mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in the field of

arithmetic algebraic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties ...

, the Manin obstruction (named after

Yuri Manin Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical log ...

) is attached to a variety ''X'' over a

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global function f ...

, which measures the failure of the

Hasse principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an diophantine equation, integer solution to an equation by using the Chinese remainder theorem to piece together solutions mod ...

for ''X''. If the value of the obstruction is non-trivial, then ''X'' may have points over all

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...

s but not over the

. The Manin obstruction is sometimes called the Brauer–Manin obstruction, as Manin used the

Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...

of X to define it. For

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...

the Manin obstruction is just the

Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of ...

and fully accounts for the failure of the local-to-global principle (under the assumption that the Tate–Shafarevich group is finite). There are however examples, due to

Alexei Skorobogatov Alexei Nikolaievich Skorobogatov (russian: Алексе́й Никола́евич Скоробога́тов) is a British-Russian mathematician and Professor in Pure Mathematics at Imperial College London specialising in algebraic geometry. His ...

, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points.

References

* * * Diophantine geometry {{algebraic-geometry-stub