Arithmetic geometry

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In mathematics, arithmetic geometry is roughly the application of techniques
from algebraic geometry to problems in number theory.[1]

Arithmetic geometry is centered around Diophantine geometry,
the study of rational points of algebraic varieties.[2][3]

In more abstract terms, arithmetic geometry can be defined as
the study of schemes of finite type over the spectrum of the ring of integers.[4]

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Introduction to arithmetic geometry - MIT Math

1. What is arithmetic geometry?

Algebraic geometry studies the set of solutions of
a multivariable polynomial equation
(or a system of such equations), usually over R or C.
For instance, x2 + xy − 5y2 = 1 defines a hyperbola.
It uses both commutative algebra
(the theory of commutative rings) and geometric intuition.

Arithmetic geometry is the same except that
one is interested instead in the solutions
where the coordinates lie in other fields that are usually
far from being algebraically closed.

Fields of special interest are Q
(the field of rational numbers) and Fp
(the finite field of p elements),
and their finite extensions.
Also of interest are solutions with coordinates
in Z (the ring of integers).