The Brumer–Stark conjecture is a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

in algebraic number theory giving a rough generalization of both the analytic class number formula for

Dedekind zeta function In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ...

s, and also of Stickelberger's theorem about the

factorization In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several ''factors'', usually smaller or simpler objects of the same kind ...

Gauss sums In algebraic number theory, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typically :G(\chi) := G(\chi, \psi)= \sum \chi(r)\cdot \psi(r) where the sum is over elements of some finite commutative ring , is a ...

. It is named after Armand Brumer and Harold Stark. It arises as a special case (abelian and first-order) of Stark's conjecture, when the

place Place may refer to: Geography * Place (United States Census Bureau), defined as any concentration of population ** Census-designated place, a populated area lacking its own municipal government * "Place", a type of street or road name ** Ofte ...

that splits completely in the extension is finite. There are very few cases where the conjecture is known to be valid. Its importance arises, for instance, from its connection with Hilbert's twelfth problem.

Statement of the conjecture

Let be an abelian extension of

global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global function fi ...

s, and let be a set of places of containing the

Archimedean place Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...

s and the prime ideals that ramify in . The -imprimitive equivariant Artin L-function is obtained from the usual equivariant Artin L-function by removing the

Euler factor In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhar ...

s corresponding to the primes in from the Artin L-functions from which the equivariant function is built. It is a function on the

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

s taking values in the complex

group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...

where is the

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...

of . It is analytic on the entire plane, excepting a lone simple pole at . Let be the group of

roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important i ...

in . The group acts on ; let be the annihilator of as a - module. An important theorem, first proved by C. L. Siegel and later independently by Takuro Shintani, states that is actually in . A deeper theorem, proved independently by

Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...

and Ken Ribet, Daniel Barsky, and Pierrette Cassou-Noguès, states that is in . In particular, is in , where is the cardinality of . The

ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a ...

of is a -module. From the above discussion, we can let act on it. The Brumer–Stark conjecture says the following: Brumer–Stark Conjecture. For each nonzero

fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...

\mathfrak

of , there is an "anti-unit" such that #

\mathfrak^ = (\varepsilon).

#The extension

K \left(\varepsilon^ \right)/k

is abelian. The first part of this conjecture is due to Armand Brumer, and Harold Stark originally suggested that the second condition might hold. The conjecture was first stated in published form by John Tate.Tate, John
Brumer–Stark–Stickelberger
Séminaire de Théorie des Nombres, Univ. Bordeaux I Talence, (1980-81), exposé no. 24. The term "anti-unit" refers to the condition that is required to be 1 for each Archimedean place .

Progress

The Brumer Stark conjecture is known to be true for extensions where * is cyclotomic: this follows from Stickelberger's theorem * is abelian over * is a quadratic extension * is a biquadratic extension Samit Dasgupta and Mahesh Kakde posted an article on

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

about the conjecture.

Function field analogue

The analogous statement in the function field case is known to be true, having been proved by John Tate and

, with a different proof by David Hayes.

References

{{DEFAULTSORT:Brumer-Stark conjecture Conjectures Unsolved problems in number theory Algebraic number theory Zeta and L-functions