Arithmetic geometry

[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.

[14] Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with Michael Artin and Jean-Louis Verdier) by 1965.

[17][18] This connection would ultimately lead to the first proof of Fermat's Last Theorem in number theory through algebraic geometry techniques of modularity lifting developed by Andrew Wiles in 1995.

[20] Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.

[27] In the 2010s, Peter Scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over p-adic fields with application to Galois representations and certain cases of the weight-monodromy conjecture.

The hyperelliptic curve defined by has only finitely many rational points (such as the points and ) by Faltings's theorem .