In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A = A(K) of K. It consists of the points of G having values in A; the definition of the appropriate topology is straightforward only in case G is a linear algebraic group. In the case of G being an abelian variety, it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms.
In case G is a linear algebraic group, it is an affine algebraic variety in affine N-space. The topology on the adelic algebraic group is taken to be the subspace topology in AN, the Cartesian product of N copies of the adele ring. In this case, is a topological group.
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algebraicgroup, treated as an adelicalgebraicgroup. It does not completely include the automorphic form idea introduced above, in that the adelic approach...
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respect to the induced quotient measure. The Tamagawa measure on the adelicalgebraicgroup G(A) is now defined as follows. Take a left-invariant n-form ω on...
technical value for dealing with the case that G is an algebraicgroup, treated as an adelicalgebraicgroup. As a result, an entire philosophy, the Langlands...
algebraic analysis. Important influences on the subject have been the technical requirements of theories of partial differential equations, and group...
parabolic cohomology (from Eichler cohomology) provided a purely algebraic setting based on group cohomology, taking account of the cusps characteristic of non-compact...
equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the classical...
Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields...
PlanetMath. Stein, William (2004), A brief introduction to classical and adelicalgebraic number theory Neukirch, Jürgen (1999). Algebraische Zahlentheorie....
functions; these representations may be of adelicalgebraicgroups. When the group is the general linear group GL 2 {\displaystyle \operatorname {GL} _{2}}...
modular forms and other automorphic forms are viewed using adelicgroups. Abstract algebra Wiles's proof of Fermat's Last Theorem Serre 1973, Ch. VII...
Parshin, A. N.; Shafarevich, I. R., eds. (1994). Algebraic Geometry IV – Linear AlgebraicGroups Invariant Theory. Encyclopaedia of Mathematical Sciences...
theory P-adic Hodge theory Adelicgroup Leong, Y. K. (July–December 2018). "Shou-Wu Zhang: Number Theory and Arithmetic Algebraic Geometry" (PDF). Imprints...
leading figure in model theory, logic, and their applications in algebra, algebraic geometry, and number theory. He is Emeritus Professor of Mathematics...
Hecke operators is by means of double cosets in the modular group. In the contemporary adelic approach, this translates to double cosets with respect to...
Grothendieck Approaches to Adelic Points" (PDF). Archived (PDF) from the original on 21 July 2018. Neukirch, Jürgen (1999). "1.13". Algebraic Number Theory. Berlin...
reciprocity laws using cohomology of groups or representations of adelicgroups or algebraic K-groups, and their relationship with the original quadratic reciprocity...
book is modern in its consistent use of adelic and idèlic methods and the simultaneous treatment of algebraic number fields and rational function fields...
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