Math 2.0

Number theory Arithmetic geometry Theorem

Gross–Zagier theorem

Builds on Elliptic curveHeegner pointQuadratic fieldHeight functionModularity theoremHilbert class field

Statement

Let EE be an elliptic curve over ℚ\Q of conductor NN, with modular parametrisation φ ⁣:X0(N)→E\varphi \colon X_0(N) \to E given by the modularity theorem. Let KK be an imaginary quadratic field of discriminant DD such that every prime dividing NN splits in KK. Let xx be a Heegner point for KK; it is defined over the Hilbert class field HH of KK. Let yK∈E(K)y_K \in E(K) be the trace of φ(x)\varphi(x) from HH to KK. Let L(E/K,s)=L(E,s) L(ED,s)L(E/K, s) = L(E, s) \, L(E^D, s), where EDE^D is the quadratic twist of EE by KK, and let h^\hat{h} be the Néron–Tate canonical height on E(K)E(K). Then L(E/K,1)=0L(E/K, 1) = 0, and there is a real constant c>0c > 0, depending on EE and KK and given explicitly by periods and other invariants, such that L′(E/K,1)=c⋅h^(yK).L'(E/K, 1) = c \cdot \hat{h}(y_K).

Remarks

In particular, yKy_K has infinite order if and only if L′(E/K,1)≠0L'(E/K, 1) \ne 0, since h^\hat{h} vanishes exactly on torsion points. The Heegner hypothesis forces the sign of the functional equation of L(E/K,s)L(E/K, s) to be −1-1, which gives L(E/K,1)=0L(E/K, 1) = 0. The proof computes both sides. The derivative is computed by the Rankin–Selberg method, and the height by local intersection theory of Heegner points on X0(N)X_0(N). Gross and Zagier then compare the two as Fourier coefficients of modular forms. Gross and Zagier proved it in 1986, with some additional conditions on DD that later work removed. Kolyvagin's Euler system argument adds the following consequence. If L(E,s)L(E, s) has a zero of order at most 11 at s=1s = 1, then the rank of E(ℚ)E(\Q) equals that order and the Tate–Shafarevich group of EE is finite.

Going deeper

For a student

This is the strongest known evidence for the Birch and Swinnerton-Dyer conjecture, covering analytic rank 00 and 11. Gross and Zagier also used it, with Goldfeld's earlier work, to give an effective lower bound for class numbers of imaginary quadratic fields. Generalisations to Shimura curves and higher-dimensional settings are an active area.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

Show the path as a graph
Congruence subgroup, concept, no evidence levelCongruence subgroupconceptCusp form, concept, no evidence levelCusp formconceptElliptic curve, concept, no evidence levelElliptic curveconceptFinite field, concept, no evidence levelFinite fieldconceptHecke operator, concept, no evidence levelHecke operatorconceptHeegner point, concept, no evidence levelHeegner pointconceptHeight function, concept, no evidence levelHeight functionconceptHilbert class field, concept, no evidence levelHilbert class fieldconceptModular form, concept, no evidence levelModular formconceptQuadratic field, concept, no evidence levelQuadratic fieldconceptModularity theorem, statement, no evidence levelModularity theoremstatementGross–Zagier theorem, statement, no evidence levelGross–Zagier theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Congruence subgroup concept
  2. Cusp form concept
  3. Elliptic curve concept
  4. Finite field concept
  5. Hecke operator concept
  6. Heegner point concept
  7. Height function concept
  8. Hilbert class field concept
  9. Modular form concept
  10. Quadratic field concept
  11. Modularity theorem statement
  12. Gross–Zagier theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Heegner_point
About this page
TagBX558DTV2D
URIm2://BX558DTV2D
Kindstatement, theorem
WritingW0 stub
MSC11G05, 11G40, 11G50
StatusThis copy of the site has no build data. Run m2 status BX558DTV2D.
AboutHeegner point, Elliptic curve
UsesElliptic curve, Heegner point, Quadratic field, Height function, Modularity theorem, Hilbert class field
Changed2 November 2026

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