Néron models
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| Néron models | |
|---|---|
| Name | Néron models |
| Field | Algebraic geometry, Number theory |
| Introduced | 1964 |
| Introduced by | André Néron |
Néron models are smooth separated group schemes over a Dedekind scheme that extend an abelian variety or a smooth commutative algebraic group defined over the function field, characterized by a universal mapping property. They play a central role in the study of arithmetic of abelian varieties, linking local arithmetic at primes with global structures in diophantine geometry and the theory of heights.
Definition and basic properties
A Néron model of an abelian variety over the function field of a Dedekind scheme is a smooth separated commutative group scheme with the Néron mapping property: for every smooth scheme over the base Dedekind scheme and every morphism from its generic fiber to the abelian variety there exists a unique extension to the whole base. Key figures and works related to the formalization include André Néron, Alexander Grothendieck, Jean-Pierre Serre, John Tate, Serge Lang, and institutions such as the Institut des Hautes Études Scientifiques, École Normale Supérieure, and Harvard University. The property ensures compatibility with morphisms from models like the spectrum of a discrete valuation ring, elliptic curve minimal models studied by Kenneth A. Ribet, and compactifications used in the construction of Néron–Severi groups; it also ties to duality theorems due to Serre duality and results of Grothendieck in SGA.
Existence and uniqueness
Existence and uniqueness results were developed by Néron and later refined by Grothendieck and others; uniqueness follows from the mapping property and representability conditions used by researchers at Université Paris-Sud and Princeton University. For abelian varieties over the function field of a Dedekind scheme such as the spectrum of the ring of integers of a number field like France's or United States's examples, existence is guaranteed, with proofs relying on reductions to discrete valuation rings studied by Jean-Pierre Serre and local analysis inspired by John Tate's work on Tate modules. Counterexamples and limits to existence arise when attempting to extend to non-commutative groups or when base schemes fail hypotheses considered in work at Institute for Advanced Study.
Construction methods
Classical constructions use prolongation of group laws and smoothening techniques from the theory developed by Alexander Grothendieck in SGA, along with Raynaud's approach via rigid-analytic geometry as in the work of Michel Raynaud, and Bosch–Lütkebohmert–Raynaud methods from non-Archimedean analytic geometry involving authors affiliated with University of Münster and Max Planck Institute. Alternate methods employ Néron–Samuel models in the style of Pierre Samuel, minimal regular models from the work of Gerd Faltings and Jean-Michel Bismut, and approaches via the theory of models over discrete valuation rings inspired by Kenji Ueno and Shigefumi Mori. Constructions often use desingularization techniques related to work at Hiroshima University and canonical extensions influenced by results of Serge Lang.
Special fibers and reduction types
The structure of the special fiber of a Néron model above a closed point of the base captures reduction behavior analogous to the Kodaira–Néron classification for elliptic curves developed by Kunihiko Kodaira and André Néron and refined by Tate. Types include good reduction, multiplicative reduction, and additive reduction, with finer invariants such as the component group, potentially tame or wild ramification studied by Jean-Pierre Serre and ramifications in the context of the Weil conjectures and Grothendieck's monodromy theorem. Component groups connect to the theory of Galois representations investigated by Pierre Deligne and to the conductors appearing in the work of John Tate and Andrew Wiles in modularity contexts.
Functoriality and base change
Néron models satisfy functoriality properties with respect to morphisms of abelian varieties and finite base change, but may fail to commute with arbitrary base change; these phenomena were analyzed by Alexander Grothendieck, Michel Raynaud, and later by researchers at University of Cambridge and Columbia University. The behavior under unramified and tamely ramified extensions interacts with the formation of the identity component and component groups, linking to specialization maps studied by Serre and duality pairings investigated by John Tate and Barry Mazur. Applications of base change properties appear in moduli problems treated at Princeton University and in descent arguments used by Jean-Pierre Serre and Gerd Faltings.
Examples and applications
Prominent examples include the Néron model of an elliptic curve over the ring of integers of a number field, studied in detail by André Néron and John Tate; Jacobians of curves such as those in the work of Riemann and Bernhard Riemann appear as primary examples with applications in the proof of the Mordell conjecture by Gerd Faltings and in the arithmetic of rational points studied by Serge Lang. Applications extend to the theory of heights in diophantine geometry developed by Joseph H. Silverman and Serge Lang, to Néron–Tate heights used by Andrew Wiles and Benedict Gross in Iwasawa theory contexts investigated by Kenkichi Iwasawa, and to explicit computational tools in the study of L-functions pursued by groups at Princeton University and Institute for Advanced Study.
Cohomological and arithmetic aspects
Cohomological analyses of Néron models connect to flat cohomology, étale cohomology, and the study of the Tate–Shafarevich group as in work by John Tate, Cassels, and Serge Lang. Duality theorems relate the Néron model to the Picard and Albanese varieties central to studies by Alexander Grothendieck and Jean-Pierre Serre; arithmetic implications include contributions to the Birch and Swinnerton-Dyer conjecture examined by Bryan Birch and Peter Swinnerton-Dyer and to the formulation of conjectures by Barry Mazur and Kolyvagin in Iwasawa theory. Advanced research on integral models, conductor formulas, and monodromy traces continues in collaborations associated with Max Planck Institute, Université Paris-Sud, and research groups led by Benedict Gross and Richard Taylor.