a precise identity of the form

geometric theta series on value of Siegel Eisenstein series on at .

Remark 3.6.2.

Kudla in fact proved a geometric Siegel–Weil formula for the generating series of special cycles of all dimensions, i.e., without assuming . Here and in the next section we focus on the case of 0-cycles and refer to Kudla’s excellent surveys Reference Kud04Reference Kud02bReference Kud02a for the general case.

4. Arithmetic Siegel–Weil formula

In this section we discuss an arithmetic version of the Siegel–Weil formula. Parallel to the previous section, we will use an example in the case considered by Gross–Keating to motivate the more general case.

4.1. Gross–Keating formula

Gross–Keating took the geometric point of view of the Hurwitz class number relation and found a remarkable generalization for arithmetic intersection numbers. As the moduli space of elliptic curves, the modular curve has a canonical integral model over such that . The integral model is an arithmetic surface fibered over the arithmetic curve , and its fiber above is a smooth curve in characteristic (see Figure 3).

Analogously, the surface has a canonical integral model over , which is an arithmetic threefold. The modular correspondence naturally extends to a divisor . Now on the arithmetic threefold , we need three (instead of two) divisors so that the intersection has the expected dimension 0. Define the arithmetic intersection number by

where encodes the intersection number supported in the fiber . This definition makes sense when these three divisors intersect properly (or more generally when their intersection is supported in finitely many fibers ).

Is there a formula for this arithmetic intersection number analogous to Equation 3.3.0.1? The index set for should be to put the three diagonal entries in , and thus we need to look at the Siegel Eisenstein series on of weight 2 (instead of on ). Then the relevant special point in the Siegel–Weil formula is , the central point. However, it is too naive to expect to be equal to : the arithmetic intersection number involves a -linear combination of ’s and hence is no longer a rational number like in Equation 3.3.0.1. Moreover, the Eisenstein series turns out to have an odd functional equation at the center , and hence automatically! This automatic vanishing suggests that it would be interesting to look at its first derivative at .

After these two appropriate modifications—replacing by and replacing the value at by the central derivative at —it turns out that we do have the following remarkable formula relating the arithmetic intersection numbers to (see a nice exposition of the proof in Reference VGW+07).

Theorem 4.1.1 (Gross–Keating Reference GK93, Gross–Kudla–Zagier).

Assume there is no positive definite binary quadratic form representing simultaneously. Then (up to an explicit constant)

Remark 4.1.2.

The assumption on is analogous to the assumption that is not a perfect square in Theorem 3.1.5, which guarantees that the three divisors intersect properly.

4.2. Arithmetic Siegel–Weil formula

Parallel to Equation 3.3.0.1, the Gross–Keating formula can be viewed as an arithmetic Siegel–Weil formula for the pair , where one replaces the theta integral on by a generating series of arithmetic intersection numbers on the arithmetic threefold ,

arithmetic theta series on central derivative of Siegel Eisenstein series on .

Of course there is nothing stopping us from considering the higher dimensional case. In fact Kudla (Reference Kud97b) and Kudla–Rapoport (Reference KR99Reference KR00aReference KR14) proposed vast conjectural generalizations of the Gross–Keating formula by

(1)

taking to be a suitable regular integral model of the Shimura variety of orthogonal/unitary type considered in §3.43.5 associated to quadratic/hermitian spaces.

(2)

defining suitable integral models of the special divisors , and more generally integral models for special divisors .

Remark 4.2.1.

Such a regular integral model is known to exist for certain level subgroups defining (cf. Reference LZ22a, §14 in the unitary case). To avoid technicalities, we will be vague about the choice of level structures and integral models, but see Remark 5.3.2(iii), (iv).

Now take (which equals ) so that the intersection of special divisors has expected dimension 0. We have a natural decomposition

index by symmetric/hermitian matrices with diagonal entries . Here denotes the fiber product over . When , it turns out that is supported in finitely many fibers , and we have a well-defined -part of the arithmetic intersection number

Now we are ready to state the conjecture on an arithmetic version of the Siegel–Weil formula (by analogy with Theorem 2.2.9 in the orthogonal case specialized to and so is the central point), which is known as the Kudla–Rapoport conjecture in the unitary case (Reference KR14, Conjecture 11.10).

Conjecture 4.2.2 (Arithmetic Sigel–Weil formula, nonsingular part).

Take . Then for any (resp., ) in the orthogonal (resp., unitary) case with diagonal entries , the following identity holds (up to a nonzero constant depending only on choices of measures).

Remark 4.2.3.

In general the special divisors do not intersect properly, and a more sophisticated definition of the arithmetic intersection numbers is needed (cf. Definition 5.1.3). In particular, with the correct definition the conjecture works even for improper intersections.

Thus the arithmetic Siegel–Weil formula is a precise conjectural identity of the form

arithmetic theta series on central derivative of Siegel Eisenstein series on ().

Now we can state one of the main results of Reference LZ22a (see Reference LZ22a, Theorem 1.3.1 for more precise technical assumptions).

Theorem 4.2.4 (Li and Zhang Reference LZ22a).

Conjecture 4.2.2 holds for arbitrary in the unitary case.

Our recent work with Zhang Reference LZ22b has also established a slightly weaker semi-global (at a good odd prime ) version of Conjecture 4.2.2 in the orthogonal case. We will discuss some key ideas of the proof in §5.

Remark 4.2.5.
(i)

When and is split (orthogonal case), the arithmetic Siegel–Weil formula recovers the Gross–Keating formula, and also treats the case of improper intersections (see Remarks 4.1.2, 4.2.3).

(ii)

Theorem 4.2.4 was previously proved when by Terstiege Reference Ter11Reference Ter13 and when the intersection is zero dimensional (Reference GK93Reference KR99Reference KR00aReference KR11Reference BY21).

(iii)

There is also an archimedean part of the arithmetic Siegel–Weil formula, relating archimedean arithmetic intersection numbers with the nonsingular but indefinite Fourier coefficients of . These Fourier coefficients are nonholomorphic, unlike the positive definite Fourier coefficients in Conjecture 4.2.2. This archimedean arithmetic Siegel–Weil formula was proved by Liu Reference Liu11a (unitary case), and Garcia–Sankaran Reference GS19 in full generality (see also Bruinier–Yang Reference BY21 for an alternative proof in the orthogonal case).

(iv)

Kudla conjectured that there should also be a singular part of the arithmetic Siegel–Weil formula, relating the singular Fourier coefficients of to certain arithmetic intersection numbers. However the singular part is more difficult to prove, or even to formulate precisely, cf. Reference Kud04, Problem 6. As a special case, the constant term of the arithmetic Siegel–Weil formula should roughly relate the arithmetic volume of to logarithmic derivatives of Dirichlet -functions. Such an explicit arithmetic volume formula was proved by Hörmann Reference Hör14 (orthogonal case) and Bruinier–Howard Reference BH21 (unitary case), though a precise comparison with the constant term of is yet to be formulated and established.

(v)

Ideally, putting all singular/nonsingular and archimedean/nonarchimedean parts together, one should arrive at a full arithmetic Siegel–Weil formula of the form in complete analogy to Equation 3.6.1.1,

Here is the conjectural arithmetic theta series in Remark 3.5.5(v), is the arithmetic degree map

and is the standard Gaussian function on the totally positive definite space over . The full arithmetic Siegel–Weil formula was established by Kudla, Rapoport, and Yang (Reference KRY99Reference Kud97bReference KR00bReference KRY06) for (orthogonal case) in great generality. However, it remains an open problem to formulate such a precise full arithmetic Siegel–Weil formula in higher dimensions.

(vi)

Recently Feng–Yun–Zhang Reference FYZ21 proved a higher Siegel–Weil formula over function fields for unitary groups, which relates nonsingular coefficients of the th derivative of Siegel Eisenstein series and intersection numbers of special cycles on moduli spaces of Drinfeld shtukas with legs. The case (resp., ) can be viewed as an analogue of the Siegel–Weil formula (resp., the arithmetic Siegel–Weil formula). Over function fields, the possibility of relating higher derivatives of analytic objects to intersection numbers was first discovered by Yun–Zhang Reference YZ17Reference YZ19 in the context of the higher Gross–Zagier formula. Over number fields, however, no analogue of such a higher Siegel–Weil formula (resp., higher Gross–Zagier formula) is currently known when . Feng–Yun–Zhang Reference FYZ21a also defined higher theta series over function fields (including all singular terms) and conjectured their modularity. Previously, an arithmetic Siegel–Weil formula over function fields was proved by Wei Reference Wei19 for special cycles on moduli spaces of Drinfeld modules of rank 2 with complex multiplication (analogue of the special case ).

5. Local arithmetic Siegel–Weil formula

In order to prove the arithmetic Siegel–Weil formula (Conjecture 4.2.2), one first notices that it can be reduced to a local identity:

(1)

Geometric side (LHS): The arithmetic intersection numbers corresponding to nonsingular matrices can be computed as a sum indexed by primes of (or the ring of integers in general). The local term at a finite prime can be further reduced to an arithmetic intersection on a Rapoport–Zink space , which is a local analogue of Shimura varieties over (or a completion of in general), via the theory of -adic uniformization of Shimura varieties (Reference RZ96).

(2)

Analytic side (RHS): The nonsingular Fourier coefficients has a product expansion indexed by primes of , and thus the derivative can also be written as a sum indexed by the primes of . The term indexed by a finite prime can be further reduced to the derivative of the local representation density of quadratic/hermitian forms over .

This reduction step is illustrated in the following diagram.

The conjectural local identity on the bottom is known as the local arithmetic Siegel–Weil formula. The local arithmetic Siegel–Weil formula has been recently proved in our work with Zhang Reference LZ22a (resp., Reference LZ22b) in the unitary (resp., orthogonal) case. Next we will make this local conjecture more precise in the unitary case. In the unitary case, this reduction step was made precise by Kudla–Rapoport (Reference KR11, Reference KR14), and the local conjecture is also known as the local Kudla–Rapoport conjecture.

5.1. Geometric side

Let be an odd prime. Let be a finite extension of with residue field and uniformer . Let be the unramified quadratic extension of (e.g., ). Associated to this datum, we have the unitary Rapoport–Zink space :

It is a formal scheme over of relative dimension , parameterizing deformations (up to quasi-isogenies) of a fixed -hermitian formal -divisible group of relative height , dimension , and signature . Here is the completion of the maximal unramified extension of .

The space of special homomorphisms

has a structure of a (nonsplit) -hermitian space of dimension , coming from the principal polarization on .

The unitary group naturally acts on via the action on .

Each vector gives rise to a special divisor or Kudla–Rapoport (KR) divisor , defined to be the locus where the homomorphism deforms. This is the local analogue of the special divisor considered in Conjecture 4.2.2.

The Rapoport–Zink space is formally smooth over , but its geometric structure is rather complicated. For example, is highly nonreduced: the reduced subscheme has dimension , near the middle dimension of . The structure of was studied by Vollaard–Wedhorn (Reference VW11), and they showed that has a nice stratification into smooth varieties, known as the Bruhat–Tits stratification. Each closed stratum of the Bruhat–Tits stratification is isomorphic to a smooth projective variety of dimension over (), and the incidence relation between the closed strata resembles the combinatorial structure of the Bruhat–Tits building for quasi-split unitary groups over . Here each is a generalized Deligne–Lusztig variety associated to the unitary group over .

Example 5.1.1.

Take . Then , and .

Example 5.1.2.

Take . Then has relative dimension 2 over , while the reduced subscheme has dimension 1 (see Figure 4). In this case only two types of Deligne–Lusztig variety show up:

(1)

, a single point.

(2)

, the Fermat curve of degree .

And is an infinite tree, where

(1)

the number of containing a given is exactly .

(2)

the number of contained in a given is exactly .

Definition 5.1.3.

Let be an -lattice of rank . Let be an -basis of . Define the special cycle or Kudla–Rapoport (KR) cycle

Define the arithmetic intersection number

where denotes the Euler–Poincaré characteristic, denotes the structure sheaf of the Kudla–Rapoport divisor , and denotes the derived tensor product of coherent sheaves on . It is known (by Terstiege Reference Ter13, as extended in Reference LZ22a, Corollary 2.8.2 or by Howard Reference How19) that is independent of the choice of the basis and hence is a well-defined invariant of itself, justifying the notation.

Remark 5.1.4.

When the intersection is zero dimensional, we have

and thus is nothing but the -length of (which equals the sum of intersection multiplicities at all points). Even though is the intersection of divisors in a -dimensional formal scheme, in general may be fail to have the expected dimension 0 due to improper intersection. In this case, the derived intersection is needed so that the intersection number is well behaved.

Example 5.1.5.

Take and (the hermitian form with respect to an -basis of is ). In this case the intersection is not zero dimensional, and in fact

The arithmetic intersection number turns out to be its topological Euler characteristic

5.2. Analytic side

Definition 5.2.1.

Let be two hermitian -lattices of rank , , respectively. Let be the -scheme such that for any -algebra ,

where denotes the set of hermitian module homomorphisms. The local density of representations of by is defined to be

It gives a quantitative measure of “how many different ways” one can embed into as a hermitian submodule.

Example 5.2.2.

Consider , the rank lattice with hermitian form given by the identity matrix. Then one can compute that

One can recognize it as the number , where is the unitary group defined over , or in fancier language, as the local -factor of the Gross motive Reference Gro97 of the quasi-split unitary group in variables.

The local density has nice compatibility when replacing by for . More precisely, it is known (Reference Hir98, Theorem II) that is a polynomial in with -coefficients.

Example 5.2.3 (Reference KR11, p.677).
Definition 5.2.4.

Define the normalized Siegel series such that

These polynomials play an important role in computing the Fourier coefficients of Siegel Eisenstein series, and many works are devoted to proving more explicit formulas for them (see, e.g., Reference Kit83Reference Kat99Reference Hir98Reference Hir12Reference IK22Reference CY20). The local Siegel series satisfies a functional equation (Reference Hir12, Theorem 5.3)

Here is the valuation of . This may be viewed as a local analogue of the functional equation of Eisenstein series.

Definition 5.2.5.

If is odd (equivalently, is a nonsplit hermitian space), then by the functional equation Equation 5.2.4.1. In this case, define the central derivative of the local density by

5.3. Local arithmetic Siegel–Weil formula

Now we are ready to state the local arithmetic Siegel–Weil formula in the unitary case, originally conjectured by Kudla–Rapoport Reference KR11, Conjecture 1.3.

Theorem 5.3.1 (Local arithmetic Siegel–Weil formula, Li and Zhang Reference LZ22a, Theorem 1.2.1).

Let be an -lattice of full rank . Then

Remark 5.3.2.
(i)

The theorem was proved by Kudla–Rapoport Reference KR11 for and Terstiege Reference Ter13 for .

(ii)

The local arithmetic Siegel–Weil formula is proved for the orthogonal case in our work with Zhang Reference LZ22b for arbitrary . The case was previously proved by Gross–Keating Reference GK93 and Terstiege Reference Ter11.

(iii)

The local arithmetic Siegel–Weil formula is proved when the quadratic extension is ramified for exotic smooth models in our work with Liu Reference LL21 for arbitrary even , and for Krämer models⁠Footnote2 by Shi and by He–Shi–Yang Reference Shi20Reference HSY20 for .

2

During the refereeing process of this article, He–Shi–Yang Reference HSY21 formulated a conjectural local arithmetic Siegel–Weil formula for Krämer models for arbitrary and proved it for the case . The case for arbitrary has been proved in our work with He–Shi–Yang Reference HLSY22.

(iv)

It is more difficult to prove or formulate the local arithmetic Siegel–Weil formula in the presence of more general level structures (even when the quadratic extension is unramified). In the unitary case Reference LZ22a formulates and proves a local arithmetic Siegel–Weil formula when the level is the parahoric subgroup given by the stabilizer of an almost self-dual lattice (the case was previously proved by Sankaran Reference San17). Recently Cho Reference Cho20 proposed a general formulation for all minuscule parahoric levels in the unitary case.

Example 5.3.3.

Take and . Specializing the formula of Cho–Yamauchi Reference CY20 (extended to the unitary case in Reference LZ22a, Theorem 3.5.1) gives

It satisfies the functional equation

It is easy to compute

So combining Example 5.1.5 we obtain in this case! It is miraculous that the purely analytic quantity secretly knows about the Euler characteristic of the Deligne–Lusztig curve .

5.4. Strategy of the proof: uncertainty principle

The previously known special cases of Theorem 5.3.1 were proved via explicit computation of both the geometric and analytic sides. Explicit computation seems infeasible for the general case. The proof in Reference LZ22a instead proceeds via induction on using the uncertainty principle, a standard tool from local harmonic analysis. Even for , this proof is different from the previous proofs.

More precisely, for a fixed -lattice of rank , consider functions on ,

Then it remains to show the equality of the two functions

By the definition of and , both functions are easily seen to vanish when is nonintegral, i.e., . Here denotes the valuation of the norm of . By utilizing the inductive structure of Rapoport–Zink spaces and local densities, it is not hard to see that if with , then

for the lattice of full rank . Thus by induction on , the difference function already vanishes on a large subset

We would like to deduce that indeed vanishes identically. To this end, we apply the following uncertainty principle.

Proposition 5.4.1 (Uncertainty Principle, Reference LZ22a, Proposition 8.1.6).

Let be a Schwartz function on . If both and its Fourier transform vanish on , then .

In other words, cannot simultaneously have small support unless . Applying the uncertainty principle to the difference function , we can then finish the proof as long as we get a good control over the support of . However, both functions have singularities along the hyperplane . Intuitively, if is closer to , then and intersect more improperly, which results in the blowup of along . These singularities cause trouble in computing the Fourier transforms or even in showing that .

5.5. Strategy of the proof: decomposition and local modularity

To overcome this difficulty, we isolate the singularities by decomposing

into horizontal and vertical parts. Here on the geometric side (resp., ) is the contribution from the horizontal (resp., vertical part) of the KR cycles, illustrated in Figure 5 (by the red (resp., blue) part in the online color version of this article). One can hope to understand the horizontal part explicitly using deformation theory, and the vertical part using algebraic geometry over the residue field .

By the uncertainty principle and the induction on , it remains to prove the following.

Theorem 5.5.1 (Key Theorem).
(KR1)

.

(KR2)

 and .

(KR3)

 and  .

In other words, the horizontal part matches , and subtracting the horizontal parts removes the singularities along so that vertical parts are indeed in . The extra invariance under Fourier transform,

can be thought of as a local modularity, by analogy with the global modularity of arithmetic generating series (such as in Bruinier–Howard–Kudla–Rapoport–Yang Reference BHK20 discussed in Remark 3.5.5(v)) encoding an extra global -symmetry.

5.6. Ingredients of the Key Theorem

Some key ingredients of the proof of the Key Theorem 5.5.1 include the following.

(KR1)

We describe explicitly the horizontal part of KR cycles in terms of Gross’s quasi-canonical liftings Reference Gro86, using the work of Tate, Grothendieck–Messing and Breuil on the deformation theory of -divisible groups.

(KR2)

On the geometric side we show Equation 5.5.1.1 by reducing to the case of intersection with Deligne–Lusztig curves . This reduction requires the Bruhat–Tits stratification of into the Deligne–Lusztig varieties , as discussed in §5.1, and the Tate conjecture Reference Tat94 for these Deligne–Lusztig varieties. We prove the latter by reducing to a cohomological computation of Lusztig Reference Lus76.

(KR3)

On the analytic side we show Equation 5.5.1.1 using Cho–Yamauchi’s explicit formula Reference CY20 for in terms of weighted lattice counting, and reduce to a (rather subtle) lattice theoretic problem. (In fact we only show directly something weaker than Equation 5.5.1.1, which is enough to imply Theorem 5.3.1, and we then deduce Equation 5.5.1.1 a posteriori.)

6. Arithmetic inner product formula

In this last section we discuss an application of the arithmetic Siegel–Weil formula to the Beilinson–Bloch conjecture and the arithmetic inner product formula.

6.1. Birch and Swinnerton-Dyer conjecture

One long-standing problem in number theory is the determination of the rational points for an elliptic curve defined over . The celebrated Birch and Swinnerton-Dyer (BSD) conjecture predicts a deep link between and its -function . Define the algebraic rank

to be the rank of the finitely generated abelian group . Define the analytic rank

to be the order of vanishing of at the central point . The BSD conjecture predicts the rank equality between these two notions of ranks of seemingly different nature,

and a refined BSD formula

for the Taylor expansion of at (here ) in terms of various important arithmetic invariants of . Among these invariants are the order of the (mysterious) Tate–Shafarevich group , and the regulator , where

is the Néron–Tate height pairing and is a basis of the free part of .

The BSD conjecture is still widely open in general, but much progress has been made in the rank 0 or 1 case. The seminal work of Gross–Zagier Reference GZ86, Kolyvagin Reference Kol90 proved the implications

confirming Equation 6.1.0.1 when . Due to the work of many people, many cases of Equation 6.1.0.2 are also known when .

The key to relate and is the Gross–Zagier formula

(up to an explicit nonzero constant, including the period of and other rational factors) relating the first derivative of the -function at and the Néron–Tate height of certain rational points on known as Heegner points (cf. Example 6.4.1). It gives one crucial implication in Equation 6.1.0.3,

The tools of Heegner points and -functions, linked via the Gross–Zagier formula, are indispensable in studying the arithmetic of elliptic curves. We refer to Zhang Reference Zha14a for an excellent recent survey on Heegner points and the Birch–Swinnerton-Dyer conjecture (see also Gross Reference Gro04 and Darmon Reference Dar04).

6.2. Beilinson–Bloch conjecture

In the 1980s, Beilinson (Reference Bem̆87, Conjecture 5.9) and Bloch (Reference Blo84a, Recurring Fantasy and Reference Blo84b, Conjecture) proposed vast generalizations of the BSD conjecture to higher dimensional varieties.

Let be a smooth projective variety of dimension over a number field . For , denote by the Chow group of algebraic cycles of codimension on defined over (up to rational equivalence), and the subgroup of geometrically cohomologically trivial cycles. Denote by the motivic -function associated to the -th étale cohomology . Then the Beilinson–Bloch (BB) conjecture predicts the equality between analytic and algebraic ranks

and a refined formula

for the leading coefficient at in terms of the determinant of the Beilinson–Bloch height pairing⁠Footnote3

3

When , the Beilinson–Bloch height pairing is only defined assuming certain conjectures on algebraic cycles on (see Reference Bem̆87, Conjectures 2.2.1 and 2.2.3). This important technical issue is addressed in Reference LL21Reference LL22 so that the RHS of Equation 6.7.1.1 in Theorem 6.7.1 can be defined unconditionally, but we will intentionally ignore it for the purpose of this article.

Example 6.2.1.

When and , we recover the BSD conjecture as

The BB conjecture is even more elusive than the BSD conjecture: in general we do not know that is finitely generated, nor do we know that can be analytically continued to the central point , so neither side of Equation 6.2.0.1 is well defined! This may be more an exciting challenge than disappointment for mathematicians—after all we were in a similar circumstance when the BSD conjecture was formulated in the 1960s: we knew neither the analytic continuation of (except when has complex multiplication) nor the finiteness of in order to make sense of either side of Equation 6.1.0.2.

A good testing ground for the BB conjecture is by taking to be Shimura varieties. By Langland’s philosophy, their -functions can be computed in terms of automorphic -functions, so the analytic rank in Equation 6.2.0.1 becomes accessible. Even though we do not known if is finitely generated, it still makes sense to test if it is nontrivial. It is thus tempting to relate special cycles on Shimura varieties to automorphic -functions, and in particular, to generalize the Gross–Zagier formula Equation 6.1.0.4 to higher dimensional Shimura varieties and prove the analogue of Equation 6.1.0.5 towards the BB conjecture

Here we use the notation to stand for the nontriviality of .

6.3. Arithmetic inner product formula

As discussed in §3.5, Shimura varieties of type and admit a rich supply of special cycles, recovering Heegner points in the case of modular curves. As explained in Reference Kud04, III, when is even, the arithmetic Siegel–Weil formula together with the doubling method of Piatetski-Shapiro–Rallis Reference PSR84 has important applications to the arithmetic inner product formula of the form

Here

is a certain cuspidal automorphic representation on or .

is the central derivative the standard -function of (cf. Reference Yam14) with global root number .

is an algebraic cycle on of codimension , constructed from holomorphic forms using the method of arithmetic theta lifting (see Definition 6.6.3).

This arithmetic inner product formula can be viewed as an arithmetic analogue of the Rallis inner product formula (see Reference GQT14) of the form

which relates the central value when and the Petersson inner product of the (usual) theta lift discussed in Remark 2.2.5. It can also be viewed as a higher dimensional generalization of the Gross–Zagier formula (the case ). Thus the arithmetic Siegel–Weil formula is intimately linked to the Birch–Swinnerton-Dyer conjecture, and more generally the Beilinson–Bloch conjecture for higher dimensional Shimura varieties.

The conjectural arithmetic inner product formula was formulated by Kudla Reference Kud97b using the Gillet–Soulé height and in more generality by Liu Reference Liu11a using the Beilinson–Bloch height (in the unitary case). In the unitary case the arithmetic inner product formula has been recently proved under mild local assumptions and Kudla’s modularity conjecture in our works with Liu Reference LL21Reference LL22, which has been applied to prove the first unconditional theorem for the Beilinson–Bloch conjecture for higher dimensional Shimura varieties. Our remaining goal is to explain some details about the main results of Reference LL21Reference LL22.

6.4. Arithmetic theta lifting on modular curves

To motivate the construction of arithmetic theta lifting, let us first consider an example of Heegner points on elliptic curves.

Example 6.4.1 (cf. Reference Zag85).

Consider the curve in Cremona’s table,

It is the rank one optimal curve over of smallest conductor (). It is isomorphic to the modular curve for ( is the smallest such that has positive genus). The Mordell-Weil group , generated by . Let

be the weight 2 newform of level associated to . It gives a modular parametrization

such that . The Shimura–Waldspurger correspondence, which can be viewed as an instance of the theta correspondence for the pair ,

gives a weight 3/2 newform in Kohnen’s plus space ,

such that .

Let be a positive integer such that . If splits in , then one can construct a rational point using Heegner points on associated to quadratic orders of discriminant . For example, when is a fundamental discriminant, we have a Heegner point defined over the Hilbert class field of , and . The point may depend on the choice of , even when . In Table 3, we compute a list of Heegner points for small ’s, and also the integer multiples such that for the generator .

From Table 3, we observe the miraculous coincidence that the integer exactly matches the coefficient of ! In other words, the generating series

is a modular form valued in . This allows us to define the arithmetic theta lift by taking the Petersson inner product of with ,

which is now a canonical element in (i.e., no need to choose any particular ). The arithmetic inner product formula in this case asserts the identity

In fact, in this case we have and we can compute explicitly

Here is the real period of . Therefore

and the equality Equation 6.4.1.1 indeed holds up to an elementary constant .

6.5. Unitary Shimura varieties

Now let us come to the setting of unitary Shimura varieties. Let be a CM extension of a totally real number field. Let be a totally definite incoherent hermitian space over of rank . Here “incoherent” means that is not the base change of a global -hermitian space, or equivalently the product of the local Hasse invariants of the local hermitian spaces is

Picking any place of gives a nearby global -hermitian space such that

Associated to any open compact subgroup , we have a unitary Shimura variety (cf. Reference Z19Reference Gro20), which has a smooth canonical model of dimension over (viewed as a subfield of via the embedding induced by a place above ) and admits complex uniformization

where