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| Néron–Tate | |
|---|---|
| Name | Néron–Tate |
| Field | Number theory; Algebraic geometry |
| Introduced | 1960s |
| Inventor | André Néron; John Tate |
Néron–Tate
The Néron–Tate height is a canonical quadratic height pairing on rational points of abelian varieties, introduced to refine Weil height constructions and to study Diophantine questions on elliptic curves and higher-dimensional abelian varieties. It links the work of André Néron and John Tate to central conjectures and theorems in Diophantine geometry, interacting with the Birch and Swinnerton-Dyer conjecture, the Mordell–Weil theorem, and the theory of heights used by researchers at institutions like Institute for Advanced Study and universities such as Harvard University and University of Cambridge. The construction leverages models like the Néron model and tools from Galois cohomology, Arakelov theory, and the study of L-functions by contributors including Gerd Faltings and Benedict Gross.
The canonical Néron–Tate height on an elliptic curve or an abelian variety is defined from a choice of an ample symmetric divisor or a line bundle, using a limiting process that averages the naive Weil height under multiplication-by-n maps, as in the approach of Joseph H. Silverman and Serge Lang. For an elliptic curve defined over a number field like Q or a global field studied by Alexander Grothendieck and Jean-Pierre Serre, one begins with local heights at places associated to p-adic number fields and archimedean embeddings such as those considered by Carl Friedrich Gauss and Bernhard Riemann, assembling them via product formulas akin to those used in Class field theory by Emil Artin and John Tate. The technical foundation exploits the existence of a Néron model over Dedekind schemes and the functoriality of the height under isogenies investigated by Goro Shimura and Igor Shafarevich.
The Néron–Tate height is a positive semidefinite quadratic form on the Mordell–Weil group modulo torsion, satisfying parallelogram laws familiar from Pythagoras-style identities and analogues used by André Weil. It vanishes precisely on the torsion subgroup characterized in the Nagell–Lutz theorem context for elliptic curves over Q and interacts with canonical pairings studied by Tate in the context of Tate modules and Weil pairing. Functorial properties include bilinearity under homomorphisms of abelian varieties as explored by Henri Poincaré and Alexander Grothendieck; the height behaves predictably under isogenies treated in the work of Jean-Pierre Serre and John Milnor.
On higher-dimensional abelian varieties such as Jacobians of curves studied by Pierre Deligne and David Mumford, the Néron–Tate height arises from symmetric ample line bundles and admits a decomposition into local contributions at places studied by Arakelov and refined by Shou-Wu Zhang. For Jacobians attached to modular curves investigated by Andrew Wiles and Barry Mazur, the height encodes arithmetic intersection numbers akin to those in Arakelov theory and links to regulators appearing in the formulation of special value conjectures by Bloch and Kato. It provides an intrinsic quadratic form on the finitely generated group whose rank is given by the Mordell–Weil theorem proved using techniques akin to those in Gerd Faltings's work on finiteness.
The bilinear pairing associated to the Néron–Tate height yields a canonical symmetric pairing on the free part of the Mordell–Weil group; its determinant is the regulator that features in the analytic side of the Birch and Swinnerton-Dyer conjecture and in conjectures of Beilinson and Bloch–Kato. The regulator interacts with periods studied by André Weil and special L-values examined by Richard Taylor and Christopher Hacon, and it is central to formulations comparing arithmetic invariants of Elliptic curves and abelian varieties with their L-functions in the work of W. N. Bailey and participants in the Langlands program such as Robert Langlands.
Effective computation of Néron–Tate heights for elliptic curves over Q has been implemented in computer algebra systems developed by teams at University of Washington and University of California, Berkeley, using algorithms from Cremona and refinements by Husemoller and Silverman. Techniques combine height pairings with explicit descent by John Cremona and point-search methods used in databases like those maintained by L-functions and Modular Forms Database contributors and projects associated with Cremona, Stein and Watkins. For higher-dimensional Jacobians, implementations rely on arithmetic intersection theory advances by Faltings, Moret-Bailly, and computational frameworks developed by David Kohel and Jeroen Sijsling.
The Néron–Tate height underpins results on rational and integral points in the traditions of Diophantus and Poincaré, enabling proofs and heuristics for finiteness results such as the Mordell conjecture proved by Gerd Faltings and effective bounds pursued by Alan Baker and Enrico Bombieri. It appears in explicit formulations of the Birch and Swinnerton-Dyer conjecture studied by John Coates and B. J. Birch, in descent arguments used by Dorian Goldfeld and Andrew Wiles, and in the study of equidistribution and unlikely intersections investigated by Zhang and Einsiedler. The pairing also informs cryptographic assessments of elliptic-curve schemes used in standards by organizations like NIST and companies such as RSA Security.
Generalizations include Néron–Tate-type canonical heights on dynamical systems as developed by Joseph H. Silverman and Matthew Baker, canonical local height pairings in non-archimedean analytic geometry advanced by Vladimir Berkovich and Robert Coleman, and height pairings in motivic cohomology pursued by Spencer Bloch and Kazuya Kato. Variants arise in Arakelov-theoretic heights for arithmetic surfaces studied by Szpiro and Faltings, and in p-adic height pairings introduced by Mazur and Tate with refinements by Bertolini and Darmon for Iwasawa-theoretic applications.