Gross–Zagier Formula on Singular Moduli

• The Gross–Zagier formula on singular moduli explicitly factors differences of CM j-invariants by leveraging local arithmetic data, including representation numbers and optimal embeddings.
• It employs both analytic methods and arithmetic intersection theory to equate norms of differences with local intersection multiplicities on modular curves, Hilbert modular surfaces, and Shimura curves.
• Recent refinements extend the classical formula to arbitrary discriminants and quaternionic settings, linking CM cycle intersections with Fourier coefficients and confirming deep conjectures.

Searching arXiv for the primary paper and closely related work on singular moduli, refinements, and Shimura-curve generalizations. Searching arXiv for "(Yang, 2010) Gross Zagier singular moduli Yang Howard Yang Hilbert modular surfaces". The Gross–Zagier formula on singular moduli is the explicit factorization formula governing differences of CM values of the modular $j$-invariant. If $\tau\in \mathfrak{H}$ is a CM point, then $j(\tau)$ is a singular modulus, an algebraic integer generating a ring or Hilbert class field. Gross and Zagier showed that products, or equivalently suitable norms, of differences $j(\tau_1)-j(\tau_2)$ admit a prime factorization controlled by local arithmetic data: representation numbers, optimal embeddings into quaternionic orders, and local intersection multiplicities on integral models of modular curves. In later work, this viewpoint was recast in the language of arithmetic intersections, refined to arbitrary discriminants, and generalized to Hilbert modular surfaces and Shimura curves, where the diagonal on $X_0(1)\times X_0(1)$ is replaced by special divisors such as Hirzebruch–Zagier divisors or quaternionic Hecke correspondences (Yang, 2010, Daas, 2023).

1. Classical statement and arithmetic meaning

For negative coprime fundamental discriminants $D_1,D_2$, with imaginary quadratic fields $K_i=\mathbb{Q}(\sqrt{D_i})$, unit counts $w_i=\#\mathcal{O}_{K_i}^\times$, and $D=D_1D_2$, the classical product of singular moduli is

$$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$

Gross–Zagier’s formula gives

$\tau\in \mathfrak{H}$0

where $\tau\in \mathfrak{H}$1 is a prime power determined by genus-character data and local parity conditions. One formulation defines

$\tau\in \mathfrak{H}$2

with $\tau\in \mathfrak{H}$3 built from the quadratic residue data of $\tau\in \mathfrak{H}$4 and $\tau\in \mathfrak{H}$5. The nontrivial contributions occur exactly when there is a unique prime $\tau\in \mathfrak{H}$6 with $\tau\in \mathfrak{H}$7 and $\tau\in \mathfrak{H}$8 odd; then $\tau\in \mathfrak{H}$9 is a positive power of $j(\tau)$0 (Nicolau, 9 Mar 2026).

An equivalent formulation, particularly natural in the Hilbert modular and Borcherds-product approach, uses the real quadratic field $j(\tau)$1, the biquadratic CM field $j(\tau)$2, and the counting function

$j(\tau)$3

Then the logarithm of the total product is expressed as a sum over totally positive $j(\tau)$4 and primes $j(\tau)$5 inert in $j(\tau)$6, with weights $j(\tau)$7 (Yang et al., 2017).

The formula explains striking numerical factorizations. One example is

$j(\tau)$8

whose prime divisors are inert in both $j(\tau)$9 and $j(\tau_1)-j(\tau_2)$0 (Daas, 2023). In the coprime fundamental setting, a recurring structural principle is that only primes with tightly constrained splitting behavior contribute to the factorization.

2. Arithmetic intersection formulation

The modern interpretation of the formula is intersection-theoretic. Gross–Zagier’s factorization can be viewed as a computation of the arithmetic intersection of CM divisors on an integral model of the modular curve, or equivalently as the intersection of the diagonal on $j(\tau_1)-j(\tau_2)$1 with a CM cycle. In this formulation, the non-archimedean valuation of a difference of singular moduli becomes a local intersection multiplicity at a prime $j(\tau_1)-j(\tau_2)$2 (Yang, 2010).

At the local level, the valuation is computed by supersingular geometry. One standard form is that

$j(\tau_1)-j(\tau_2)$3

is a sum over supersingular elliptic curves $j(\tau_1)-j(\tau_2)$4 of non-negative integers counting compatible optimal embeddings of the two quadratic orders into $j(\tau_1)-j(\tau_2)$5, equivalently representation numbers attached to ternary quadratic forms or Gross lattices (Li, 2018, Howard et al., 2012). This is the source of the prime-by-prime factorization.

A stack-theoretic formulation makes the intersection precise. For the Deligne–Mumford stack $j(\tau_1)-j(\tau_2)$6 of elliptic curves and finite étale CM stacks $j(\tau_1)-j(\tau_2)$7, the arithmetic intersection multiplicity is defined by summing lengths of strictly Henselian local rings at geometric intersection points, weighted by automorphism groups and $j(\tau_1)-j(\tau_2)$8 (Phillips, 15 Sep 2025). In this language, the classical quantity

$j(\tau_1)-j(\tau_2)$9

encodes the arithmetic degree of the CM intersection.

This perspective also clarifies a common point of confusion. The fundamental object is often not a single bare difference $X_0(1)\times X_0(1)$0, but rather a norm or a product over Galois conjugates, or an intersection number on a stack. The local factors are geometric multiplicities, not merely ad hoc exponents.

3. Refined formulas and arbitrary discriminants

The original Gross–Zagier theorem assumed coprime fundamental discriminants. Subsequent work refined both the indexing of local contributions and the range of admissible discriminants. A decisive refinement is due to Howard–Yang, who decomposed the Hecke correspondence into zero-dimensional stacks $X_0(1)\times X_0(1)$1 indexed by totally positive $X_0(1)\times X_0(1)$2, with $X_0(1)\times X_0(1)$3. If $X_0(1)\times X_0(1)$4 and $X_0(1)\times X_0(1)$5 is a singleton, then $X_0(1)\times X_0(1)$6 is supported in characteristic $X_0(1)\times X_0(1)$7 and

$X_0(1)\times X_0(1)$8

while the degree vanishes if $X_0(1)\times X_0(1)$9 or $D_1,D_2$0. The total Hecke intersection is recovered by summing $D_1,D_2$1 over $D_1,D_2$2 of fixed trace (Phillips, 15 Sep 2025).

Howard–Yang also proved an arithmetic Siegel–Weil identity identifying these arithmetic degrees with Fourier coefficients of the central derivative of Hecke’s normalized Eisenstein series. In the notation of that work, if

$D_1,D_2$3

then

$D_1,D_2$4

and $D_1,D_2$5 is explicitly $D_1,D_2$6 when $D_1,D_2$7, and $D_1,D_2$8 otherwise (Howard et al., 2012). This refines Gross–Zagier by separating the local intersection according to the auxiliary index $D_1,D_2$9, rather than only after summation over all $K_i=\mathbb{Q}(\sqrt{D_i})$0 of a given trace.

An independent direction is the extension to arbitrary discriminants, including non-maximal orders. For distinct discriminants $K_i=\mathbb{Q}(\sqrt{D_i})$1, one still has

$K_i=\mathbb{Q}(\sqrt{D_i})$2

but now $K_i=\mathbb{Q}(\sqrt{D_i})$3 is allowed to be a possibly fractional prime power in the degenerate $K_i=\mathbb{Q}(\sqrt{D_i})$4 case, and for $K_i=\mathbb{Q}(\sqrt{D_i})$5 it is supported at a single rational prime $K_i=\mathbb{Q}(\sqrt{D_i})$6 characterized by Hilbert-symbol conditions. For $K_i=\mathbb{Q}(\sqrt{D_i})$7, the valuation $K_i=\mathbb{Q}(\sqrt{D_i})$8 is computed by counting invertible ideals of prescribed norm in $K_i=\mathbb{Q}(\sqrt{D_i})$9, subject to local optimality conditions (Lauter et al., 2012). This removes the maximal-order restriction and is motivated in part by denominator questions for genus-two invariants.

4. Hilbert modular surfaces and Yang’s generalization

A substantial generalization replaces the modular curve by a Hilbert modular surface. Let $w_i=\#\mathcal{O}_{K_i}^\times$0 with $w_i=\#\mathcal{O}_{K_i}^\times$1 prime, and let $w_i=\#\mathcal{O}_{K_i}^\times$2 be the Hilbert modular stack parametrizing abelian surfaces with real multiplication by $w_i=\#\mathcal{O}_{K_i}^\times$3 and a $w_i=\#\mathcal{O}_{K_i}^\times$4-polarization in the sense of Deligne–Pappas. The Hirzebruch–Zagier divisors $w_i=\#\mathcal{O}_{K_i}^\times$5 on $w_i=\#\mathcal{O}_{K_i}^\times$6 play the role of special divisors, and for $w_i=\#\mathcal{O}_{K_i}^\times$7 the divisor $w_i=\#\mathcal{O}_{K_i}^\times$8 identifies with the diagonal embedding of the moduli stack of elliptic curves into $w_i=\#\mathcal{O}_{K_i}^\times$9 (Yang, 2010).

Yang considers a non-biquadratic quartic CM field $D=D_1D_2$0 and the associated CM cycle $D=D_1D_2$1 on $D=D_1D_2$2. Under the assumptions

$D=D_1D_2$3

and that the discriminant $D=D_1D_2$4 is prime, the main theorem asserts

$D=D_1D_2$5

for all $D=D_1D_2$6, where $D=D_1D_2$7 is defined from local data of the reflex extension $D=D_1D_2$8 and the counting function

$D=D_1D_2$9

This confirms a special case of the Bruinier–Yang conjecture and makes the local structure of the intersection completely explicit (Yang, 2010).

The relationship with the classical Gross–Zagier formula is exact when $$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$0. In that case $$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$1 is the diagonal, $$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$2, and Yang recovers

$$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$3

The local intersection terms match the Gross–Zagier exponents, so the Hilbert modular-surface formula is literally a higher-dimensional generalization of the singular-moduli factorization (Yang, 2010).

This framework also yields a height-theoretic consequence: under the same hypotheses, Yang proves the first non-trivial non-abelian Chowla–Selberg formula, a special case of Colmez’s conjecture, namely

$$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$4

for a principally polarized CM abelian surface of type $$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$5 (Yang, 2010).

5. Quaternionic and Shimura-curve extensions

A distinct family of generalizations replaces modular curves by Shimura curves attached to indefinite quaternion algebras. For genus-zero Shimura curves $$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$6 with $$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$7, one chooses a Hauptmodul $$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$8 and forms a cross-ratio invariant built from CM points $$\mathcal{J}(D_1,D_2) \coloneqq \prod_{\substack{[\tau_1],[\tau_2]\ \mathrm{disc}(\tau_i)=D_i}} \bigl(j(\tau_1)-j(\tau_2)\bigr)^{\frac{4}{w_1w_2}}.$$9 and their Atkin–Lehner transforms. Giampietro–Darmon conjectured, and Daas proved, that the norm of this cross-ratio factors as

$\tau\in \mathfrak{H}$00

where $\tau\in \mathfrak{H}$01 depends on the residue class of $\tau\in \mathfrak{H}$02 modulo $\tau\in \mathfrak{H}$03. The proof uses $\tau\in \mathfrak{H}$04-adic $\tau\in \mathfrak{H}$05-functions on the Cerednik–Drinfeld uniformization of $\tau\in \mathfrak{H}$06, together with CM theory and an analytic argument via $\tau\in \mathfrak{H}$07-adic infinitesimal deformations of Hilbert Eisenstein series and an $\tau\in \mathfrak{H}$08 theorem (Daas, 2023).

An archimedean proof of the same factorization was later obtained by evaluating automorphic Green’s functions at CM points. In that approach, the logarithm of the cross-ratio of $\tau\in \mathfrak{H}$09-values is expressed as a linear combination of Green’s function values, and the first Fourier coefficient of a holomorphic projection of a diagonal-restricted Hilbert Eisenstein series is shown to match the finite factorization term. For $\tau\in \mathfrak{H}$10, the relevant cusp-form space vanishes or is annihilated by Atkin–Lehner symmetry, so the archimedean and finite contributions coincide exactly (Nicolau, 9 Mar 2026).

There is also an intersection-theoretic Shimura-curve generalization. For an indefinite quaternion algebra $\tau\in \mathfrak{H}$11 with maximal order $\tau\in \mathfrak{H}$12, one considers the stack $\tau\in \mathfrak{H}$13 of QM abelian surfaces, the CM divisors $\tau\in \mathfrak{H}$14, and the Hecke correspondence $\tau\in \mathfrak{H}$15. After decomposing the CM stacks by congruence types $\tau\in \mathfrak{H}$16, one defines zero-dimensional stacks $\tau\in \mathfrak{H}$17. If $\tau\in \mathfrak{H}$18 is totally positive and $\tau\in \mathfrak{H}$19, then

$\tau\in \mathfrak{H}$20

and

$\tau\in \mathfrak{H}$21

When $\tau\in \mathfrak{H}$22 is split, this reduces to the Howard–Yang refinement on modular curves (Phillips, 15 Sep 2025).

6. Consequences, related structures, and higher-dimensional context

One direct arithmetic consequence is that differences of singular moduli are never units. Using Gross–Zagier, Gross–Kohnen–Zagier, Schofer, and Bruinier–Kudla–Yang, Li gave a short proof that

$\tau\in \mathfrak{H}$23

hence $\tau\in \mathfrak{H}$24 cannot be a unit. The same method implies that for any $\tau\in \mathfrak{H}$25 there exists a prime $\tau\in \mathfrak{H}$26 such that the reductions modulo $\tau\in \mathfrak{H}$27 of two CM elliptic curves are $\tau\in \mathfrak{H}$28-isogenous, via the modular polynomial criterion $\tau\in \mathfrak{H}$29 (Li, 2018).

A related but distinct development concerns traces rather than products. In the language of generalized Jacobians with cuspidal modulus, Bruinier and Li proved that the generating series of Heegner divisor classes is a weakly holomorphic modular form of weight $\tau\in \mathfrak{H}$30, extending Gross–Kohnen–Zagier and yielding new proofs of Zagier’s modularity of traces of singular moduli. In the classical case $\tau\in \mathfrak{H}$31, this recovers

$\tau\in \mathfrak{H}$32

thus placing the trace formulas and the Gross-style Jacobian interpretation in a common framework (Bruinier et al., 2015).

Higher-dimensional quaternionic analogues also exist over totally real fields of strict class number $\tau\in \mathfrak{H}$33. In that setting, the role of optimal embeddings into $\tau\in \mathfrak{H}$34 is played by simultaneous embeddings of primitive CM fields into superspecial orders in definite quaternion algebras over the totally real base field. The resulting counting formulas determine which superspecial primes can occur in factorizations of differences of Siegel modular function values at CM points and provide bounds relevant to Igusa invariants and genus-two CM constructions (Goren et al., 2011). This suggests that the singular-moduli formula is best understood not as an isolated identity for $\tau\in \mathfrak{H}$35-values, but as the one-dimensional prototype of a much broader arithmetic intersection and automorphic framework.

Across these developments, the persistent structural principle is the same: a seemingly elementary factorization of special values is governed by local deformation theory, representation densities, and special cycles on Shimura varieties. The classical Gross–Zagier formula remains the basic example, but its later refinements show that its natural home is arithmetic geometry rather than elementary algebraic number theory alone.