Arithmetic geometry

Arithmetic geometry studies polynomial equations through the arithmetic of their solution sets over number fields, finite fields, and related structures. It combines methods from algebraic geometry and number theory.

Algebraic geometry: The study of geometric objects defined by polynomial equations and the maps between them. Its geometric language supplies the varieties and schemes studied arithmetically.
Local field: A field complete with respect to a discrete valuation or an archimedean absolute value. Studying solutions over completions gives local tests for arithmetic properties.
Mordell–Weil theorem: The theorem that the rational points of an abelian variety over a number field form a finitely generated group. It constrains the structure of rational points on elliptic curves and related varieties.
Pierre de Fermat: A seventeenth-century French mathematician known for foundational work in number theory. His questions about integer solutions anticipated arithmetic study of polynomial equations.
Number field: A finite extension of the rational numbers. Number fields provide central domains over which arithmetic varieties are studied.
p-adic number: An element of the completion of the rational numbers under the p-adic absolute value. P-adic solutions capture behavior near a chosen prime.
Faltings's theorem: The theorem that a curve of genus greater than one over a number field has finitely many rational points. It resolves a central finiteness question about rational solutions to curve equations.
Diophantine equation: A polynomial equation whose solutions are sought in integers or other specified arithmetic domains. The subject generalizes classical questions about integer solutions to geometric settings.
Finite field: A field containing finitely many elements, with size a power of a prime. Counting solutions over finite fields often reveals information about varieties over number fields.
Reduction modulo a prime: The passage from equations over a ring of integers to equations over a residue field. Reducing a variety can expose finite-field point counts and the behavior of its fibers.

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