Gross–Zagier Log-Algebraicity Conjecture

• Gross–Zagier log‐algebraicity conjecture is a statement that normalized values of higher automorphic Green’s functions at CM points equal rational multiples of logarithms of algebraic numbers.
• The conjecture interrelates weakly holomorphic modular forms, Hecke translates, and regularized theta lifts to connect analytic behaviors with algebraic properties.
• Recent advances utilizing orthogonal Shimura varieties and deformed theta integrals provide a breakthrough, reinforcing the link between automorphic objects and motivic regulators.

The Gross–Zagier log-algebraicity conjecture is the assertion that suitably normalized values of higher automorphic Green’s functions on products of modular curves at pairs of CM points are rational multiples of logarithms of algebraic numbers. In a standard formulation, one starts from a weakly holomorphic modular form $f$ and the associated Hecke combination $G_k^f(z_1,z_2)$; for CM points $z_1,z_2$ of discriminants $d_1,d_2$, the conjecture predicts an identity of the shape

$$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$

with $\kappa\in \mathbb Q^\times$ and $\alpha$ algebraic in the compositum of the relevant ring class fields, together with a natural Galois equivariance statement (Sreekantan, 7 Feb 2025). Originating in work of Gross–Zagier and Gross–Kohnen–Zagier, the conjecture now forms part of a broader structure linking regularized theta lifts, Borcherds products, CM cycles, motivic regulators, and single-valued periods of mixed modular motives (Brown et al., 6 Aug 2025).

1. Classical formulation

Let $\Gamma$ be a congruence subgroup and $X=\overline{\mathfrak H/\Gamma}$. For $k\ge 1$, the higher Green’s function of weight $G_k^f(z_1,z_2)$0 on $G_k^f(z_1,z_2)$1 is

$G_k^f(z_1,z_2)$2

where $G_k^f(z_1,z_2)$3 is the image of $G_k^f(z_1,z_2)$4 in $G_k^f(z_1,z_2)$5, and

$G_k^f(z_1,z_2)$6

If $G_k^f(z_1,z_2)$7 has $G_k^f(z_1,z_2)$8, then $G_k^f(z_1,z_2)$9, and the Hecke translate is

$z_1,z_2$0

For a weakly holomorphic modular form

$z_1,z_2$1

of weight $z_1,z_2$2 with integral principal part, one defines

$z_1,z_2$3

Gross–Zagier and Gross–Kohnen–Zagier originally formulated the conjecture in terms of relations among Fourier coefficients of modular forms of weight $z_1,z_2$4; the weakly holomorphic modular form formulation is equivalent to that description (Sreekantan, 7 Feb 2025).

For CM points $z_1,z_2$5 of discriminants $z_1,z_2$6, with $z_1,z_2$7 the corresponding ring class fields and $z_1,z_2$8, the conjecture states that, when one of $z_1,z_2$9 is fundamental if $d_1,d_2$0 is even, there exist

$d_1,d_2$1

depending only on $d_1,d_2$2, such that

$d_1,d_2$3

If

$d_1,d_2$4

then for $d_1,d_2$5,

$d_1,d_2$6

In this formulation, “log-algebraicity” means precisely that the Green’s value is a rational multiple of the logarithm of the absolute value of an algebraic number (Sreekantan, 7 Feb 2025).

A closely related diagonal version concerns renormalized Green’s functions. In a cusp-form-free case, Gross–Zagier’s renormalized algebraicity conjecture asserts

$d_1,d_2$7

for CM $d_1,d_2$8; this is the diagonal “self-energy” analogue of the off-diagonal CM-value conjecture (Zhou, 2013).

2. Analytic development and proof

Before the full pointwise theorem, several partial results established the expected algebraicity pattern in special settings. For even $d_1,d_2$9, an averaged theorem on the CM cycle $$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$0 proved that there exist $$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$1 and $$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$2, where $$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$3, such that

$$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$4

together with an explicit ideal factorization formula for the fractional ideal generated by $$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$5 (Li, 2018). In a different direction, the weight-$$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$6 renormalized self-energy on $$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$7 was computed exactly: $$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$8 which verifies the corresponding renormalized Gross–Zagier algebraicity conjecture and shows directly that the value is the logarithm of a modular unit (Zhou, 2013).

The first full pointwise breakthrough at level $$(d_1d_2)^{(k-1)/2}G_k^f(z_1,z_2)=\frac{1}{\kappa}\log|\alpha|,$$9 was Li’s theorem. For $\kappa\in \mathbb Q^\times$0, $\kappa\in \mathbb Q^\times$1 with integral Fourier coefficients, and CM points $\kappa\in \mathbb Q^\times$2 of discriminants $\kappa\in \mathbb Q^\times$3, Li proved that

$\kappa\in \mathbb Q^\times$4

with $\kappa\in \mathbb Q^\times$5 in the compositum of the corresponding ring class fields and with Galois equivariance

$\kappa\in \mathbb Q^\times$6

The same work placed the classical product-of-modular-curves problem into the framework of orthogonal Shimura varieties and regularized theta lifts (Li, 2021).

The $\kappa\in \mathbb Q^\times$7 case was then completed analytically by the theory of deformations of theta integrals. In that approach, the classical product $\kappa\in \mathbb Q^\times$8 is treated as an orthogonal Shimura variety of signature $\kappa\in \mathbb Q^\times$9, and principal higher Green functions are realized as regularized theta lifts. The new ingredient is an analogue of the incoherent Eisenstein series over a real quadratic field, constructed as the Doi–Naganuma theta lift of a deformed theta integral on hyperbolic space. This new Hilbert modular object controls the antisymmetric CM combination not seen by the usual incoherent Eisenstein series, and its Fourier coefficients are logarithms of algebraic numbers with explicit valuation formulas. In signature $\alpha$0, the resulting parity simplification yields the pointwise formula predicted by Gross–Zagier (Bruinier et al., 2022).

Recent work records the overall status succinctly: the conjecture has been settled in the case of congruence subgroups of the form $\alpha$1 by analytic methods (Brown et al., 6 Aug 2025).

3. Orthogonal and theta-lift reformulations

A decisive conceptual step is the reinterpretation of higher Green functions as regularized theta lifts on orthogonal Shimura varieties. Let $\alpha$2 be a rational quadratic space of signature $\alpha$3, $\alpha$4 the corresponding orthogonal Shimura variety, and $\alpha$5 a harmonic Maass form of weight $\alpha$6. The associated higher Green function is

$\alpha$7

When $\alpha$8 with determinant form, the associated Shimura variety is $\alpha$9, and the classical higher Green functions are recovered from the orthogonal theta lift. In the level-$\Gamma$0 normalization used by Li,

$\Gamma$1

so the conjecture on products of modular curves is a special case of an orthogonal CM-value problem (Li, 2021).

This orthogonal reformulation also clarifies the role of CM cycles. If $\Gamma$2 is a binary CM quadratic space over a totally real field $\Gamma$3, an embedding $\Gamma$4 defines a CM $\Gamma$5-cycle $\Gamma$6. The higher Green value at a CM point becomes a regularized theta-lift value on such a CM cycle. In the biquadratic CM case, the averaged CM value theorem takes the form

$\Gamma$7

and in signature $\Gamma$8 one parity disappears, leaving a single logarithm and recovering the original Gross–Zagier statement (Bruinier et al., 2022).

The orthogonal framework also supplies the mechanism behind the new proof for $\Gamma$9. The classical incoherent Eisenstein series accounts for the symmetric combination

$X=\overline{\mathfrak H/\Gamma}$0

but not for the antisymmetric combination

$X=\overline{\mathfrak H/\Gamma}$1

The deformed-theta construction fills that gap. A plausible implication is that the log-algebraicity phenomenon is inseparable from the interaction between coherent and incoherent automorphic objects, rather than from Green functions alone.

4. Motivic cycles and regulator formulas

A distinct line of work gives a geometric explanation for why logarithms of algebraic numbers should appear. The basic setting is the universal family of products of elliptic curves over a modular curve, or equivalently the associated Kummer $X=\overline{\mathfrak H/\Gamma}$2 family. For a surface $X=\overline{\mathfrak H/\Gamma}$3,

$X=\overline{\mathfrak H/\Gamma}$4

and a class is represented by

$X=\overline{\mathfrak H/\Gamma}$5

with $X=\overline{\mathfrak H/\Gamma}$6. Nodal rational curves on Kummer surfaces yield explicit classes of this type after blowing up the node and adjoining the exceptional divisor; this is the basic engine behind the construction of higher Chow cycles (Sreekantan, 7 Feb 2025).

In the modular-curve situation, one studies classes $X=\overline{\mathfrak H/\Gamma}$7 on the generic fiber of the universal abelian surface whose boundary is a sum of CM cycles: $X=\overline{\mathfrak H/\Gamma}$8 If $X=\overline{\mathfrak H/\Gamma}$9 is a CM point outside the support of the boundary, then the real regulator satisfies

$k\ge 1$0

and one has the algebraicity identity

$k\ge 1$1

Because the intersection points are algebraic and the $k\ge 1$2 are algebraic functions, the right-hand side is the logarithm of an algebraic number (Sreekantan, 7 Feb 2025).

This mechanism was already visible in earlier work on algebraic cycles and values of Green’s functions. There, explicit indecomposable classes were constructed in

$k\ge 1$3

for universal abelian surfaces and their Kummer $k\ge 1$4 surfaces, and their regulators were shown to produce linear combinations of weight-$k\ge 1$5 higher Green’s functions at CM points that are logarithms of algebraic numbers. That work formulated a conjectural correspondence between weakly holomorphic modular forms and motivic classes whose regulators recover Borcherds lifts and Green functions (Sreekantan, 2022).

The more recent construction on products of elliptic curves strengthens this picture. It constructs infinitely many indecomposable classes on Kummer $k\ge 1$6 surfaces attached to odd-degree isogenies, proves a boundary formula

$k\ge 1$7

up to decomposable boundary terms, and deduces that

$k\ge 1$8

has infinite rank (Sreekantan, 7 Feb 2025). These results do not by themselves re-prove the full conjecture, but they furnish a geometric explanation for the logarithmic-algebraic output.

5. Single-valued periods and mixed modular motives

A further reformulation identifies higher Green functions with single-valued periods. For any $k\ge 1$9, one has a matrix-valued higher Green function $G_k^f(z_1,z_2)$00, and for even $G_k^f(z_1,z_2)$01 its central entry recovers the classical scalar Green function: $G_k^f(z_1,z_2)$02 The entire matrix is realized as a block of the single-valued period matrix of

$G_k^f(z_1,z_2)$03

where $G_k^f(z_1,z_2)$04 for the universal elliptic curve over the modular curve stack (Brown et al., 6 Aug 2025).

For CM points $G_k^f(z_1,z_2)$05 and a Hecke correspondence $G_k^f(z_1,z_2)$06 annihilating the cusp-form contribution, one extracts a rank-two extension

$G_k^f(z_1,z_2)$07

The Gross–Zagier conjecture becomes the assertion that this extension is Kummer. If

$G_k^f(z_1,z_2)$08

for some algebraic $G_k^f(z_1,z_2)$09, then the off-diagonal single-valued period equals a rational multiple of $G_k^f(z_1,z_2)$10, and one obtains

$G_k^f(z_1,z_2)$11

equivalently

$G_k^f(z_1,z_2)$12

In this framework, the appearance of logarithms of algebraic numbers is a direct consequence of the single-valued period matrix of a Kummer extension (Brown et al., 6 Aug 2025).

The same work constructs genuine motives from moduli stacks of pointed elliptic curves. In level $G_k^f(z_1,z_2)$13, the relevant cusp-form motive of weight $G_k^f(z_1,z_2)$14 is realized by the motive of $G_k^f(z_1,z_2)$15. Since that motive is mixed Tate, its Betti realization is conservative; because there are no level-$G_k^f(z_1,z_2)$16 cusp forms of weight $G_k^f(z_1,z_2)$17, the cusp-form motive vanishes. This yields a completely geometric proof of the Gross–Zagier conjecture in level $G_k^f(z_1,z_2)$18, weight $G_k^f(z_1,z_2)$19 (Brown et al., 6 Aug 2025).

This suggests a broad generalization: matrix-valued higher Green functions at non-CM points should correspond not to Kummer extensions and logarithms of algebraic numbers, but to more general mixed modular motives whose single-valued periods are expected to encode special values of $G_k^f(z_1,z_2)$20-functions.

6. Terminology, adjacent theories, and scope

The expression “Gross–Zagier conjecture” is genuinely ambiguous. In the context of higher Green functions, it denotes the log-algebraicity statement just described. A different conjecture of Gross and Zagier, proved in a separate arithmetic setting, concerns divisibility of $G_k^f(z_1,z_2)$21 by the product of the Manin constant, Tamagawa factors, and $G_k^f(z_1,z_2)$22 in analytic rank $G_k^f(z_1,z_2)$23; that conjecture is unrelated to higher Green functions and does not concern logarithms of algebraic numbers (Byeon et al., 2015).

There are also adjacent Gross–Zagier theories in which the basic output is a height rather than a logarithm of an algebraic number. The $G_k^f(z_1,z_2)$24-adic Iwasawa-theoretic Gross–Zagier theorem identifies the cyclotomic linear term of a two-variable $G_k^f(z_1,z_2)$25-adic $G_k^f(z_1,z_2)$26-function with $G_k^f(z_1,z_2)$27-adic height pairings of norm-compatible Heegner points, not with a direct logarithm formula (Howard, 2012). Explicit Gross–Zagier formulas for cube-sum elliptic curves and for twisted CM points on Shimura curves similarly express period-normalized central derivatives as Néron–Tate heights of algebraic CM points or twisted CM divisors [(Cai et al., 2014); (Howard, 2012); (Yin, 2 Jul 2026)]. In higher-dimensional Shimura varieties, the Arithmetic Gan–Gross–Prasad framework replaces Heegner points by arithmetic diagonal cycles and canonical heights by Beilinson–Bloch or Gillet–Soulé pairings, with central derivatives of normalized $G_k^f(z_1,z_2)$28-functions on the analytic side (Zhang, 2024).

These neighboring theories are not instances of the Gross–Zagier log-algebraicity conjecture, but they show that the conjecture belongs to a larger Gross–Zagier paradigm in which special values or derivatives of automorphic objects are controlled by arithmetic invariants of algebraic cycles. A plausible implication is that the logarithm-of-algebraic-number phenomenon for higher Green functions is one visible boundary case of a broader regulator-and-period formalism.