local-global principles
Generated by GPT-5-miniExpansion Funnel Raw 53 → Dedup 0 → NER 0 → Enqueued 0
| local-global principles | |
|---|---|
| Name | Local–global principles |
| Field | Number theory, Algebraic geometry, Arithmetic geometry |
| Introduced | 19th century |
| Notable | Helmut Hasse, John Tate, Yuri Manin, Jean-Pierre Serre, Alexander Grothendieck |
local-global principles
Local–global principles assert that global solvability of arithmetical or geometric problems can be detected from their solvability in all completions or localizations. Originating in the work of nineteenth- and twentieth-century mathematicians, these principles connect arithmetic properties over fields like Q and number fields with properties over completions such as Q_p and R, and they underpin major results in Diophantine geometry and the arithmetic of algebraic varieties.
Definition and motivation
A local–global principle posits that an arithmetic object defined over a global field, for example an equation over Q or a variety over a number field like Q(√2), has a rational point (or other global property) if and only if it has points over every completion such as R and every Q_p for primes p. Motivations include classical problems studied by mathematicians like Carl Friedrich Gauss, Adrien-Marie Legendre, and Ernst Eduard Kummer and modern structural frameworks developed by figures such as Helmut Hasse, John Tate, and Alexander Grothendieck to link local arithmetic data with global arithmetic invariants.
Classical instances (number theory and quadratic forms)
The prototypical classical instance is the Hasse–Minkowski theorem for quadratic forms: a quadratic form over a number field represents zero nontrivially over the global field precisely when it does so over all completions, a theorem proved by Helmut Hasse building on work of Hermann Minkowski and Emil Artin. Related results include the local-to-global behavior of norm equations studied by David Hilbert in his reciprocity law and the reciprocity formulations by Ernst Eduard Kummer and Richard Dedekind. Classical explicit criteria involve invariants such as Hilbert symbols and local invariants tied to places of number fields like those studied by Friedrich Hirzebruch and used by Yuri Manin in later contexts.
Hasse principle and counterexamples
The Hasse principle is the assertion that local solvability implies global solvability; it holds for some classes of varieties but fails in many others. Famous counterexamples include certain cubic curves and diagonal cubic surfaces discovered by Ernst Selmer and those constructed by Manjul Bhargava in statistical studies of failures; other notable failures arise for principal homogeneous spaces under tori investigated by Colliot-Thélène and Sansuc. Explicit counterexamples often involve torsors under algebraic groups such as elliptic curves with nontrivial Шafarevich–Tate groups studied by B. J. Birch and John Tate, and higher-dimensional failures appear in constructions related to work of Jean-Pierre Serre and Yu. I. Manin.
Cohomological and adelic formulations
Cohomological frameworks recast local–global statements in terms of Galois cohomology groups H^i(G_K, M) for the absolute Galois group of a global field, a viewpoint developed by John Tate and Serre and systematized in Alexander Grothendieck's cohomological machinery. Adelic formulations use the adele ring A_K of a number field K and the space of adelic points of an algebraic variety, linking to classical objects like the idele class group studied by Emil Artin and Claude Chevalley. These formulations allow the use of duality theorems such as Tate local and global duality and Poitou–Tate sequences attributed to J. Tate and G. Poitou to detect obstructions to lifting local solutions to global ones.
Methods and obstructions (Brauer–Manin, descent, patching)
Key obstructions to the Hasse principle and methods to study them include the Brauer–Manin obstruction introduced by Yuri Manin, which uses elements of the Brauer group Br(X) and pairings with adelic points to explain failures; the descent method pioneered by Louis Mordell and refined by Evgeny Kani and Jean-Louis Colliot-Thélène examines covers and torsors under algebraic groups; and patching techniques developed in the work of David Harbater, Moshe Jarden, and Armand Brumer that construct global objects from compatible local data. The Brauer–Manin obstruction relates to the arithmetic of algebraic groups studied by James S. Milne and to duality theorems of John Tate, while descent connects to the arithmetic of principal homogeneous spaces and Galois cohomology treated by Serre.
Applications and generalizations
Local–global principles and their obstructions inform broad areas: rational points on curves and higher-dimensional varieties such as K3 surfaces and Fano varieties; the arithmetic of elliptic curves and the analysis of their Mordell–Weil theorem ranks; the study of automorphic forms and reciprocity conjectures linked to Robert Langlands; and algorithmic questions in computational number theory as pursued by researchers at institutions such as Institute for Advanced Study and Mathematical Sciences Research Institute. Generalizations extend to global fields of positive characteristic like F_q(t), to arithmetic of algebraic stacks developed by Jacob Lurie and Max Lieblich, and to patching in geometric contexts influenced by Michael Artin and Harvard University research groups.