Weil conjecture on Tamagawa numbers
Encyclopedia
In mathematics
, the Weil conjecture on Tamagawa numbers is a result about algebraic group
s formulated by André Weil
in the late 1950s and proved in 1989. It states that the Tamagawa number τ(G) is 1 for any simply connected semisimple algebraic group
G defined over a number field K.
Here simply connected is in the algebraic group theory
sense of not having a proper algebraic covering, which is not always the topologists' meaning
.
cases to propose the conjecture. In particular for spin groups it implies the known Smith–Minkowski–Siegel mass formula
.
Robert Langlands
(1966) introduced harmonic analysis
methods to show it for Chevalley groups. J. G. M. Mars gave further results during the 1960s.
K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. proved it for all groups satisfying the Hasse principle
, which at the time was known for all groups without E8 factors. V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant E8 case (see strong approximation in algebraic groups), thus completing the proof of Weil's conjecture.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Weil conjecture on Tamagawa numbers is a result about algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
s formulated by André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
in the late 1950s and proved in 1989. It states that the Tamagawa number τ(G) is 1 for any simply connected semisimple algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
G defined over a number field K.
Here simply connected is in the algebraic group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
sense of not having a proper algebraic covering, which is not always the topologists' meaning
Simply connected space
In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...
.
History
Weil checked this in enough classical groupClassical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
cases to propose the conjecture. In particular for spin groups it implies the known Smith–Minkowski–Siegel mass formula
Smith–Minkowski–Siegel mass formula
In mathematics, the Smith–Minkowski–Siegel mass formula is a formula for the sum of the weights of the lattices in a genus, weighted by the reciprocals of the orders of their automorphism groups...
.
Robert Langlands
Robert Langlands
Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...
(1966) introduced harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
methods to show it for Chevalley groups. J. G. M. Mars gave further results during the 1960s.
K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. proved it for all groups satisfying the Hasse principle
Hasse principle
In mathematics, Helmut Hasse's local-global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number...
, which at the time was known for all groups without E8 factors. V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant E8 case (see strong approximation in algebraic groups), thus completing the proof of Weil's conjecture.