arithmetic geometry

English

Alternative forms

• arithmetical geometry

Noun

arithmetic geometry (uncountable)

1. (mathematics) A developing branch of mathematics in which techniques of algebraic geometry are applied to number theory, in particular being concerned with schemes of finite type over the spectrum of the ring of integers, Spec(ℤ).
   • 1994, Nancy Childress, John W. Jones (editors), Arithmetic Geometry: Conference on Arithmetic Geometry with an Emphasis on Iwasawa Theory, American Mathematical Society, back cover,

     This book resulted from a research conference in arithmetic geometry held at Arizona State University in March 1993.

   • 2010, Jean-Louis Colliot-Thélène, Peter Swinnerton-Dyer, Paul Vojta, Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School, Springer, page v:

     Arithmetic Geometry can be defined as the part of Algebraic Geometry connected with the study of algebraic varieties over arbitrary rings, in particular over nonalgebraically closed fields. It lies at the intersection between classical algebraic geometry and number theory.

   • 2013, Matthieu Romagny, “Models of Curves”, in Pierre Dèbes, Michel Emsalem, Matthieu Romagny, A. Muhammed Uludağ, editors, Arithmetic and Geometry Around Galois Theory, Springer, page 149:

     Their^{[semistable models']} occurrence in arithmetic geometry is ubiquitous for the study of ℓ-adic or p-adic cohomology, and of Galois representations.

   • 2014, Atsushi Moriwaki, Arakelov Geometry, American Mathematical Society, page vii:

     Arakelov geometry is one of the branches in arithmetic geometry.

Synonyms

• (branch of mathematics): arithmetic algebraic geometry, arithmetical algebraic geometry

Translations

Further reading

• Glossary of arithmetic and diophantine geometry on Wikipedia.Wikipedia
• Diophantine geometry on Wikipedia.Wikipedia
• Algebraic geometry on Wikipedia.Wikipedia
• Arithmetic Geometry on Wolfram MathWorld