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| Gross–Zagier formula | |
|---|---|
| Name | Gross–Zagier formula |
| Field | Number theory, Algebraic geometry, Arithmetic geometry |
| Introduced | 1980 |
| Authors | Benedict Gross, Don Zagier |
| Related | Birch and Swinnerton-Dyer conjecture, Heegner point, L-function |
Gross–Zagier formula The Gross–Zagier formula is a result linking heights of special points on elliptic curves to derivatives of L-functions of modular forms, establishing a bridge between arithmetic of Mordell–Weil groups and analytic behavior of Dirichlet series. It provides explicit relations that played a central role in progress on the Birch and Swinnerton-Dyer conjecture and the proof of certain cases of Fermat's Last Theorem. The formula originates from work of Benedict Gross and Don Zagier in 1986.
Gross and Zagier prove that for an optimal elliptic curve E over Q associated to a newform f of weight 2 on Γ0(N), the Néron–Tate height of a certain Heegner point P_K constructed from an imaginary quadratic field K is proportional to the first derivative at s=1 of the complex L-series L(E,s), i.e. L'(E,1). Their formula explicitly relates the canonical height
The formula arose amid efforts to understand the Birch and Swinnerton-Dyer conjecture and the role of special values of L-functions, following numerical and theoretical work by John Tate, Bryan Birch, Peter Swinnerton-Dyer, and conceptual frameworks from Iwasawa theory and Galois representations. Influences include the theory of complex multiplication developed by Kurt Heegner and refined in work of Atkin–Lehner theory, the modularity conjectures culminating in results by Gerhard Frey, Ken Ribet, and Andrew Wiles, and earlier explicit formulae linking heights and L-values by Heegner and Gross's prior investigations. The Gross–Zagier formula provided evidence for analytic ranks and was instrumental in conditional proofs by Kolyvagin and subsequent unconditional results combining with modularity theorems by Breuil, Conrad, Diamond, and Taylor.
Understanding the statement and proof requires familiarity with several advanced concepts: the theory of modular forms and newforms on SL2(Z), the arithmetic of elliptic curves over Q and the structure of their Mordell–Weil theorem groups, the construction of Heegner points via complex multiplication on modular curves X0(N), the analytic continuation and functional equation of L-series attached to newforms from work of Hecke and Atkin, height pairings and the Néron model, local and global root numbers as in Tate's thesis, and intersection theory on arithmetic surfaces building on techniques of Arakelov and Faltings. The proof also uses the representation-theoretic language of automorphic forms, the theory of Rankin–Selberg convolution, and properties of Shimura curves in certain generalizations.
Gross and Zagier's approach combines analytic and geometric techniques. They realize Heegner divisors on X0(N) and compute the height pairing via intersection theory on modular curves, relating it to Fourier coefficients of derivatives of certain Eisenstein series through the Rankin–Selberg method and the Maass–Shimura operators. On the analytic side, explicit formulas for the derivative L'(f,1) arise from the Petersson inner product and period integrals of f against theta series attached to binary quadratic forms from K. The proof synthesizes work of Petersson, Rankin, Selberg, and uses explicit evaluation of local factors akin to computations by Gross and Zagier, together with harmonic analysis on adele groups inspired by Tate and Weil.
The Gross–Zagier formula has numerous consequences: it gives criteria for positive analytic rank of E when Heegner points are non-torsion, supplies explicit generators of the Mordell–Weil group in rank one cases, and, together with Euler system techniques of Victor Kolyvagin, leads to proofs of finiteness of Shafarevich–Tate group components and verification of the Birch–Swinnerton-Dyer conjecture for many rank one elliptic curves. It influenced progress on modularity lifting theorems used by Andrew Wiles and Richard Taylor, contributed to constructions in Iwasawa theory for anticyclotomic extensions by Bertolini and Darmon, and found applications in explicit computation of heights in works of Cremona and Silverman. The formula also plays a role in modern developments in Gross–Prasad conjecture contexts and reciprocity laws envisioned by Langlands.
Subsequent work generalized Gross–Zagier phenomena to higher-weight forms, higher-dimensional abelian varieties, and non-compact Shimura varieties. Extensions include the Gross–Zagier formula on Shimura curves by Shou-Wu Zhang, p-adic Gross–Zagier formulas by Perrin-Riou, Bertolini, and Darmon, and Gross–Zagier type results in the context of the Gan–Gross–Prasad conjecture by Gan, Gross, and Prasad. Further refinements involve explicit formulae for higher derivatives related to Beilinson–Bloch conjectures by Beilinson and Bloch, and relations to Kato's Euler systems and Skinner–Urban Iwasawa main conjecture progress.