Algebraicity of higher Green functions at a CM point

In this paper, we investigate the algebraic nature of the value of a higher Green function on an orthogonal Shimura variety at a single CM point. This is motivated by a conjecture of Gross and Zagier in the setting of higher Green functions on the product of two modular curves. In the process, we will study analogue of harmonic Maass forms in the setting of Hilbert modular forms, and obtain results concerning the arithmetic of their holomorphic part Fourier coefficients. As a consequence, we answer a question of Zagier in his 1986 ICM proceeding.


Introduction
Let j(z) be the modular j-invariant on the modular curve X 0 (1) := SL 2 (Z)\H with H the upper-half plane. Its values at CM points are algebraic integers called singular moduli. They play an important role in the explicit construction of class fields of imaginary quadratic fields.
1.1. Conjecture and Results. The function G 1 (z 1 , z 2 ) := 2 log |j(z 1 ) − j(z 2 )| is the automorphic Green function on X 0 (1) × X 0 (1), and the limiting member of a family of automorphic functions G s (z 1 , z 2 ) := −2 γ∈Γ Q s−1 1 + |z 1 − γz 2 | 2 2ℑ(z 1 )ℑ(γz 2 ) , ℜ(s) > 1, that are eigenfunctions with respect to the Laplacians in z 1 and z 2 . For integral parameters s = r + 1 ∈ N, these functions are called higher Green functions, and played an important role in calculating arithmetic intersections of Heegner cycles on Kuga-Sato varieties [40]. Given a weakly holomorphic modular form f = m≫−∞ c(m)q m ∈ M ! −2r on X 0 (1), one can associate to it a higher Green function  Although the theory of complex multiplication does not directly apply, the values of G r+1,f at CM points on X 0 (1) 2 should be algebraic in nature, as in the case of the automorphic Green function. More precisely, these values should be algebraic multiples of logarithm of algebraic numbers. This was conjectured in [25,Conjecture (4.4)] when z 1 , z 2 have the same discriminant, and mentioned as a question in [23, section V.1] for the general case (see also [31] and [35]). In this paper, we prove the following result, which in particular solves problem (ii) raised by Don Zagier at the end of his 1986 ICM proceeding [39].
Theorem 1.1. Let r ∈ N and f ∈ M ! −2r with integral Fourier coefficients. Suppose d 1 , d 2 are negative discriminants, such that one of them is fundamental when r is odd. For any CM point z i with discriminant d i , there exist κ ∈ N depending on d 1 , d 2 , r and f , and α = α(z 1 , z 2 ) ∈ H such that where H = H 1 H 2 with H i the ring class field extension of E i := Q( √ d i ) associated to z i . Furthermore, we have (1.1.4) α(z σ 1 , z σ 2 ) = σ(α(z 1 , z 2 )) for any σ ∈ Gal(H/E), where E = E 1 E 2 .
Remark 1.2. The group Gal(H/E) can be embedded as a subgroup of Gal(H 1 /E 1 )×Gal(H 2 /E 2 ), which then acts on the CM point (z 1 , z 2 ).
There has been a lot of previous works concerning this question. The first such result is due to Gross, Kohnen and Zagier [23], where r is even 1 and one considers average of the whole Gal(H/E)-orbit of (z 1 , z 2 ). In that case, the value is a rational multiple of the logarithm of a rational number. When E 1 = E 2 , this conjecture follows from the work of Zhang [40], under the assumption of the non-degeneracy of certain height pairing of Heegner cycles on Kuga-Sato varieties. In [35], Viazovska gave an analytic proof without this assumption. When E 1 = E 2 , Mellit [31] gave a strategy to systematically verify this conjecture with one of the points fixed, and carried it out for z 1 = i. In [29], we considered the average over the whole Gal(H/E)-orbit with r odd, and were able to show that α ∈ Q( √ d 1 d 2 ) and give an explicit factorization of the ideal it generates in the spirit of the seminal work of Gross and Zagier on singular moduli [24]. Very recently, Bruinier, Ehlen and Yang made significant progress and proved algebraicity result in the sense of Theorem 1.1 by averaging over the Galois orbit of one of the two CM points with fundamental discriminant [12]. We have now removed this averaging in Theorem 1.1 to obtain an algebraicity result at an individual CM point.
It is important to mention that one can replace SL 2 (Z) with a congruence subgroup Γ 0 (N), define higher Green functions on X 0 (N) 2 analogously, and ask the same question. This was in fact the setting that [23] and [40] were in. By viewing X 0 (N) 2 as the Shimura variety for the Q-split group O (2,2), it is natural to generalize the setting to an arbitrary orthogonal Shimura variety, and ask the question about the algebraic nature of the CM-values of higher Green functions on such varieties. This framework, which was adopted in [12], will be the one we work in.
To be more precise, let V be a rational quadratic space of signature (n, 2), and Given an even, integral lattice L ⊂ V such that K fixesL = L ⊗Ẑ and acts trivially on the finite abelian groupL ′ /L, one can associate a higher Green function Φ r L (z, h, f ) on X K to each weakly holomorphic modular form f ∈ M ! 1−n/2−2r,ρ L and r ∈ N (see Equation (3.3.4)). It has logarithmic singularity along special divisors on X K .
For a totally real field F of degree d, a quadratic CM extension E/F becomes a binary F -quadratic space W with respect to a quadratic form αNm E/F for some α ∈ F × . Suppose W has signature ((0, 2), (2, 0), . . . , (2,0)) with respect to the real embeddings σ 1 , . . . , σ d of F and there is an isometric embedding W Q := Res F/Q W ֒→ V . This not only implies (1.1.6) n + 2 ≥ 2d, but also gives CM points Z(W Q ) ⊂ X K (see Equation (2.5.2)). This 0-cycle is defined over F , and each individual point (z 0 , h) ∈ Z(W Q ) is defined over certain abelian extension of E. We will prove the following result concerning the algebraic nature of Φ r L at CM points in Z(W Q ). Theorem 1.3. In the setting above, suppose f has integral Fourier coefficients. Then there exist algebraic numbers λ j ∈ F and α j ∈ E ab for 1 ≤ j ≤ d such that for any (z 0 , h i ) ∈ Z(W Q ). Here σ h ∈ Gal(E ab /E) is the element associated to h ∈ E × \Ê × via class field theory. In particular when F = Q( √ D) is real quadratic and n + 2 = 2d = 4, we can take λ 1 = D r/2 /κ for some κ ∈ N and λ 2 = 0.
-quadratic space. These are called "big CM points", resp. "small CM points", when F is real quadratic, resp. F = Q. Remark 1.5. Theorem 1.3 applies even when Z(W Q ) intersects the singularity of Φ r L , In that case, the function Φ r (z, 1, f ) − Φ r (z, h, f ) in z can be continued to a real-analytic function in the neighborhood of the singularity, and its value at z = z 0 defines the quantity on the left hand side of (1.1.7).
Previous results concerning any linear combinations of CM values of higher Green function either assume n ≤ 2 or d = 1. Theorem 1.3 is the first result where the cases with n ≥ 3 and d ≥ 2 are addressed. In such cases, there is no known results even when one averages the higher Green function over all the CM points in all Galois conjugates of Z(W Q ), unlike for Green functions studied in [14,16]. Together with Theorem 1.1, Theorem 1.3 naturally leads one to expect the following. Conjecture 1.6. In the setting of Theorem 1.3, suppose f has integral Fourier coefficients and the singularity of Φ r L (z, h, f ) does not intersect Z(W Q ). Then there exists λ j ∈ F and α j ∈ E ab for 1 ≤ j ≤ d such that When F is real quadratic, i.e. d = 2, we can confirm it in the following case.
Theorem 1.7. Conjecture 1.6 holds when F is a real quadratic field, r is even, n = 4, and Z(W Q ) is defined over Q, in which case we can take λ 1 ∈ Q and λ 2 = 0.
Remark 1.8. When E/Q is Galois, there are many instances when Z(W Q ) is defined over Q (see e.g. Lemma 3.4 in [16]). In particular, the CM points on X 0 (1) 2 satisfy this condition (see Example 2.7). Therefore, the case for even r in Theorem 1.1 follows from Theorem 1.7.
Remark 1.9. In a recent joint work [9], we have proved Conjecture 1.6 when E/Q is biquadratic.
1.2. General Proof Strategy. When F = Q, Conjecture 1.6 follows from Theorem 5.5 in [12]. Here we give a sketch of its proof, which is analytic in nature. First, one expresses Φ r L (z, h, f ) as an integral of f against a suitable theta kernel R r τ Θ L (τ, z, h), where R τ is the raising operator (see (2.2.1)). Then a CM point (z 0 , h) ∈ Z(W Q ) leads to a rational splitting of V since F = Q. Suppose it leads to an integral splitting of L into L =L ⊕ N withL and N definite lattices of signature (n, 0) and (0, 2) respectively. Then the theta kernel becomes Note that θ N is non-holomorphic and has weight −1. One can then construct a preimageθ N of θ N under the lowering operator L τ . It is a harmonic Maass form of weight 1. The notion of harmonic Maass form was introduced in the seminal work of Bruinier and Funke [13], and studied around the same time by Zwegers in the context of modular completion of Ramanujan's mock theta functions [41].
Applying the Rankin-Cohen operator (see (2.2.2)) to θL andθ N then gives us a preimage of R r τ (θL(τ )θ N (τ )) under the L τ . Putting these together and applying Stokes' theorem gives us , where {, } is a pairing of formal Fourier series (see (4.2.5)) and CT denotes the constant term of a Fourier series. The functionθ + N is the holomorphic part ofθ N , and the modular formf in the last expression is weakly holomorphic with weight −1 and rational Fourier coefficients.
The harmonic Maass formθ N of weight 1 was studied in [20,21,36]. It was shown that the term CT(f (τ ) ·θ + N (τ )) is the logarithm of an algebraic number. To see this, let P 1 , P 2 be positive definite, unimodular lattices such that θ P 1 − θ P 2 is holomorphic on H. One can rewind the process above (with r = 0) and write are the regularized Borcherds lifts off and are logarithms of rational functions on Shimura varieties associated to L i . Their values at CM points z i are logarithms of algebraic numbers by the theory of complex multiplication. This finishes the sketch of the proof. The process of multiplying and dividing by θ P 1 − θ P 2 a manifestation of the embedding trick (see [6, section 8]). The partial averaging result in [12,Theorem 1.2] used the coincidence that the average of G r+1,f (z 1 , z) over the Galois orbit of z 1 is a higher Green function in z on the modular curve, i.e. n + 2 = 3. This is a rather special phenomenon that only happens when E/Q is biquadratic. By (1.1.6), one is reduced to the case of d = 1 in Conjecture 1.6.
For d ≥ 2, the lattice L splits asL ⊕ Res F/Q N with N ⊂ W an O-lattice of signature ((0, 2), (2, 0) . . . , (2, 0)), and the analogue of (1.2.1) is is the diagonal restriction of the Hilbert theta function θ N (τ 1 , . . . , τ d ) of weight (−1, 1 . . . , 1) associated to N. When one tries to execute the above strategy to construct a preimage of θ ∆ N under the lowering operator, it is necessary to work with Hilbert modular forms, and there are some serious obstacles.
• The analogousθ N should be a Hilbert modular form that satisfies suitable properties similar to those of harmonic Maass forms in the setting of elliptic modular forms.
However, there is no suitable extension of the notion of harmonic Maass forms to higher rank groups. • As harmonic Maass forms have singularities at the cusps, one would expect the same for the analogousθ N . However, Koecher's principle would imply that sucĥ θ N could not have singularity only at the cusps, but in the interior of the Hilbert modular variety as well. This also holds for its diagonal restriction and complicates the application of Stokes' theorem. • To extract information about the Fourier coefficient ofθ N , one needs the generalization of Borcherds' lift over totally real fields. In a large part, this has been accomplished in [11] by considering regularized theta lifts of Whittaker forms. However, as the Shimura varieties appeared loc. cit. are compact, there is no Fourier expansion and one has limited information about the rationality of the lift. Furthermore, it seems hopeless to direct generalize the embedding trick in [6] to totally real fields.
Instead of studying the value at an individual CM point, one can average over CM points in Z(W Q ), and those in Z(W (j) Q ) for 2 ≤ j ≤ d, where each W (j) is a neighboring Fquadratic space of W (see section 2.3). Then the rational quadratic spaces Res F/Q W (j) are all isomorphic and with N(j) ⊂ W (j) suitable lattices. The Hilbert modular form θ N (j 0 ) is holomorphic in τ j for j = j 0 and has weight (1, . . . , 1, −1, 1, . . . , 1) with −1 at the j 0 -th place. One can now explicitly construct an incoherent Hilbert Eisenstein series E N of parallel weight 1 that maps to E N (j) under the lowering operator in τ j for all 1 ≤ j ≤ d. For d = 2, this is the real-analytic Eisenstein series that appeared in the seminal works of Gross and Zagier on singular moduli and the Gross-Zagier formula [24,25]. It also appeared in [16], and has been combined with the regularized theta lifting of Borcherds to give fruitful generalizations of [24,25] in [14,17]. The advantage of E N is that its Fourier coefficients can be computed explicitly, and shown to be logarithms of rational numbers. They furthermore can be interpreted as arithmetic intersection numbers. On the other hand, it provides limited information about the arithmetic of higher Green function at a single CM point, as the differential operator in the strategy for d = 1 does not readily generalize except in the case n = 2d = 4 and r even (see the discussion at the end of section 5 in [12]). The higher Green functions studied by Gross, Kohnen and Zagier in [23] happen to be in this single case.
1.3. Ideas. We now describe some ideas and observations that help to overcome the obstacles mentioned in the previous section: • For any holomorphic Hilbert cusp form g(τ ) and ℓ ∈ N sufficiently large, the product g(τ ) ℓ θ N (τ ) has a modular preimage under the lowering operator in τ 1 with no singularity in H d . Furthermore, this preimage is harmonic in τ 1 and holomorphic in τ 2 , . . . , τ d . • The generalization of Borcherds' lift in [11] differs from the logarithm of an F -rational function by a locally constant function, which can be canceled out when considering differences of linear combinations of CM values. • The embedding trick only needs to work along the diagonal of H d , and one can apply the Siegel-Weil formula to replace the difficult task of finding suitable positive definite O-lattice P to the simpler one of analyzing Eisenstein series. The first idea is inspired by Zwegers' work [41], where the product of a mock theta function and a classical theta function is completed to become a real-analytic modular form without singularity in H. Such products are also called "mixed mock-modular forms" in [19] and are natural objects to consider. Since the differential operators in τ 1 , . . . , τ d are all independent, this idea can be applied in the setting of Hilbert modular forms. The existence of the modular preimage will be proved using complex geometry (see section 4.1), as done in the elliptic case in [13]. The parameter ℓ serves to ensure certain cohomology group vanishes (see Theorem 4.1). The Rankin-Cohen differential operator can also be generalized to be applied on such functions (see the differential operator D κ,r in (2.2.9)).
The second idea is a compromise so that one can still use the generalization of Borcherds' lift in [11] to deduce algebraicity results. Considering differences is quite effective in removing the so-called "normalizing constant" in the regularized theta lift (see Theorem 1.1 in [10]), as different linear combinations could give rise to the same normalizing constant. Furthermore, considering the difference turns out to simplify many other situations as well. For example, it is enough to construct a preimage of g(τ ) ℓ (θ N 1 (τ ) − θ N 2 (τ )) with g(τ ) a holomorphic Hilbert cusp form. This is accomplished in Theorem 4.3, using the ampleness of twists of determinant of the Hodge bundle on toroidal compactifications of Hilbert modular varieties, which is contained in Theorem 4.1 and a result of independent interest 3 . Also, one does not need to worry so much about the singularity of Φ r L (see Remark 1.5 and Lemma 4.5). The linear combination we take will come from multiplying this preimage with an Eisenstein series EP . This leads to the crucial algebraicity result in Theorem 4.10, which is of independent interest.
For the embedding trick, the last idea reduces the problem of dividing by a Hilbert cusp form g, which is constructed from theta series, to dividing by its diagonal restriction g ∆ , which is an elliptic modular form. Using the Siegel-Weil formula, we can relate g to Hilbert Eisenstein series. By varying the weight, we will show that for any finite set of points in H d , there is a Hilbert Eisenstein series that does not vanish on this set (see Lemma 3.2). This observation has its root in the classical work [32] of Rankin and Swinnerton-Dyer on zeros of elliptic Eisenstein series, and leads to the "partition of unity" result in Proposition 3.4.
By putting these ideas together, we are able to overcome the obstacles and prove Theorem 1.3. When it is specialized to the case in Theorem 1.1, we can combine this result about differences with the result about partial averages in [12] to complete the proof.
1.4. Outlook and Organization. To prove Conjecture 1.6, one needs algebraicity results concerning sums of CM points, in addition to the "difference result" in Theorem 1.3. For real quadratic F , we have worked out such a "sum result" when E/Q is biquadratic in [9], which has led to a proof of Conjecture 1.6 in this case. When d ≥ 3, one can try to relate the (in)coherent Eisenstein series to Eisenstein series on O(2, 1) over F , and realize them as suitable theta lifts from SL 2 over F . We plan to pursue this idea in a future work.
The paper is organized as follows. In sections 2 and 3, we setup notations and collect various preliminary notions from the literature. Results such as Lemma 3. Acknowledgement: We thank Jan H. Bruinier, Ben Howard, Steve Kudla, Jakob Stix Torsten Wedhorn, and Tonghai Yang for helpful discussions. We also appreciate fruitful conversations with Stephan Ehlen and Maryna Viazovska over the years concerning CM values of higher Green functions. We are thankful for the thorough reading and helpful comments by the anonymous referee. Finally, we thank Don Zagier for drawing our attention to this problem and many encouraging discussions.

Preliminary
Fix an embedding Q ֒→ C. Throughout the paper, F will be a totally real field of degree d with ring of integers O, different d and discriminant D. For 1 ≤ j ≤ d and m ∈ F , denote σ j : F ֒→ R the real embeddings of F and m j : Given For a semigroup G and a G-graded ring R = ⊕ i∈G R i , we use .
Given f, g ∈ A κ,ρ (Γ) such that at least one of them has exponential decay near the cusps, we can define their Petersson inner product where dµ(τ ) := dµ(τ 1 ) . . . dµ(τ d ) is the invariant measure on H d (see Equation (4.21) in [11]). For κ = (k 1 , . . . , k d ), denote the following related weights We omit Γ, resp. ρ, from the notation when F is fixed and Γ = Γ F := SL 2 (O), resp. it is trivial. When F = Q, we will use the superscript ! to indicate modular forms with singularities at the cusps.
is the subspace of modular forms with rational Fourier coefficients. This is also the case for M κ,ρ when ρ is a Weil representation defined below (see [30], [11, section 7]).
For later purposes, we will be interested in the (N d -)graded ring

Differential Operators.
For k ∈ Z, we have the usual raising, lowering and hyperbolic Laplacian operators on H They change the weight by 2, −2 and 0 respectively. For κ = (k 1 , k 2 ) ∈ Q 2 and r ∈ N 0 , we can define the Rankin-Cohen operator on a real-analytic function f (τ 1 , τ 2 ) ∈ H 2 by is the binomial coefficient. The equality in the second line can be proved by considering the generating series constructed from the differential operators ∂ τ and R τ . The details are contained in section 5.2 of [15], in particular Propositions 18 and 19. The first expression shows that the operator preserves holomorphicity. When κ ∈ Z 2 , the second expression shows that it preserves modularity in the sense that for any γ ∈ SL 2 (R). The same result holds in the metaplectic setting when κ ∈ 1 2 Z 2 .
and P r (X) the r-th Legendre polynomial, which has parity (−1) r . The last equality is a consequence of Rodrigues' formula (see (8.6.18) in [1]). This example will be used in the proof of Theorem 1.3.
]. If f 1 is harmonic of weight k 1 and f 2 is holomorphic of weight k 2 , then we have For real-analytic functions f, g : H → C, rational numbers k, ℓ and an integer Proof. This is done by induction on r. The base case of r = 0 is trivial. For the inductive step to prove the case r +1, we have k +ℓ ∈ {−2r, −2r +1, . . . , 0}. Denote x a :=R a fR r+1−a g for 0 ≤ a ≤ r + 1. ApplyingR to (2.2.5) shows that x a + x a+1 is a rational linear combination Therefore, the right hand side below is a rational linear combination ofR r+1−j [f, g] j for 0 ≤ j ≤ r + 1 and it suffices to show that the square matrix on the left is invertible The right kernel of A is spanned by the vector ((−1) a ) 0≤a≤r+1 . On the other hand which is zero precisely when k + ℓ ∈ {−2r, −2r + 1, . . . , −r}. This is not possible by the condition imposed on k + ℓ. Therefore the matrix Now we will extend the Rankin-Cohen operator to functions on H d for any d ≥ 2 by first restricting it to H 2 , before applying the usual Rankin-Cohen operator. This can be expressed as a linear combination of the generalized Rankin-Cohen operators studied in [27].
with a e ∈ Z explicit constants given by From the definition, one sees that D κ,r (f, g) is real-analytic on H and satisfies for κ, λ ∈ Z d and γ ∈ SL 2 (R). The upshot of this operator is the following result.
be a real-analytic function that is harmonic in τ 1 of weight k 1 and holomorphic in τ 2 , . . . , τ d . For any holomorphic function g : Proof. This follows directly from the definition and equation (2.2.4).
We can also componentwisely apply D κ,r when f is vector-valued, in which case we also write D κ,r (f, g), and the result above holds as well.
2.3. Quadratic Space and Shimura Variety. Let V be a finite dimensional F -vector space of dimension n + 2 ≥ 0 with a non-degenerate quadratic form Q. For our purpose, n is even when is positive definite. The symmetric domain D associated to SO(V (R)) is realized as the Grassmannian of q-dimensional negative definite oriented subspaces of V (R). It consists of 2 components unless q = 0, in which case it is a point.
Let GSpin V be the general spin group of V . We will be interested in the Q-algebraic group which fits into the exact sequence Then we have e 0 + e 3 = 1 in C(M) and an R-algebra isomorphism The even Clifford algebra C 0 (M) corresponds precisely to the diagonal matrices in M 2 (R). The group GSpin then consists of invertible diagonal matrices, and the spinor norm ν is just the determinant.
Then the hermitian symmetric space associated to H can be realized as the Grassmannian and extend the quadratic form C-bilinearly to V C , then we can identify D with the quadric in the projective space P(V C ) by sending [Z = X +iY ] to the oriented 2-plane spanned by the ordered basis {X, Y } ⊂ V σ 1 . This endows D with a complex structure. We can furthermore identify the tube domain For z ∈ D ± , denotez ∈ D ∓ the 2-plane with the opposite orientation. The subgroup of H(R) fixing D + is the subgroup H(R) + consisting of elements with totally positive spinor norm. For a compact open K ⊂ H(Q), the C-points of the Shimura variety associated to H is a complex quasi-projective variety of dimension n, and has a canonical model over σ 1 (F ) [33]. When V is anisotropic over F , the variety X K is projective.
such that the connected component of the Shimura variety X V,K Γ can be identified with the product of modular curves X Γ × X Γ . See section 3.1 in [38] for more details.
A meromorphic modular form on X K of weight w ∈ Z is a collection of meromorphic functions Ψ(·, h) : and are meromorphic at the boundary 5 . For such a meromorphic modular form, we also denote which is a real-analytic function on X K (see section 2 of [11]). To describe the connected components of X K , we write [26] gives us When restricted to the center Z in (2.3.2), the map ν above is simply the square map and its image consists of square elements in Gal(E K /F ). When n = 0, the domain D has two points and the group GSpin V can be identified with E × W for a totally imaginary, quadratic extension E W over F , the norm from E W to F is simply the spinor norm, and sig(V (j)) = ((n + 2, 0), . . . , (n + 2, 0), (n, 2), (n + 2, 0), . . . , (n + 2, 0)) and isomorphic to V at all finite places. They are neighboring quadratic spaces of an admissible incoherent quadratic space (V, Q) overF (see [10, section 7]). One can carry out the construction before (2.3.5) to define X V (j),K , which is the C-points of a Shimura variety defined over σ j (F ). There is a quasi-projective variety X K defined over F such that the base change to σ j (F ) is X V (j),K , and the union of X V (j),K over all j is the C-points of X K considered as a scheme over Q (see Lemma 7.1 of [10]). which is empty if m ∈ d −1 + Q(µ). Also, we denote We say that a lattice L is Z-unimodular if L ′ = L. Then the set  In this case, L is called unimodular. Satz 1 in [18] gives a necessary and sufficient condition for the existence of definite unimodular O-lattices, which is easily seen to be satisfied when V = W ⊕4 with dim F (W ) is divisible by 2. Furthermore, for any one of the 2 d possible signatures for definite spaces, there is a space V having this signature and containing a unimodular O-lattice. So for any α ∈ F × , there is a definite space V such that it becomes totally positive definite after scaling its quadratic form by α.
It is a well-known result of Hecke (see the last Theorem in [37]) that the class of d in Cl(F ) is a square. So we can write d −1 = a 2 (δ) with a ⊂ O and δ ∈ F . Let (L, Q) be a non-trivial, integral unimodular O-lattice such that δQ is totally positive definite. Then (aL, δQ) is an even O-lattice and δ(λ, aL) ⊂ d −1 = a 2 (δ) ⇔ (λ, a −1 L) ⊂ O for all λ ∈ (aL) ′ . So λ ∈ aL # = aL and (aL, δQ) is non-trivial, Z-unimodular and totally positive definite.
2.5. Special Cycles. Now suppose V decomposes as W ⊕ U such that U is totally positive subspace of dimension r. Then the Grassmannian D U of U consists of one point z U , and D W can be realized as an analytic submanifold of D via Similarly, the algebraic group H W := Res F/Q GSpin W , resp. H U := Res F/Q GSpin U , is isomorphic to the pointwise stabilizer of U, resp. W , in H V , which induces defines a codimension-r cycle on X K , denoted by Z(W, h). A word of caution about the notation: in [26], the items D W , H W and Z(W, h) were defined with W replaced by U. We decide to change the notation here as U will be varying later and it is important to keep track of W .
When r = n, the set D W = {z ± W } consists of two elements and points of Z(W, h) are called (small) CM points. For a subfield F 0 ⊂ F , we can consider the F 0 -quadratic space Its C-points are given by . These were called "big CM points" in [14] when F 0 = Q and U 0 is trivial. We omit h from the notation when it is trivial.
To obtain a 0-cycle defined over Q, one considers the 0-cycle where W is the admissible incoherent quadratic space with neighbors W (j). Note that Z(W) is Z(W ) in Equation (2.13) of [14]. The 0-cycles τ j (Z(W )) can be constructed as above with W replaced by W (j) for 1 ≤ j ≤ d (see [14,Lemma 2.2]).
is an isometry with N = a 1 a 2 . The CM 0-cycle Z(W Q ) defined in (2.5.2) is given by Note that H 0 = Q when d 1 , d 2 are co-prime. On the other hand, we have Lemma 3.2 in [28] tells us that H 0 /Q is abelian. Its proof even implies that every element in Gal(H 0 /Q) has order dividing 2. From these, we then know that the element σ 2 ∈ Gal(H 2 /E 2 ) satisfying z σ 2 2 = −z 2 is a square and hence trivial when restricted to H 0 . Therefore, Z(W Q ) = Z(W (2) Q ), and Z(W Q ) is already defined over Q.
On the other extreme, when r = 1, we have W = (F x 0 ) ⊥ for some x 0 ∈ F with Q(x 0 ) = m ≫ 0, and the cycle Z(W, h) is a divisor. We define a weighted divisor by the finite sum for any φ ∈ S(V (F )) K . We also write for µ ∈ L ′ /L and L ⊂ V an even O F -lattice.
2.6. A helpful Lemma. In this section, we record a result that will be helpful in studying zeros of definite theta functions. for all 1 ≤ j ≤ N.
In the general case, we can first use Kronecker's approximation theorem to produce a sequence {n ′ i } ⊂ Z such that lim i→∞ e(n ′ i θ j ) = e(α j ) for all 1 ≤ j ≤ N. For each i, we can find i ′ > i such that n ′′ i := n ′ i + n i ′ forms an increasing sequence in N, where {n i } is the sequence we have constructed in the case all α j 's are 0. Then the new sequence {n ′′ i } satisfies the condition of the lemma. Proof. Without loss of generality, we take s 0 = 0. After rearranging and scaling all the c i 's, we can suppose that |c i | ≥ |c i+1 | for all i ∈ N, and 1 := |c 1 | = |c m | > |c m+1 |. Denote Using induction, it is then enough to prove the lemma for φ 1 (s). The condition φ(s) = 0 for all but finitely many s ∈ N implies that For any β ∈ (R/Z) N , Lemma 2.8 implies that there exists an infinite subsequence {n k : k ∈ N} ⊂ N such that lim k→∞ (e((Mn k + b)θ 1 ), . . . , e((Mn k + b)θ N )) = β.
Since f b is continuous, we then have This implies that the polynomial f b is identically zero, or equivalently for all r ∈ Z N and b ∈ Z/MZ. This then implies 1≤i≤m, (r i,j ) 0≤j≤N =r ′ α i = 0 for all r ′ ∈ Z/MZ × Z N and φ 1 (s) is identically 0. So the lemma holds for φ 1 , and hence also for φ by induction.
As an immediate consequence, we have the following result.  Also denote Γ f := SL 2 (Ô) ⊂ G(Q).
For an F -quadratic space (V, Q) of even dimension, let ω = ω ψ be the Weil representation of G(A) on the space of Schwartz functions S(V (A F )) = S(V (F )) ⊗ 1≤j≤d S(V σ j ), where ψ is the standard additive character on F \A F . At the infinite local place, suppose (W, Q W ) is an R-quadratic space of signature (p, q) with 2 | (p + q). For a point w in the symmetric space D W associated to SO(W ), we obtain an orthogonal decomposition W = w ⊥ ⊕ w and a Schwartz function in S(W ), which is acted on by SL 2 (R) via the Weil representation ω W to produce Note that as it is independent of the orientation of w. If q = 0, then the expressions in (3.1.1) and (3.1.2) are independent of w and we omit them from the notation. In addition, we will also be interested in the following "singular Schwartz function" when q ≥ 1 Here Γ(s, x) := ∞ x t s−1 e −t dt is the incomplete Gamma function. Direct calculations yield for all w ∈ D W and λ ∈ W . To describe the finite local place, let L ⊂ V be an even lattice and S L ⊂ S(V (F )) the subspace as in (2.4.3). Then Γ acts on the space S L via ω, whose complex conjugate we denote by ρ L . Its explicit values on the generators of Γ can be found in section 3.2 of [11]. Furthermore, it is unitary with respect to the hermitian pairing on S L (3.1.6) φ, ψ := More generally, for lattices L, M and φ ∈ S M ⊕L , ψ ∈ S M , we define φ, ψ L ∈ S L by These pairings are then naturally defined for functions valued in S L . For the rest of the section, suppose V σ j is positive definite whenever j ≥ 2. Then D Vσ j is D for j = 1 and a point otherwise. For τ = (τ j ) ∈ H d , z ∈ D, h ∈ H(Q) and φ f ∈ S(V (F )), we define the Siegel theta function For any even lattice L ⊂ V , we also denote the associated theta function by which is valued in S L . When h = 1, we omit it from the notation. The definition of Θ L implies that where L h is defined in (2.4.4) and L ′ /L ∼ = L ′ h /L h via (2.4.5). If V σ 1 is definite, then Θ L (τ, z, h) is independent of z and we write (3.1.12) θ L (τ, h) := Θ L (τ, z, h).
In particular when L = P is positive definite and Z-unimodular, this is a scalar-valued, holomorphic Hilbert modular form on Γ F of parallel weight n/2 + 1. We denote the graded subring (3.1.13) M θ F := Span{θ P (τ ) : P ∈ U + F } ⊂ M F . For future convenience, we also define The sums converge absolutely since the singular Schwartz function decays as a Schwartz function and z ⊥ ∩ L ′ is contained in a positive definite lattice. For fixed (z, h), it defines a real-analytic function in τ , which satisfies the analogue of (3.1.11) as well as (3.1.15) L τ 1 Θ * L (τ, z, h) = Θ L (τ, z, h) by (3.1.5). When V σ 1 is negative definite, we have q 1 = n + 2 and the function θ * L (τ, h) := Θ * L (τ, z, h) can be written explicitly as Although θ * L is not modular in τ , difference of such functions will become modular after adding suitable holomorphic functions (see Theorem 4.3).
Remark 3.1. When dim F V is odd, all the constructions above still hold by working with metaplectic covers. As this is not needed for most of the applications, we refrain from introducing more notations and refer the readers to [11].  , (n+2, 0)). For a standard section Φ ∈ I(s, χ), the Eisenstein series

Eisenstein Series and
converges absolutely for ℜ(s) ≫ 0 and has meromorphic continuation to s ∈ C. When s 0 := n/2, the map λ : S(V (A F )) → I(s 0 , χ) defined by is SL 2 (A F )-equivariant, and λ(φ) can be extended uniquely to all s and produce a standard sections in I(s, χ). The infinite part ⊗ 1≤j≤d I(s, χ σ j ) is generated by functions Φ κ Lemma 4.1]. For φ f ∈ S(V (F )) andκ =κ(V ) as in (3.1.9), the Eisenstein series is a Hilbert modular form of weightκ. For any even lattice L ⊂ V , we have where Ω is the Kähler form on X K normalized as in [11] and vol(X K ) = X K Ω n . For n = 0, the Siegel-Weil formula yields When q = 0, we haveκ = −−−→ s 0 + 1 and the lattice P = L is totally positive definite. The classical Siegel-Weil formula yields Though θ P ℓ = θ ℓ P , the Eisenstein series E P ℓ is almost never the same as E ℓ P ! Furthermore suppose that P ∈ U + F with rank 2r. Then E P (τ ) is in M θ F and coincides with the Hecke Eisenstein series for F of parallel weight r = n/2 + 1. It can be written as (see section 2 in [5]) for certain non-zero integers A β depending only on β. Note that since P ℓ ∈ U + F for any ℓ ∈ N, we have E P ℓ (τ ) ∈ M θ F . Applying Lemma 2.9, we can deduce the following results about zeros of theta functions. Proof. Since SL 2 (R) d acts transitively on H d , we can write S = {τ 0 , g 1 τ 0 , . . . , g N τ 0 } with τ 0 ∈ H d and g i ∈ SL 2 (R) d , where we set g 0 to be the identity. Assume that for every ℓ ≥ 1, there exists 0 ≤ i ≤ N such that E P ℓ (g i τ 0 ) = 0. Then the function vanishes at τ = τ 0 for all ℓ ≥ 1. Using the expression (3.2.8) we can write By Remark 2.10, we have N i=0 c(β i , g i τ 0 ) = 0 for all β 1 , . . . , β N ∈ P 1 (F ), which is clearly a contradiction since N i=0 c(∞, g i τ 0 ) = 1. Lemma 3.3. For any τ 0 ∈ H d , there exist P 1 , P 2 ∈ U + F in the same genus such that θ P 1 (τ 0 ) = θ P 2 (τ 0 ).
Proof. Assume otherwise. Then for any P ∈ U + F , the function θ P (τ, h) takes the same value at τ 0 for all h ∈ H(Q) and we have which contradicts Corollary 2.11 since c(β, τ 0 ) = 0 for all β ∈ P 1 (F ). Proposition 3.4 (Partition of Unity). Let F be a totally real field. Then there exist m ∈ N, P i , P 1,i , P 2,i ∈ U + F with ranks 2r i and 2r i for 1 ≤ i ≤ m such that P 1,i and P 2,i are in the same genus for all i and there is no τ 0 ∈ H such that (EP i (θ P 1,i − θ P 2,i )) ∆ (τ 0 ) = 0 for all i.
Furthermore, for any d 1 , . . . , d m , e 1 , . . . , e m ∈ N, there exists elliptic modular forms g i ∈ M ! −(r i d i +r i e i )d with rational Fourier coefficients such Proof. Start with a point τ 0 ∈ H, we can find P 1 , P 2 ∈ U + F in the same genus satisfying (θ P 1 − θ P 2 ) ∆ (τ 0 ) = 0 by Lemma 3.3. If (θ P 1 − θ P 2 ) ∆ has no zero on H, then we can apply Lemma 3.2 to takeP 1 , do not have common zero on H. The forms EP 1 · (θ P 1 − θ P 2 ) and EP 2 · (θ P 1 − θ P 2 ) satisfy the first claim. Otherwise, let τ 2 , . . . , τ m ∈ H be the zeros of (θ P 1 − θ P 2 ) ∆ in a fundamental domain F of SL 2 (Z)\H. By Lemma 3.3, there exists P 1,i and P 2,i such that θ ∆ P 1,i (τ i ) = θ ∆ P 2,i (τ i ) for all 2 ≤ i ≤ m. Let S ⊂ F be the finite set of the zeros of 1≤i≤m (θ ∆ P 1,i −θ ∆ P 2,i ). Applying Lemma 3.2, we can findP ∈ U + F such that EP does not vanish on S. Let S ′ ⊂ F be the finite set of the zeros of E ∆ P 1≤i≤m (θ ∆ P 1,i − θ ∆ P 2,i ). We can apply Lemma 3.2 to find EP i for 2 ≤ i ≤ m such that they do not vanish on S ′ . Now the forms EP · (θ P 1 − θ P 2 ) and EP i · (θ P 1,i − θ P 2,i ) with 2 ≤ i ≤ m satisfy the first claim.
To prove the second claim, we can write Setting 3.3. Higher Green Function. We follow [12] to recall higher Green function on the Shimura variety X K for F = Q. Let V /Q be a quadratic space of signature (n, 2) and L ⊂ V an even lattice with K an open compact stabilizingL. Also, we denote For µ ∈ L ′ /L and m ∈ Z + Q(µ), the automorphic Green function is defined by where F (a, b, c; z) is the Gauss hypergeometric function [1,Chapter 15]. The sum converges normally on X K \Z(m, µ) for s > σ 0 + 1 and defines an eigenfunction of the Laplacian ∆ on D, normalized as in [7], with eigenvalue 1 2 (s − σ 0 − 1)(s + σ 0 ). Furthermore, it has a meromorphic continuation to s ∈ C with a simple pole at s = σ 0 + 1, whose constant term is denoted by Φ m,µ (z, h, σ 0 + 1) and the regularized theta lift of Hejhal-Poincaré series of index (m, µ) [7].
At s = σ 0 + 1 + r with r ∈ N, the function Φ m,µ (z, h, s) is called a higher Green function. For the unimodular lattice L = M 2 (Z) and z = Z(z 1 , z 2 ) as in Example 2.5, we have where G m s is defined in (1.1.2). For a harmonic Maass form f = m,µ c(m, µ)q −m φ µ +O(1) ∈ H k−2r,ρ L with k := −2σ 0 , define to be the associated higher Green function. Following from the work of Borcherds [6] and generalization by Bruinier [8] (also see [12,35]), the function Φ r L has the following integral representation (3.3.5) where F T is the truncated fundamental domain of Γ Q \H at height T > 1.

Real-analytic Hilbert Modular Forms and Algebraicity of Pairing
In this section, we will prove the existence of certain real-analytic Hilbert modular forms by generalizing the proof of Theorem 3.7 in [13], and give some results concerning their Fourier coefficients. The notations F, D, O, d, d are the same as in section 2.

4.1.
Certain real-analytic Hilbert modular forms. Let ρ = ρ L be a Weil representation, κ = (k 1 , . . . , k d ) ∈ Z d , Γ ⊂ Γ F a congruence subgroup, and X = X(Γ) = Γ\H d be the open Hilbert modular variety. By adding finitely many cusps to X, we obtain the Baily-Borel compactification X BB . It can also be constructed as the Proj of the ring of holomorphic modular forms on X, and is a normal, Noetherian scheme over C. When Γ is neat, we fix a smooth toroidal compactificationX of X. It is a compact complex manifold, and a desingularization of X BB . We also have the natural map π :X → X BB and let E be the boundary divisor onX. SupposeX is associated to a projective Γ-admissible decomposition (see [4,

section II.2]).
Denote O and E p,q the sheaf of holomorphic functions and smooth differential forms of type (p, q) onX respectively, and take the subsheaf E ′ := ker(E 0,1∂ → E 0,2 ). Then the Dolbeault resolution of O gives us the short exact sequence For a Cartier divisor D and quasi-coherent sheaf F onX, we write F (D) for the corresponding twisting sheaf. Also, let L κ,ρ be the sheaf of modular forms of weight κ and representation ρ on X. It extends to X BB andX by Koecher's principle, and we use L κ,ρ andL κ,ρ to denote these extensions. In particular, Note that π * Lκ,ρ = L κ,ρ , π * L κ,ρ =L κ,ρ . When ρ is trivial and κ = (1, . . . , 1), L = L κ,ρ is the determinant of the Hodge bundle and ample on X BB . However, the extensionL is trivial at the fiber of a cusp, and in general not ample onX. Nevertheless, we can use it along with twisting by E to prove the following result.
Theorem 4.1. In the notations above, for any N ∈ N, there exists n 0 , k ∈ N >N such that the following map is surjective for all n ≥ n 0 .
Proof. For simplicity, suppose X BB has only one cusp x. By Theorem 2.2 in Chapter IV of [4]),X is the normalization of the blowing-up of X BB at certain coherent sheaf I of ideals concentrated at x. As X BB is Noetherian, so isX and π is quasi-compact. We claim that  For κ ∈ Z d , letκ,κ ∈ Z d be as in (2.1.3). Suppose f ∈ Aκ ,ρ (Γ) is holomorphic in τ j for 2 ≤ j ≤ d. Given any g ∈ S κ ′ (Γ) of parallel weight, there exists ℓ 0 ∈ N and functionsĜ ℓ ∈ Aκ +ℓκ ′ ,ρ (Γ) for all ℓ ≥ ℓ 0 such that they are holomorphic in τ j for 2 ≤ j ≤ d, Proof. Suppose Γ is neat and g is non-zero. Let κ ′ = − → k ′ with k ′ ∈ N and fix some N > k ′ . Now choose n 0 , k > N as in Theorem 4.1 and set ℓ 0 = n 0 k. Given f ∈ Aκ ,ρ (Γ), the differential form v −2 1 g(τ ) ℓ 0 f (τ )dτ 1 is in the kernel of∂ ⊗ 1 ⊗ 1 since f is holomorphic in τ 2 , . . . , τ d . Furthermore, it is orders of vanishing at the cusps are at least ℓ 0 since f is bounded near the cusps. Therefore, it is a global section of E ′ ⊗Lκ ,ρ ⊗L κ ′ (−E) ⊗ℓ 0 . Note thatL 1 g ℓ 0 f dτ 1 . As f, g are real-analytic, so isĜ ℓ 0 . So for any ℓ ≥ ℓ 0 , the real-analytic modular formĜ ℓ := g ℓ−ℓ 0Ĝ ℓ 0 ∈ Aκ +ℓκ ′ ,ρ is holomorphic in τ 2 , . . . , τ d and satisfies From the construction, the last condition is also satisfied. Finally for any congruence subgroup Γ ⊂ Γ F , there exists a neat, normal subgroup Γ ′ ⊂ Γ of finite index. Averaging the functionĜ ℓ ∈ Aκ +ℓκ ′ ,ρ (Γ ′ ) constructed above over Γ/Γ ′ then gives the desired function in level Γ. Now, we will apply this result to the case when f is the special value of a theta kernel and g is the holomorphic theta function for a positive definite lattice. (1) It is holomorphic in τ 2 , . . . , τ d and has exponential decay near the cusps.
(2) It satisfies for all m ∈ F and µ ∈ L ′ /L. (5) For any r 1 , . . . , r d ∈ N 0 , we have Remark 4.4. Up to holomorphic cusp forms of parallel weight rℓ + 1, the holomorphic part δ + is uniquely determined by the conditions above. We will show later in Theorem 4.10 that certain rational linear combinations of the Fourier coefficients ofδ + are logarithms of algebraic numbers.

4.2.
Whittaker Forms. Suppose d ≥ 2 for this section. We follow [10,11] is called a harmonic Whittaker form of weight κ in the sense of [11]. The space generated by such forms is denoted by H κ,ρ L . Given f = µ∈L ′ /L, m≫0 c(m, µ)f m,µ ∈ H κ,ρ L , the Fourier polynomial Conversely, given any polynomial of the above form, there is a unique harmonic Whittaker form f P ∈ H κ,ρ L with this principal part. Note that such polynomial only depends on the finite quadratic modular L ′ /L. We say that f or P(f ) is rational if the polynomial P(f ) has rational coefficients.
Letκ be the dual weight of κ as in (2.1.3). There is a natural surjection ξ = ξ (1) where the sum above converges absolutely as n > 2. This induces a bilinear pairing between g = n,ν b(n, ν)q n φ ν ∈ Mκ ,ρ L and f ∈ H κ,ρ L given by A harmonic Whittaker form f is called weakly holomorphic if ξ(f ) vanishes identically, i.e. {g, f } = 0 for all g ∈ Sκ ,ρ L . We use M ! κ,ρ L ⊂ H κ,ρ L to denote the subspace of such forms. Using the last expression in (4.2.5), we can extend {, } to formal Fourier series in the parameter e(tr(mu)). For a subfield F ⊂ C, let M ! κ,ρ L (F ) denote the subspace of M ! κ,ρ L with Fourier coefficients in F . Then the fact that Mκ ,ρ L (F ) = Mκ ,ρ L (Q) ⊗ F implies that

Regularized Theta Lifts.
For each f = f P = m,µ c(m, µ)f m,µ ∈ H κ,ρ L , Bruinier computed its regularized theta lift in [11] and constructed an Arakelov Green function Φ(z, h, f ) for the divisor for any Γ ∞ -invariant function G on H d such that the integral converges.
Proof. Without loss of generality, we can suppose f = f m,µ . From (4.3.6), we have Then convergence of the integral at z ∈ Z(m, µ) follows directly from Proposition 5.2 of [10]. For such z, we can apply the unfolding calculation in the proof of Theorem 5.3 of [11] while evaluating at s = s 0 . Then the following integrals are identically equal for any From the second expression, we see that the integral in (4.3.7) also converges for z = z 0 .
To evaluate the regularized theta integral, one can apply Stokes' theorem when certain primitive exist. The following lemma distilled from Theorems 6.3 and 7.2 in [10] will be helpful for this purpose. Lemma 4.6. Suppose n = 2s 0 > 2 and f = m≫0, µ∈L ′ /L c(m, µ)f m,µ is a harmonic Whittaker form in H κ,ρ L . Let η ∈ Aκ ,ρ L such that Suppose there existsη (τ ) = m∈F, µ∈L ′ /Lb m,µ (v 1 )e(tr(mτ ))φ µ ∈ Aκ ,ρ L 9 Via the diagonal embedding, we tacitly view elements in V (Q) as in V (Q). The set of z 0 in D such that V ∩ z ⊥ 0 is non-trivial has measure 0.
Since ξ(f ) is a cusp form, the integral defining the Petersson inner product exists, and so does the limit in T .
We can now apply Lemmas 4.5 and 4.6 to the function Proof. It is easy to see that h 0 = 1, h ′ 0 = h W satisfy the condition (4.3.6). Sinceδ has exponential decay near the cusps, the function ηP (τ ) Pet is bounded on H d . Furthermore, equation (4.4.3) and the holomorphicity of θP implies that If we denoteb m,µ (v 1 ) andb + m,µ the Fourier coefficients ofηP andη + P respectively, then we have by Theorem 4.3. Therefore Lemmas 4.5 and 4.6 together with (4.4.2) imply This finishes the proof.
Remark 4.9. If we replaceP byPh withh ∈ HŨ (Q), thenPh is in the same genus asP and N ⊕Ph ⊕ P i The proposition above and Equation (3.1.11) imply that the functionηPh satisfies (4.4.6) If c(m, µ) ∈ Z for all m and µ, then Theorem 6.8 of [11] implies that there exists a meromorphic modular form Ψ L (z, h, f ) on X K with weight −B(f ) and a finite order multiplier system such that (4.4.7) − log Ψ L (z, h, f ) 2 Pet = Φ L (z, h, f ). Furthermore, the divisor of Ψ L (z, h, f ) is the special cycle Z(f ), which is defined over F . Let M ∈ N be the order of the multiplier system. Then up to a locally constant function C L (z, h, f ) on X K , the form Ψ L (z, h, f ) M equals to a meromorphic modular form R L (z, h, f ) of weight −B(M · f ) ∈ Z on X K defined over σ 1 (F ), and we can write Now we let L = L i for 0 ≤ i ≤ ℓ as in Proposition 4.8. Since the spinor norm ν : HŨ → T is surjective, the subgroup HŨ (Q) ⊂ H V (Q) acts transitively on the connected components of X K by (2.3.8). Therefore for any open compact KŨ ⊂ K ∩ HŨ (Q), the quantity is independent of h W ∈ H W (Q) and z ∈ D. Now, we are ready to state and prove the main result of this section.
Theorem 4.10. Let N, h, P 1 , P 2 , ℓ andδ(τ ; N, h, P 1 , P 2 , ℓ) ∈ A −−→ rℓ+1,ρ L be the same as in Theorem 4.3 andP , L i , V, κ the same as in Proposition 4.8. For any weakly holomorphic Whittaker form f = f P ∈ M ! κ,ρ N with coefficients in a number field F ⊂ C, there exists λ 1 , . . . , λ m ∈ F and α 1 , . . . , α m ∈ E ab W independent of h W such that Remark 4.11. We can choose λ i , α i above such that the index m is bounded by the degree of F /Q. Due to the order the multiplier system of the meromorphic modular form Ψ(z, h, f ) in (4.4.7), the denominator of λ i depends on f even when it has integral coefficients and suppose that F = Q. By replacing f with M · f for some M ∈ N, we can suppose that f has integral Fourier coefficients and the modular form Ψ L i (z, h, f ) in (4.4.7) has trivial character for all 0 ≤ i ≤ ℓ. Let R L (z, h, f ) and C L (z, h, f ) be the same as in (4.4.8). Let where Q(z, h) is a meromorphic function on X K ′ defined over σ 1 (F ) given by Since H W is abelian, we have K N × KŨ ⊂ K ′ . Therefore, the CM 0-cycle Z(W ) also lies on X K ′ and each CM point is defined over a number field E K N ⊂ E ab W . The function R L i is defined over σ 1 (F ), as well as the natural map X K ′ → X K given by right multiplication with h 0 ∈ H W (Q) ⊂ H V (Q) (see [26, page 46]). Therefore, the modular function Q(z, h) is also defined over σ 1 (F ). Furthermore, it is non-zero at (z, h) = (z 0 , (h W , 1)) for all h W ∈ H W (Q). These values are algebraic numbers satisfying σ(Q(z 0 , (1,h))) = Q(z 0 , (h W ,h)) as σ fixes the function Q(z, h), which is defined over σ 1 (F ), and acts on CM points in the CM 0-cycle Z(W,h) by Shimura's reciprocity law (see sections 3.1 and 5.3 in [2] for the relevant case here). Setting α = h ∈HŨ (Q)\HŨ (Q)/KŨ Q(z 0 , (1,h)) and λ = − 2 c KŨ ·#S W finishes the proof.
When d = 2 = n/2, we already have L = N 0 and the reduction step in the first paragraph above is not necessary. The functionη + r is simply (G ∆ ) r+1 C (1,1),r (η + /G), and the last claim follows from Example 2.1.

5.2.
Proof of Theorem 1.7. As Z(W Q ) is defined over Q, i.e. Z(W Q ) = Z(W (2) Q ) on X V,K , we can apply Theorem 5.10 in [12] to conclude that 2 deg(Z(W Q )) Φ r L (Z(W Q ), f ) = CT( f, C (1,1),r (E + L ) ), where E + L is the holomorphic part of the derivative of an incoherent Eisenstein series, which is a real-analytic Hilbert modular form of weight (1, 1). From Example 2.1, we see that the constant term of C (1,1),r (E + L ) vanishes when r ≥ 1. Furthermore, since f has rational Fourier coefficients, the term CT( f, C (1,1),r (E + L ) ) is a rational linear combinations of the non-zero Fourier coefficients of E + L , which are rational multiple of logarithms of integers by Proposition 4.6 in [14]. Therefore, we have for some c ′ ∈ N and a ∈ Q.

5.3.
Proof of Theorem 1.1. When r is even, this follows from the discussion in Example 2.7 and Theorem 1.7. When r is odd and d 1 is fundamental, Theorem 7.13 of [12] gives the algebraicity analogous to (5.2.1) with the left hand side replaced by certain partial average. Using this and proceeding with the rest of the argument in the proof of Theorem 1.7 gives Theorem 1.1.