mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 ...

\tau(G)

of a simply connected simple

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that 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 In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

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 \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric an ...

s and observed that it is an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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 investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

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 phra ...

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 each d ...

, 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), 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. In 2014, Lurie received a MacArthur Fellowship. Lurie's research interests are algebraic geometry, topology, and ...

and

Dennis Gaitsgory Dennis Gaitsgory (born 17 November 1973) is an Israeli-American mathematician. He is a mathematician at Max Planck Institute for Mathematics (MPIM) at Bonn and is known for his research on the geometric Langlands program. Life and career Born ...

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

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

s, with part of the argument published in , and planned to be completed in a second volume using the Grothendieck- Lefschetz trace formula and the Ran space.

Applications

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

spin group In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathbb_2 \to \o ...

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 , last1=Weil , first1=André , author1-link=André Weil , title=Adeles and algebraic groups , orig-year=1961 , url=https://books.google.com/books?id=vQvvAAAAMAAJ , publisher=Birkhäuser Boston , location=Boston, MA , series=Progress in Mathematics , isbn=978-3-7643-3092-7 , mr=670072 , year=1982 , volume=23

History

Applications

See also

References

Further reading