arithmetic geometry

arithmetic geometry
Name	arithmetic geometry
Field	Mathematics
Related	Algebraic number theory, Algebraic geometry, Diophantine geometry

Contents

arithmetic geometry is a field at the interface of number theory and algebraic geometry that studies solutions of polynomial equations with arithmetic constraints. It combines techniques from Diophantine equations, Galois theory, and schemes to address problems originating with figures such as Pierre de Fermat, Carl Friedrich Gauss, and Évariste Galois. Applications and influences reach into work of André Weil, Alexander Grothendieck, and Andrew Wiles.

Overview and Scope

Arithmetic geometry treats geometric objects defined over arithmetic base fields such as rational numbers, number fields, and finite fields, and it examines their rational points, integral points, and reduction behavior. Central objects include algebraic curves, abelian varietys, elliptic curves, and moduli spaces studied by researchers like Maxwell Rosenlicht and institutions such as the Institut des Hautes Études Scientifiques and the Clay Mathematics Institute. The field connects to conjectures and theorems associated with Shimura varieties, the Langlands program, and the Birch and Swinnerton-Dyer conjecture.

Foundational algebraic geometry notions, developed by Alexander Grothendieck, Jean-Pierre Serre, and Oscar Zariski, such as schemes, étale cohomology, and cohomology theories, pair with arithmetic concepts from algebraic number theory including local fields, p-adic numbers, ideal class groups, and adelic methods. Galois-theoretic frameworks from Évariste Galois and modern formulations by John Tate and Serre inform the study of Galois representations, inertia groups, and Frobenius endomorphism actions. Foundational results by André Weil on Weil conjectures and by Pierre Deligne on cohomological purity underpin many arithmetic arguments.

Major thematic pillars include rational and integral points on varieties as in Mordell conjecture (proved by Gerd Faltings), distribution of rational points explored in work of Yuri Manin and Paul Vojta, Diophantine approximation tied to Thue–Siegel–Roth theorem and Wolfgang Schmidt, and height functions developed by Shouwu Zhang and Enrico Bombieri. The study of L-functions and zeta functions of varieties features the Hasse–Weil L-function and connects to automorphic forms studied by Robert Langlands and Harish-Chandra. The arithmetic of elliptic curves and modular forms, crystallized in the proof of the Taniyama–Shimura conjecture (modularity theorem) by teams including Andrew Wiles and Richard Taylor, illustrates deep links among these themes.

Techniques include cohomological methods such as étale cohomology and crystalline cohomology advanced by Luc Illusie and Pierre Berthelot, Arakelov-theoretic approaches introduced by Suren Arakelov and expanded by Henri Gillet and Christophe Soulé, and model-theoretic tools from model theory contributors like Alexander Grothendieck's school and modern logicians such as Ehud Hrushovski. p-adic analytic techniques developed by Kurt Hensel's successors, including Jean-Marc Fontaine and Barry Mazur, study p-adic Hodge theory and (φ,Γ)-modules. Geometric invariant theory from David Mumford, deformation theory from Michael Artin, and intersection theory as in work by William Fulton are operational across many proofs. Computational methods using algorithms from Henri Cohen and packages developed by institutions like SageMath assist explicit calculations on Jacobians and Mordell–Weil groups.

Highlights include Faltings's proof of the Mordell conjecture; Deligne's proof of the last of the Weil conjectures; the proof of the Modularity theorem for semistable elliptic curves by Andrew Wiles and Richard Taylor; the proof of the Taniyama–Shimura conjecture implications for Fermat's Last Theorem by Gerhard Frey and Ken Ribet; and the development of the Langlands correspondence conjectures advanced by Robert Langlands and contributors like Michael Harris and Richard Taylor. Other central theorems include the Chebotarev density theorem originating in work of Nikolai Chebotarev and broadened by John Tate, Jean-Pierre Serre's open image results for Galois representations, and Barry Mazur's theorems on rational isogenies of elliptic curves.

Active directions include progress on the Birch and Swinnerton-Dyer conjecture pursued by researchers at institutions like Princeton University and funded projects of the Clay Mathematics Institute; refinements of the ABC conjecture proposed by Joseph Oesterlé and David Masser and claims by Shinichi Mochizuki; extensions of the Langlands program in work by Edward Frenkel and Peter Scholze; p-adic and motivic cohomology advances by Vladimir Voevodsky and Alexander Beilinson; and the study of rational points on higher-dimensional varieties by teams around Bjorn Poonen and Arne Brumer. Computational and explicit methods continue through efforts by John Cremona and groups at Max Planck Institute for Mathematics and Institut Henri Poincaré. Open problems include the full resolution of the Birch and Swinnerton-Dyer conjecture, proofs relating to the full Langlands reciprocity in number fields, effective bounds in Diophantine geometry sought by Paul Vojta and Enrico Bombieri, and verification of Mochizuki's claims on inter-universal Teichmüller theory by specialists such as Go Yamashita and Haruzo Hida.