Gross–Zagier

Gross–Zagier
Name	Gross–Zagier
Field	Number theory
Authors	Benedict H. Gross; Don B. Zagier
First published	1986
Known for	Relation between heights of Heegner points and derivatives of L-series

Gross–Zagier

The Gross–Zagier result is a foundational theorem in modern Number theory connecting special values of L-functions to arithmetic geometry via heights of Heegner points on elliptic curves and modular parametrizations. Proven by Benedict H. Gross and Don Zagier in 1986, the theorem forged deep links between the Birch and Swinnerton-Dyer conjecture, modular forms, and the arithmetic of quadratic imaginary fields, influencing later work of Andrew Wiles, Richard Taylor, Karl Rubin, and Kolyvagin.

Introduction

The Gross–Zagier theorem relates the first derivative at the central critical point of the Hasse–Weil L-function of an elliptic curve over Q to the Néron–Tate height of a canonical Heegner point constructed using a modular parametrization by a newform of weight two on SL2(Z). Gross and Zagier’s work builds on classical results of Heegner, Birch and Swinnerton-Dyer, Goro Shimura, and Yutaka Taniyama, and it plays a pivotal role alongside the Modularity theorem proved by Wiles and Breuil, Conrad, Diamond, Taylor.

Statement of the Gross–Zagier theorem

Let E be an elliptic curve over Q of conductor N that is modular via a normalized newform f in S2(Γ0(N)), and let K be an imaginary quadratic field satisfying the Heegner hypothesis for N. The theorem asserts that L'(E/K,1) (equivalently the derivative at s=1 of the Rankin–Selberg convolution L(f,χ,s) for a ring class character χ of K) is proportional to the Néron–Tate height of a Heegner point P_K on E(K), with explicit nonzero constant given in terms of periods and local factors. This formula ties the analytic invariant L'(E,1) to the arithmetic invariant rank(E(Q)) and the regulator appearing in the Birch and Swinnerton-Dyer conjecture, and refines prior formulas of Gross in the context of Hecke characters and complex multiplication by linking heights to central derivatives.

Historical context and development

The theorem emerged from a confluence of threads in 20th-century arithmetic. Early motivations trace to Heegner’s construction of rational points on certain elliptic curves via complex multiplication, and to conjectures by Birch and Swinnerton-Dyer relating ranks to central L-values. Foundational analytic input came from the theory of modular forms developed by Atkin, Lehner, Petersson, and Rankin; algebraic geometry frameworks were shaped by Néron, Tate, and Weil. Gross and Zagier synthesized ideas from Gross’s work on local constants, Zagier’s insights on heights and modular symbols, and methods reminiscent of Poincaré and Selberg trace formulas. Subsequent advances by Kolyvagin used Euler system techniques to convert nonvanishing of L' into finiteness of Sha and determination of rank, while later modularity lifting by Wiles and Taylor–Wiles cemented applicability to wide classes of elliptic curves. Influential contributors in extensions include Zhang, Perrin-Riou, Gross–Kudla, Kato, and Skinner–Urban.

Key ideas and proof outline

Gross and Zagier combine analytic, algebraic, and geometric inputs. Analytically, they employ Rankin–Selberg convolution of the newform f with theta series attached to K, invoking properties of Rankin–Selberg L-functions, the functional equation, and explicit evaluation of Fourier coefficients via Waldspurger-type formulas. Geometrically, they construct Heegner points via the modular parametrization X0(N) → E and analyze their Néron–Tate heights using intersecting arithmetic divisors on modular curves and Arakelov theory-style pairings. The proof features explicit calculations of local intersection numbers at primes dividing N, invokes the theory of Gross–Zagier formula for derivatives of L-series, and uses the interplay of complex multiplication on elliptic curve with CM and the action of the Galois group of ring class fields. Auxiliary techniques include use of modular symbols, computations of Petersson inner products, analysis of Green’s functions on modular curves, and deformation arguments reminiscent of Iwasawa theory.

Applications and consequences

The Gross–Zagier theorem has immediate consequences for the Birch and Swinnerton-Dyer conjecture in analytic rank one: combined with Kolyvagin’s Euler system of Heegner points, it implies finiteness of the Shafarevich–Tate group for many elliptic curves and determines the rank as one when L'(E,1) ≠ 0. It underpins work on explicit construction of rational points by methods of Darmon, Cremona, and Silverman and informs algorithms implemented by John Cremona and others for computing ranks and regulators. Gross–Zagier techniques inform proofs and heuristics in the study of central values of Rankin–Selberg L-functions, the nonvanishing results of Bump–Friedberg type, and progress on the Bloch–Kato conjecture for modular motives. The theorem influenced breakthroughs in the proof of modularity lifting theorems by Wiles and subsequent developments by Breuil, Conrad, Diamond, and Taylor.

Extensions and generalizations

Extensions include higher-weight and higher-rank analogues by Zhang and Yuan–Zhang–Zhang relating derivatives of L-functions to heights of algebraic cycles on Shimura varieties, p-adic Gross–Zagier formulas by Perrin-Riou, Kobayashi, and Bertolini–Darmon–Prasanna, and generalizations to Hilbert modular forms by Zhang and Saito–Tunnell-type results linking central values and toric periods. Work by Kato on Euler systems, by Skinner–Urban on Iwasawa main conjectures, and by Nekovář on Selmer complexes expands the conceptual framework, while computational and explicit refinements by Gross–Kudla, Zagier himself, and Zagier–Kohnen yield concrete arithmetic data. The panorama now includes connections to the Langlands program, to arithmetic intersections on Shimura varietys, and to conjectures of Beilinson and Bloch on special values of L-functions.

Category:Number theory