Faltings height
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| Faltings height | |
|---|---|
| Name | Gerd Faltings |
| Birth date | 1954 |
| Nationality | German |
| Known for | Proof of Mordell conjecture, contributions to arithmetic geometry |
Faltings height is a numerical invariant attached to abelian varieties and algebraic curves over number fields that measures arithmetic complexity. Introduced in the context of Arakelov theory and Diophantine geometry, it plays a central role in studies by Gerd Faltings, Jean-Benoît Bost, Serge Lang, Paul Vojta, and others concerning rational points, moduli, and heights on varieties. The invariant interacts with the theories of Arakelov theory, Néron models, Tate conjecture, and the Mordell conjecture proved by Faltings.
Definition and background
The invariant arises from an intersection of ideas in Arakelov theory, Néron model constructions, and metrics on determinant line bundles over moduli spaces such as Siegel modular variety and Shimura variety. For an abelian variety A defined over a number field K, one constructs a metrized line bundle on the base Spec of the ring of integers of K using a choice of symmetric ample line bundle and a canonical invariant metric at archimedean places inspired by work of André Weil, Armand Borel, and Shouwu Zhang. The resulting Arakelov degree of the determinant of cohomology defines the height. The background includes foundational contributions from David Mumford on moduli of abelian varieties, Alexander Grothendieck on schemes, and Jean-Pierre Serre on Galois representations.
Properties and basic examples
The Faltings height has several structural properties proved in the literature: functoriality under isogenies between abelian varieties, behavior under base change of number fields, and additivity in products of varieties. For principally polarized abelian varieties, normalization choices lead to invariance under twist by CM field actions or by Weil pairing-preserving automorphisms studied by Shimura and Taniyama. Basic examples include elliptic curves where the invariant coincides, up to explicit constants, with the stable Faltings height computed via the modular lambda invariant and the discriminant of the minimal Weierstrass equation; such computations relate to classical work by John Tate, Goro Shimura, and Christophe Soulé. For Jacobians of curves coming from families studied by Alexandre Grothendieck-style deformation theory, the height reflects the geometry of the underlying curve and its stable reduction as classified in work related to Deligne–Mumford compactification.
Computation and formulas
Explicit formulas for the height in special cases involve analytic torsion, determinants of Laplacians, and Arakelov intersection numbers developed by Raymond Bott, Isadore Singer, and Ribet-style arguments. For elliptic curves over Q, the stable Faltings height can be expressed in terms of the Petersson norm of the modular discriminant Δ and the modular parameter τ in the upper half-plane, relating to formulas appearing in the work of Niels Henrik Abel-inspired modular theory and classical modular forms studied by Kurt Gödel-no, by Émile Picard and Pierre Deligne. More generally, analytic expressions employ the determinant of cohomology (Quillen metric) introduced by Daniel Quillen and refined by Jean-Michel Bismut and Jeffrey Cheeger through analytic torsion techniques. Computational approaches use period matrices from Siegel modular form theory, comparisons of Néron models via Raynaud's criteria, and p-adic approximations anchored in Serre–Tate theory.
Relation to arithmetic invariants
The height connects to discriminants of number fields in the spirit of results like the Hermite–Minkowski theorem and interacts with conductor exponents appearing in the Artin conductor and Tate module local invariants. It appears in inequalities linking the height to the minimal discriminant of a model, echoing comparisons in the work of Szpiro and conjectures by Jean-Pierre Serre and André Oort on isogeny estimates. Through the arithmetic of Galois representations arising from Tate modules, the height influences the study of the Sato–Tate conjecture and the Hodge conjecture-adjacent structures on Hodge bundles in families over Shimura varieties studied by Deligne, Milne, and Mumford.
Applications and significance in diophantine geometry
Faltings height figures prominently in the proof of the Mordell conjecture by Gerd Faltings, via reduction to finiteness statements for isomorphism classes and moduli finiteness on Siegel modular variety-type parameter spaces. It provides effective and non-effective finiteness bounds for rational points on curves through height inequalities and the theory of canonical heights pioneered by John Tate and extended by Joseph Silverman. Applications include finiteness results for isogeny classes of abelian varieties used in Faltings's proof, uniformity conjectures for rational points inspired by Lucien Szpiro and Paul Vojta, and investigations of rational and integral points on moduli spaces by Bost, Zhang, and Caporaso.
Extensions and generalizations
Generalizations extend the notion to motives and variations of Hodge structure on higher-dimensional varieties, with contributions from Pierre Deligne, Kazuya Kato, Shinichi Mochizuki, and Shouwu Zhang proposing heights for mixed motives and adelic metrized line bundles. Non-archimedean analogues and p-adic constructions adapt methods from Bertrand Toën-style derived geometry and Fontaine-theory for p-adic Hodge structures. The framework also informs conjectures connecting heights to special values of L-functions as in the Beilinson conjectures and Bloch–Kato conjecture explorations by Spencer Bloch and Kazuya Kato.