Explore: Arithmetic Dynamics On General Algebraic Varieties

Diophantine geometry, the study of rational points of algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of schemes

from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study

algebraic dynamics, where a polynomial or rational function is iterated. In geometric terms, that amounts to iterating a mapping from some algebraic variety

Mordell–Weil theorem Mazur's torsion theorem Congruent number Arithmetic of abelian varieties Elliptic divisibility sequences Mordell curve Fermat's Last

of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools

various levels of generality. Diophantine geometry in general is the study of algebraic varieties V over fields K that are finitely generated over their

local ring at (x-c), a point together with a neighborhood around it. Algebraic varieties also have a well-defined notion of tangent space called the Zariski

in the algebra, one has {x, y ⋅ z} = {x, y} ⋅ z + y ⋅ {x, z}. The last property often allows a variety of different formulations of the algebra to be given

calculus Nonarchimedean dynamics also known as p-adic analysis or local arithmetic dynamics Noncommutative algebra Noncommutative algebraic geometry a direction

may be computed in terms of division polynomials. Analytic torsion Arithmetic dynamics Flat module Annihilator (ring theory) Localization of a module Rank