In algebraic geometry, a Néron model (or Néron minimal model, or minimal model) for an Abelian variety AK defined over a local field K is the "best possible" group scheme AO defined over the ring of integers R of the local field K that becomes isomorphic to AK after base change from R to K. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety In Mathematics, a local field is a special type of field that is a Locally compact Topological field with respect to a non-discrete topology
They were introduced by André Néron (1964). André Néron was a French Mathematician at the Université de Poitiers who worked on Elliptic curves and Abelian varieties.
Definition
Suppose that R is a Dedekind domain with field of fractions K, and suppose that AK is an Abelian variety over K. In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper Then a Néron model is defined to be a universal smooth scheme AR over R with a rational map to AK. This is a glossary of scheme theory. For an introduction to the theory of schemes in Algebraic geometry, see Affine scheme, Projective space, sheaf In Mathematics, in particular the subfield of Algebraic geometry, a rational map is a kind of Partial function between Algebraic varieties. More precisely, this means that AR is a smooth scheme over R with general fiber AK, such that any rational map from a smooth scheme over R to AR can be extended to a unique morphism.
Néron models exist and are unique (up to unique isomorphism) and are group schemes of finite type over R. In Mathematics, a group scheme is a Group object in the Category of schemes. The fiber of a Néron model over a closed point of Spec(R) is an algebraic group, but need not be an Abelian variety: for example, it may be disconnected, or unipotent, or a torus. In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse
References
- Artin, Michael (1986), “Néron models”, in Cornell, G. Michael Artin (born 1934 is an American Mathematician and a professor at MIT, known for his contributions to Algebraic & Silverman, Joseph H., Arithmetic geometry (Storrs, Conn. Joseph H Silverman (born 1955 is currently a professor of mathematics at Brown University. , 1984), Berlin, New York: Springer-Verlag, pp. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic 213–230, MR861977
- Bosch, Siegfried; Lütkebohmert, Werner & Raynaud, Michel (1990), Néron models, vol. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Michel Raynaud (born 16 June 1938) is a French Mathematician working in Algebraic geometry. 21, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Berlin, New York: Springer-Verlag, MR1045822, ISBN 978-3-540-50587-7
- I. Ergebnisse der Mathematik und ihrer Grenzgebiete /A Series of Modern Surveys in Mathematics is a series of scholarly monographs published by Springer-Verlag. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many V. Dolgachev (2001), “Néron model”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Néron, André (1964), “Modèles minimaux des variétes abèliennes sur les corps locaux et globaux”, Publications Mathématiques de l'IHÉS 21: 5–128, MR0179172, ISSN 1618-1913, <http://www.numdam.org/item?id=PMIHES_1964__21__5_0>
- W. The Encyclopaedia of Mathematics is a large reference work in Mathematics. André Néron was a French Mathematician at the Université de Poitiers who worked on Elliptic curves and Abelian varieties. Publications Mathématiques de l'IHÉS is a Mathematics journal Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. Stein, What are Néron models? (2003)
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