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The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem. [9] Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to k 1 and k 2 we often denote the resulting number as k 1, k 2 . [citation needed]
In arithmetic geometry, the Tate–Shafarevich group Ш(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group (/) = (,), where = (/) is the absolute Galois group of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the p-adic fields obtained from ...
The Weil pairing is an important concept in elliptic curve cryptography; e.g., it may be used to attack certain elliptic curves (see MOV attack). It and other pairings have been used to develop identity-based encryption schemes.
Among other statements, Poitou–Tate duality establishes a perfect pairing between certain Shafarevich groups.Given a global field , a set S of primes, and the maximal extension which is unramified outside S, the Shafarevich groups capture, broadly speaking, those elements in the cohomology of (/) which vanish in the Galois cohomology of the local fields pertaining to the primes in S.
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
an object is a pair (,) of a scheme U in C ... Lecture Notes in Mathematics. Lecture Notes in Mathematics. Vol. 1776. ... This page was last edited on 22 May 2024, ...
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Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.Geometric, algebraic, and arithmetic objects are assigned objects called K-groups.