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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 11G: Arithmetic algebraic geometry (Diophantine geometry)

Introduction

However, for simplicity we have placed most materials regarding this topic with the corresponding section of 14: Algebraic Geometry.

Attached below are a few topics on a related theme: what number-theoretic questions can we ask (and answer) regarding geometric figures?

History

Applications and related fields

Material on elliptic curves is collected in 14H52.

see also 11Dxx, 14-XX, 14Gxx, 14Kxx

Subfields

• 11G05: Elliptic curves over global fields, See also 14H52
• 11G07: Elliptic curves over local fields, See also 14G20, 14H52
• 11G09: Drinfel´d modules; higher-dimensional motives, etc., See also 14L05
• 11G10: Abelian varieties of dimension greater than 1, See also 14Kxx
• 11G15: Complex multiplication and moduli of abelian varieties, See also 14K22
• 11G16: Elliptic and modular units, See also 11R27
• 11G18: Arithmetic aspects of modular and Shimura varieties, See also 14G35
• 11G20: Curves over finite and local fields, See also 14H25
• 11G25: Varieties over finite and local fields, See also 14G15, 14G20
• 11G30: Curves of arbitrary genus or genus not equal to 1 over global fields, See also 14H25
• 11G35: Varieties over global fields, See also 14G25
• 11G40: L-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, See also 14G10
• 11G45: Geometric class field theory, See also 11R37, 14C35, 19F05
• 11G50: Heights [See also 14G40] [new in 2000]
• 11G55: Polylogarithms and relations with K-theory [new in 2000]
• 11G99: None of the above but in this section

Parent field: 11: Number Theory

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Most textbooks in this area are limited to elliptic curves; see e.g. Coates, John: "Elliptic curves and Iwasawa theory", in Modular forms (Durham, 1983), 51--73; Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1984. Somewhat more focussed in this area are

• Lang, Serge: "Fundamentals of Diophantine geometry", Springer-Verlag, New York-Berlin, 1983. 370 pp. ISBN 0-387-90837-4
• Faltings, Gerd, "Neuere Entwicklungen in der arithmetischen algebraischen Geometrie", Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 55--61, Amer. Math. Soc., Providence, RI, 1987.
• Faltings, Gerd, "Recent progress in Diophantine geometry", Lecture Notes in Math., 1525 (pp. 78-86), Springer-Verlag, Berlin, 1992. ISBN 3-540-56011-4

See also the references for number theory in general.

Software and tables

Other web sites with this focus

Selected topics at this site

• How many triangles with all vertices lying in a square portion of Z^2? (up to similarity,...)
• How many triangles with all vertices lying in a square portion of Z^2? (up to similarity,...); asked again, and this time answered! There's a little follow-up information, too (never posted).
• How many triangles are there on a Geoboard (tm)?
• How many lattice points in a circle of radius r ? (pi*r^2; But error estimate = ?)
• Integrality questions concerning triangles.
• Is there a regular n-simplex in R^n with integer coordinates?
• Are there 7 points in the plane which are a rational distance apart?