Arithmetic geometry in the context of Rational point

In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Diophantine geometry is part of the broader field of arithmetic geometry.

Four theorems in Diophantine geometry that are of fundamental importance include:

Barry Charles Mazur (/ˈmeɪzər/; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.

Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field ${\displaystyle \mathbb {Q} }$ of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing ${\displaystyle \mathbb {Q} }$ by any number field.

Christophe Breuil (French: [kʁistɔf bʁœj]; born 1968) is a French mathematician, who works in arithmetic geometry and algebraic number theory.