number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, the Stark conjectures, introduced by and later expanded by , give

conjectural 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 ...

information about the

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...

of the leading term in the

Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

of an

Artin L-function In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. T ...

associated with a

Galois extension In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ...

''K''/''k'' of

algebraic 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 f ...

s. The conjectures generalize the

analytic class number formula In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function. General statement of the class number formula We start with the following data: * is a number field. ...

expressing the leading coefficient of the Taylor series for the

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 ca ...

of a number field as the product of a regulator related to

S-units In mathematics, in the field of algebraic number theory, an ''S''-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for ''S''-units. Definition Let ''K'' be a numb ...

of the field and a

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...

. When ''K''/''k'' is an

abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvabl ...

and the order of vanishing of the L-function at ''s'' = 0 is one, Stark gave a refinement of his conjecture, predicting the existence of certain S-units, called Stark units. and

Cristian Dumitru Popescu Cristian Dumitru Popescu (born 1964) is a Romanian-American mathematician at the University of California, San Diego. His research interests are in algebraic number theory, and in particular, in special values of L-functions. Education and c ...

gave extensions of this refined conjecture to higher orders of vanishing.

Formulation

The Stark conjectures, in the most general form, predict that the leading coefficient of an Artin L-function is the product of a type of regulator, the Stark regulator, with an

algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...

. When the extension is abelian and the order of vanishing of an L-function at ''s'' = 0 is one, Stark's refined conjecture predicts the existence of the Stark units, whose roots generate

Kummer extension In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer a ...

s of ''K'' that are abelian over the base field ''k'' (and not just abelian over ''K'', as Kummer theory implies). As such, this refinement of his conjecture has theoretical implications for solving

Hilbert's twelfth problem Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogue ...

. Also, it is possible to compute Stark units in specific examples, allowing verification of the veracity of his refined conjecture as well as providing an important computational tool for generating abelian extensions of number fields. In fact, some standard algorithms for computing abelian extensions of number fields involve producing Stark units that generate the extensions (see below).

Computation

The first order zero conjectures are used in recent versions of the PARI/GP computer algebra system to compute

Hilbert class field In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonicall ...

s of totally real number fields, and the conjectures provide one solution to Hilbert's twelfth problem, which challenged mathematicians to show how class fields may be constructed over any number field by the methods of

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...

Progress

Stark's principal conjecture has been proven in various special cases, including the case where the character defining the ''L''-function takes on only rational values. Except when the base field is the field of rational numbers or an imaginary

quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 an ...

, the abelian Stark conjectures are still unproved in number fields, and more progress has been made in function fields of an algebraic variety. related Stark's conjectures to the

noncommutative geometry Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some ge ...

Alain Connes Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vande ...

. This provides a conceptual framework for studying the conjectures, although at the moment it is unclear whether Manin's techniques will yield the actual proof. Recent progress has been made by Dasgupta and Kakd

Notes

References

* * * * * * * * * *

External links

* {{Authority control Conjectures Unsolved problems in number theory Field (mathematics) Algebraic number theory Zeta and L-functions