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11G: Arithmetic algebraic geometry (Diophantine geometry)


This section is the intersection of fields 11 (Number Theory) and 14 (Algebraic Geometry). The typical question in this area asks, "Are there any points on this variety (i.e., whose coordinates satisfy certain polynomial equations) whose coordinates are rational?" For example, the answer is "yes" when the coordinates are to satisfy the equation x^2+y^2=1 but "no" when the coordinates are to satisfy y^2=x^3-5.

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?


Applications and related fields

Material on elliptic curves is collected in 14H52.

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


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

See also the references for number theory in general.

Software and tables

Other web sites with this focus

Selected topics at this site

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Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org