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 ...

, the Weil conjecture on Tamagawa numbers is the statement that the

Tamagawa number In mathematics, the Tamagawa number \tau(G) of a semisimple algebraic group defined over a global field is the measure of G(\mathbb)/G(k), where \mathbb is the adele ring of . Tamagawa numbers were introduced by , and named after him by . Tsuneo T ...

\tau(G)

of a

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

simple

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

defined over a number field is 1. In this case, ''simply connected'' means "not having a proper ''algebraic'' covering" in the algebraic

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

sense, which is not always the topologists' meaning.

History

calculated the Tamagawa number in many cases of

classical group In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or ske ...

s and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. The first observation does not hold for all groups: found examples where the Tamagawa numbers are not integers. The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture.

Robert Langlands Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study o ...

(1966) introduced

harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...

methods to show it for

Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...

s. K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. proved it for all groups satisfying 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 integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of eac ...

, which at the time was known for all groups without ''E''₈ factors. V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant ''E''₈ case (see

strong approximation in algebraic groups In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several int ...

), thus completing the proof of Weil's conjecture. In 2011,

Jacob Lurie Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow. Life When he was a student in the Science, Mathematics, and Computer Science ...

and

Dennis Gaitsgory Dennis Gaitsgory is a professor of mathematics at Harvard University known for his research on the geometric Langlands program. Born in Chișinău, now in Moldova, he grew up in Tajikistan, before studying at Tel Aviv University under Joseph Ber ...

announced a proof of the conjecture for algebraic groups over function fields over finite fields.

Applications

used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups. For

spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a L ...

s, the conjecture implies the known

Smith–Minkowski–Siegel mass formula In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism gr ...

References

* * *. * * * * * * * * *{{Citation , last=Lurie , first=Jacob , author-link=Jacob Lurie , title=Tamagawa Numbers via Nonabelian Poincaré Duality , year=2014 , url=http://www.math.harvard.edu/~lurie/282y.html

History

Applications

See also

References

Further reading