Gross–Zagier Conjecture

• Gross–Zagier conjecture is a set of interrelated assertions that connect the central derivative of L-functions with the Néron–Tate height of Heegner points and predict arithmetic phenomena in elliptic curves.
• It encompasses various formulations, including the classical rank-one setting, p-adic analogues, and theories involving higher Green functions and singular moduli, illustrating its broad methodological scope.
• Recent work confirms key divisibility results in specific torsion cases and offers explicit formulas in CM families, while extensions to higher dimensions and arithmetic Gan–Gross–Prasad remain active research areas.

The Gross–Zagier conjecture is a name attached to several closely related conjectural statements and formulas originating in the work of Benedict Gross and Don Zagier. In its classical rank-one form, it concerns the relation between the central derivative of an $L$-function and the height of a Heegner point, together with the arithmetic consequences predicted by the Birch–Swinnerton-Dyer philosophy. In later usage, the same name also refers to Gross–Zagier’s conjecture on the algebraicity of higher Green functions at CM points, to singular-moduli intersection formulas, and to a large $p$-adic and higher-dimensional generalization program on Shimura varieties and arithmetic cycles (Byeon et al., 2015, Bruinier et al., 2022, Zhang, 2024).

1. Classical rank-one formulation

A standard arithmetic setting is the following. Let $E/\mathbf{Q}$ be an elliptic curve of conductor $N$, let $K=\mathbf{Q}(\sqrt d)$ be an imaginary quadratic field in which every prime dividing $N$ splits completely, and let $P_K\in E(K)$ be the Heegner point arising from the modular parametrization of $E$. Write $M$ for the Manin constant, $C=\prod_{p\mid N} C_p$ for the product of local Tamagawa numbers, and $p$0 for the Tate–Shafarevich group over $p$1. If $p$2 is the number of roots of unity in $p$3, then Gross and Zagier relate the derivative $p$4 to the Néron–Tate height of $p$5 by

$p$6

where $p$7 is the period normalization appearing in the paper. On the other hand, the Birch–Swinnerton-Dyer formula in analytic rank $p$8 predicts

$p$9

Equating the two expressions yields the “strong” Gross–Zagier conjecture

$E/\mathbf{Q}$0

provided $E/\mathbf{Q}$1 has infinite order. Since $E/\mathbf{Q}$2 divides $E/\mathbf{Q}$3, one obtains the weaker divisibility statement

$E/\mathbf{Q}$4

This is the conjecture explicitly identified as the Gross–Zagier conjecture in the rank-one Heegner setting in Byeon–Kim–Yhee’s terminology (Byeon et al., 2015).

The same circle of ideas is presented in the higher-dimensional survey literature as the prototype of the principle that “arithmetic equals derivative of $E/\mathbf{Q}$5-function.” In the modular-curve case, one has a formula of the shape

$E/\mathbf{Q}$6

and a BSD comparison

$E/\mathbf{Q}$7

so that the Gross–Zagier identity becomes the analytic input for rank-one BSD statements (Zhang, 2024).

2. The divisibility conjecture and its proof in rank one

Byeon, Kim, and Yhee prove the weak Gross–Zagier divisibility conjecture for all rational elliptic curves whose rational torsion subgroup is one of Mazur’s six remaining “small” possibilities: $E/\mathbf{Q}$8 More precisely, if $E/\mathbf{Q}$9 has one of these torsion structures and $N$0 satisfies the Heegner hypothesis with $N$1 of analytic rank $N$2, then

$N$3

The result verifies, in these cases, the predicted compatibility between rational torsion, local reduction data encoded by Tamagawa numbers, and the global obstruction measured by $N$4 (Byeon et al., 2015).

The proof is case-by-case but structurally uniform. The main ingredients are Tamagawa-number computations via Tate’s algorithm, Kramer’s formula for the $N$5-primary part of $N$6, Cassels’ theorem and a descent analysis for the $N$7-torsion cases, isogeny invariance of the conjecture, and results on optimal curves, étale minimal curves, and Manin constants in isogeny classes. In the $N$8-power torsion cases, direct Tamagawa computations often suffice, while in cases such as $N$9 and $K=\mathbf{Q}(\sqrt d)$0 one may need nontriviality of $K=\mathbf{Q}(\sqrt d)$1 via Kramer’s formula or passage to an isogenous curve. For torsion involving $K=\mathbf{Q}(\sqrt d)$2, the analysis separates the $K=\mathbf{Q}(\sqrt d)$3 case, where Tamagawa factors already force divisibility, from the $K=\mathbf{Q}(\sqrt d)$4 case, where Tamagawa numbers, the Manin constant, and Cassels’ theorem together control the factor $K=\mathbf{Q}(\sqrt d)$5 (Byeon et al., 2015).

In this formulation, the conjecture is not an independent height formula but an integrality consequence extracted from the Gross–Zagier formula and BSD. Its mathematical content is therefore arithmetically sharper than a bare proportionality statement: it asserts divisibility among torsion order, local Tamagawa contributions, the Manin constant, and the square root of the Tate–Shafarevich order.

3. Higher Green functions, singular moduli, and other statements bearing the same name

A distinct Gross–Zagier conjecture concerns the algebraicity of higher Green functions at CM points on $K=\mathbf{Q}(\sqrt d)$6. For

$K=\mathbf{Q}(\sqrt d)$7

the associated principal higher Green function is

$K=\mathbf{Q}(\sqrt d)$8

Gross and Zagier conjectured that for any CM point $K=\mathbf{Q}(\sqrt d)$9, there exists $N$0 such that

$N$1

where $N$2 are the discriminants of the CM points. Bruinier, Ehlen, and Yang complete the proof by proving a factorization theorem for CM values of higher Green functions on orthogonal Shimura varieties and specializing to the product of modular curves. A key 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 (Bruinier et al., 2022).

Subsequent work gives this conjecture a geometric and motivic interpretation. Brown, Castaño-Bernard, and Gangl reinterpret higher Green functions as single-valued periods of motives built from suitable moduli stacks of elliptic curves with marked points, with a biextension structure involving symmetric powers of elliptic-curve motives. In their formulation, if a Hecke operator $N$3 kills the cuspidal motive, then the Gross–Zagier value at CM points becomes the single-valued period of a Kummer extension, hence a rational multiple of $N$4 for algebraic $N$5. They also give a geometric proof in level $N$6 and weight $N$7 by showing that the motive of $N$8 is mixed Tate (Brown et al., 6 Aug 2025). In a parallel direction, Kerr, Li, and others give a cycle-theoretic proof in weight $N$9 for several genus-zero modular curves by realizing the relevant higher Green functions as regulator pairings of higher Chow cycles with a canonical $P_K\in E(K)$0-class (Doran et al., 2 Sep 2025).

The singular-moduli side of Gross–Zagier leads to another branch of the subject. The classical formula on singular moduli is interpreted as an arithmetic intersection calculation for CM divisors on the integral model of a modular curve. Goren and Lauter generalize this viewpoint to higher-dimensional CM abelian varieties over totally real fields by counting simultaneous embeddings of CM orders into superspecial orders in quaternion algebras, thereby extending the factorization/counting philosophy of Gross–Zagier to Hilbert modular varieties (Goren et al., 2011). Phillips later proves a Shimura-curve analogue: for CM divisors $P_K\in E(K)$1 on the integral model $P_K\in E(K)$2 of a quaternionic Shimura curve,

$P_K\in E(K)$3

and each zero-dimensional term is evaluated by an explicit formula of the form

$P_K\in E(K)$4

under the stated local conditions (Phillips, 15 Sep 2025).

4. $P_K\in E(K)$5-adic Gross–Zagier theory

The $P_K\in E(K)$6-adic Gross–Zagier theorem replaces complex heights and central derivatives by $P_K\in E(K)$7-adic heights and derivatives or special values of $P_K\in E(K)$8-adic $P_K\in E(K)$9-functions. Disegni proves a general formula on Shimura curves for modular abelian varieties $E$0 of $E$1-type over a totally real field, assuming potentially ordinary reduction at the primes above $E$2. In the root number $E$3 case, the cyclotomic derivative of a Rankin–Selberg $E$4-adic $E$5-function equals the $E$6-adic height pairing of Heegner points; in the root number $E$7 case, the corresponding statement is a $E$8-adic Waldspurger formula for toric periods. The family version gives an anticyclotomic Gross–Zagier formula in which the derivative transverse to the anticyclotomic locus controls the height pairing of big Heegner-point families (Disegni, 2015).

Howard proves an Iwasawa-theoretic Gross–Zagier theorem for ordinary weight-$E$9 forms over the anticyclotomic tower $M$0. Writing Hida–Perrin-Riou’s two-variable $M$1-adic $M$2-function as

$M$3

he identifies the linear term $M$4 with a generating series of $M$5-adic heights of regularized Heegner points. In the elliptic-curve case this becomes the Mazur–Rubin Iwasawa-theoretic Gross–Zagier conjecture, expressed as an equality between the Heegner $M$6-adic $M$7-function and the linear term of the two-variable analytic $M$8-adic $M$9-function (Howard, 2012).

A critical-slope refinement is established by Kobayashi. For a $C=\prod_{p\mid N} C_p$0-ordinary newform $C=\prod_{p\mid N} C_p$1 with critical-slope stabilization $C=\prod_{p\mid N} C_p$2, assuming $C=\prod_{p\mid N} C_p$3 is not $C=\prod_{p\mid N} C_p$4-critical and $C=\prod_{p\mid N} C_p$5 satisfies the Heegner hypothesis, the derivative of the critical-slope $C=\prod_{p\mid N} C_p$6-adic $C=\prod_{p\mid N} C_p$7-function at the center is proportional to Nekovář’s $C=\prod_{p\mid N} C_p$8-adic height of the Heegner class: $C=\prod_{p\mid N} C_p$9 In weight $p$00, this yields consequences for Perrin-Riou’s conjecture and the $p$01-part of BSD in analytic rank one (Büyükboduk et al., 2018).

Two distinct family-theoretic enlargements have appeared. Disegni and collaborators prove a universal $p$02-adic Gross–Zagier formula over a Hida family $p$03, constructing a universal Heegner class $p$04 whose $p$05-adic height is given by the cyclotomic derivative of a multivariable $p$06-adic $p$07-function. At classical points this specializes to the expected Gross–Zagier formulas for Hilbert modular forms and CM twists (Disegni, 2019). Meanwhile, Kobayashi–Venjakob–Xiao obtain a finite-slope, three-variable $p$08-adic Gross–Zagier formula for triple product $p$09-adic $p$10-functions attached to Coleman families: $p$11 so that a special value of a finite-slope $p$12-adic $p$13-function is identified with the $p$14-adic Abel–Jacobi image of a generalized diagonal cycle (Huang, 2024).

A further development is a new proof of the $p$15-adic Gross–Zagier theorem via the BDP formula and Beilinson–Flach elements. In this approach, one compares Beilinson–Flach classes with Heegner cycles by a wall-crossing argument, treating both ordinary and non-ordinary scenarios, including cases with

$p$16

The resulting theorem gives

$p$17

with full explicit normalization in the nonvanishing range (Büyükboduk et al., 15 Apr 2026).

5. Higher-dimensional generalizations and arithmetic Gan–Gross–Prasad

In higher dimensions, the Gross–Zagier formula is treated as the first case of a much broader pattern: special cycles on Shimura varieties should correspond to central values or central derivatives of automorphic $p$18-functions. The modern formulation is the Arithmetic Gan–Gross–Prasad conjecture. For a special pair of Shimura data

$p$19

one obtains special cycles $p$20. When $p$21 is odd, these cycles lie just below middle dimension, and their Beilinson–Bloch heights are conjecturally related to first derivatives of Rankin–Selberg-type $p$22-functions. In the refined conjecture, the arithmetic intersection decomposition

$p$23

is expected to satisfy

$p$24

for tempered $p$25. The relative trace formula then compares this global arithmetic distribution with derivatives of orbital-integral distributions, reducing the conjecture to local identities such as the Arithmetic Fundamental Lemma and arithmetic transfer conjectures (Zhang, 2024).

The unitary Shimura-curve case provides a concrete realization of this philosophy. Nakayama studies the arithmetic Gan–Gross–Prasad conjecture in the modified Rapoport–Smithling–Zhang framework and reinterprets Xue’s numerical theorem, originally proved via the Gross–Zagier formula for quaternionic Shimura curves, in terms of RSZ Shimura curves. The main identity relates the Néron–Tate height pairing of a diagonal cycle $p$26 to the central value of

$p$27

and adjoint $p$28-values, and under the stated cohomological hypothesis proves the expected equivalence between nonvanishing of the height pairing and a simple zero of the Rankin–Selberg $p$29-function together with the multiplicity-one condition (Nakayama, 2022).

At the same time, the survey literature emphasizes that this higher-dimensional program remains incomplete. The main obstructions explicitly listed are the lack of unconditional height pairings in general, the absence of regular integral models at deeper levels, unresolved arithmetic transfer conjectures for ramified higher-rank cases, and the fact that the full AGGP conjecture depends on poorly understood Chow-group automorphy (Zhang, 2024).

6. Explicit arithmetic applications and current status

One of the most concrete uses of explicit Gross–Zagier formulas is the cube-sum and Sylvester-conjecture literature. Yuan–Zhang–Zhang’s framework is specialized in work on the curves

$p$30

using the CM curve $p$31. For the Heegner divisor $p$32, an explicit formula of the shape

$p$33

is used to prove that for any odd integer $p$34, there are infinitely many cube-free odd integers $p$35 with exactly $p$36 distinct prime factors such that $p$37 is a cube sum and infinitely many such that $p$38 is not a cube sum; for $p$39, the same argument yields rank and BSD consequences for the relevant cubic twists (Cai et al., 2014).

In a different CM family, Cai–Shu–Tian’s explicit formula is specialized to elliptic curves

$p$40

and used to prove

$p$41

for primes $p$42 with $p$43 not a cubic residue mod $p$44. This explicit Gross–Zagier identity is then used to establish the $p$45-part of the product BSD formula for $p$46 and $p$47 (Hu et al., 2017). Very recent work returns to the $p$48 cases of Sylvester’s conjecture and proves an explicit formula for the CM point $p$49 on

$p$50

namely

$p$51

The paper identifies the exact local factors at $p$52 and $p$53 responsible for the factor $p$54 (Yin, 2 Jul 2026).

The present status is therefore mixed. Several statements traditionally called the Gross–Zagier conjecture are now theorems: the rank-one divisibility conjecture in the torsion cases treated by Byeon–Kim–Yhee, the higher Green function log-algebraicity conjecture on $p$55, broad $p$56-adic Gross–Zagier formulas in ordinary, critical-slope, finite-slope, and family settings, and explicit Shimura-curve singular-moduli analogues (Byeon et al., 2015, Bruinier et al., 2022, Disegni, 2015, Büyükboduk et al., 2018, Huang, 2024, Phillips, 15 Sep 2025). At the same time, the higher-dimensional AGGP program, the full arithmetic transfer framework, and the more speculative motivic extensions beyond CM points remain open-ended directions rather than completed theories (Zhang, 2024, Brown et al., 6 Aug 2025).