Frédéric Déglise

Heights and the Gross-Zagier Formula

Working Group ReMoLD
2025–26
________

Schedule: Tuesday, 10:15 AM
Location: Room 435, M7-411, Zoom connection available

Overview: The class number formula, due to Gauss (quadratic case) and Dirichlet, has certainly been one of the main driving force of the geometric theory of numbers/arithmetic geometry up to the present day. Among the milestones, one can note the BSD conjecture in 1963 and the Bloch–Beilinson conjectures in the 1980s, whose Beilinson paper in 1984 stands as a cornerstone. In this landscape, the Gross–Zagier formula, published in 1984, appears both as a precursor and a justification for current conjectures on special values of L-functions. In particular, it constitutes one of the two major 20th-century cases that justify the conjectural generalization of Dirichlet’s class number formula beyond the number field setting. It also leads to the first cases beyond the CM situation where the BSD conjecture is known, thanks to the work of Kolyvagin (1990). The working group will focus on reading the paper by Gross and Zagier, in particular the proof of their main theorem, with possible extensions toward Arakelov theory.

Working Group Schedule:

04/11	Introduction	Olivier Taïbi	Overview and scheduling of talks.
18/11	Modular curve and Heegner points	tba	Definition of the modular curve X₀(N), cusps, and relations with modular forms. (Moduli space and rational model.) Atkin–Lehner involutions, Hecke operators. Heegner points. Rationality over the associated class field H. Galois action G = Gal(H/K). Associated divisors and action of the algebra T[G_K]. Ref: Gross-Zagier, §§1–2, Silverman, Ch. II, Darmon, Ch. 1–2.
25/11	Weil heights	tba	Cover Serre, §2 (except 2.12 if time is short) (Heights, properties and Northcott’s Theorem, line bundles and positivity.)
02/12	Néron–Tate heights	tba	Cover Serre, §3 (except 3.6, 3.9, 3.10, 3.11) Weil heights on abelian varieties (quadraticity), Néron–Tate heights. Fundamental properties (positivity and torsion detection, height pairing and positivity).
09/12	Local heights and Néron’s formula	tba	Cover Serre, §6 (except 3.6, 3.9, 3.10, 3.11) Definition of local heights and their relation with “global” heights.
16/12	Local heights on the modular curve	tba	Sections II.1 and II.2 of Gross-Zagier.
06/01	Computation of local heights I	tba	Sections II.3 and II.4 of Gross-Zagier.
13/01	Computation of local heights II (case of non-disjoint supports)	tba	Computation of heights in the case of non-disjoint supports and Section II.5 of Gross-Zagier.
20/01	Non-archimedean height computations I	tba	The goal of this series of three talks is to compute the local heights appearing in the Gross–Zagier formula, section III of Gross-Zagier. Preliminaries: integral model of X₀(N), Serre–Tate theorem, canonical liftings Gross.
27/01	Non-archimedean height computations II	tba	Sections III.2–5 of Gross-Zagier.
27/01	Non-archimedean height computations III	tba	Sections III.6–9 of Gross-Zagier.
27/01	L-functions and Rankin’s method	tba	Preliminaries: Rankin’s method, trace computations and Fourier transform. Sections IV.1–3 of Gross-Zagier.
03/02	Functional equation and special values	tba	Explicit computations of the L-series and its derivative I: Sections IV.4–5 of Gross-Zagier.
10/02	Special values in the critical case	tba	Explicit computations of the derivative series in the critical case: Section IV.6 of Gross-Zagier.
17/02	Final proof and applications	tba	A summary of the computations obtained in the previous talks will be made to derive the proof of the Gross–Zagier formula. The main applications will be presented. Section V of Gross-Zagier.
Date	Title	Speaker	Description	Notes

References:

Gauss: Gauss, Carl Friedrich. Disquisitiones Arithmeticae. Leipzig: Fleischer, 1801.
Dirichlet: Dirichlet, Johann Peter. Beweis des Satzes, daß jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Faktor sind, unendlich viele Primzahlen enthält. Abhandlungen der Königlichen Preußischen Akademie der Wissenschaften zu Berlin (1837–1839).
Birch–Swinnerton-Dyer: Birch, Bryan J. and Swinnerton-Dyer, Peter. Notes on elliptic curves. I–II. J. Reine Angew. Math. 212 (1963), 7–25; 218 (1965), 79–108.
Bloch: Bloch, Spencer. Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 545–548; voir aussi Ann. Sci. Éc. Norm. Sup. (4) 14 (1981), 171–202.
Beilinson (HLF): Beilinson, Alexander A. Higher regulators and values of L-functions. J. Soviet Math. 30 (1985), 2036–2070.
Beilinson (Heights): Beilinson, Alexander A. Height pairings between algebraic cycles. In Arithmetic Geometry (Storrs, 1984), Contemporary Mathematics 67, AMS, 1987, 1–25.
Gross–Zagier: Gross, Benedict H. and Zagier, Don B. Heegner points and derivatives of L-series. Invent. Math. 84 (1986), 225–320.
Serre: Serre, Jean-Pierre. Lectures on the Mordell-Weil Theorem. Translated and edited by Martin Brown from notes by Michel Waldschmidt. 3rd edition. Springer Fachmedien Wiesbaden (Vieweg+Teubner), Aspects of Mathematics, vol. 15, 2013.
Gross: Gross, Benedict H. On canonical and quasi-canonical liftings. Inventiones Mathematicae 84 (1986), 321–326.
Kolyvagin: Kolyvagin, Victor A. Finiteness of E(ℚ) and Ш(E,ℚ) for a subclass of Weil curves. Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 522–540; English transl. in Math. USSR Izv. 32 (1989), 523–541.
Gillet–Soulé: Gillet, Henri and Soulé, Christophe. Arithmetic intersection theory. Publ. Math. IHÉS 72 (1990), 93–174.
Bost–Gillet–Soulé: Bost, Jean-Benoît; Gillet, Henri; and Soulé, Christophe. Heights of projective varieties and positive Green forms. J. Amer. Math. Soc. 7 (1994), 903–1027.
Yuan-Zhang-Zhang: Yuan, Xinyi ; Zhang, Shou-Wu ; Zhang, Wei. The Gross–Zagier Formula on Shimura Curves. Annals of Mathematics Studies, no. 184. Princeton University Press, 2012. Exposé moderne et complet de la formule de Gross–Zagier.