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# 11G: Arithmetic algebraic geometry (Diophantine geometry)

## Introduction

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.

## Subfields

• 11G10: Abelian varieties of dimension greater than 1, See also 14Kxx
• 11G15: Complex multiplication and moduli of abelian varieties, See also 14K22
• 11G18: Arithmetic aspects of modular and Shimura varieties, See also 14G35
• 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
• 11G40: L-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, See also 14G10
• 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