number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, the Stark conjectures, introduced by and later expanded by , give conjectural information about the

coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...

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 ser ...

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 theo ...

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 ...

s. The conjectures generalize the

analytic class number formula In number theory, the class number formula relates many important invariants of an algebraic 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 nu ...

expressing the leading coefficient of the

Taylor series 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 ser ...

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 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 (for example, The set of all ...

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

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

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, which generate abelian extensions of number fields.

Formulation

General case

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

Abelian rank-one case

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 Stark units, whose roots generate

Kummer extension Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Chris ...

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 Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity t ...

Computation

Stark units in the abelian rank-one case have been computed in specific examples, allowing verification of the veracity of his refined conjecture. These also provide an important computational tool for generating abelian extensions of number fields, forming the basis for some standard algorithms for computing abelian extensions of number fields. The first rank-zero cases 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 extension, maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' ...

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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

Progress

Stark's principal conjecture has been proven in a few special cases, such as when 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 of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...

, which were covered in the work of Stark, the abelian Stark conjectures is still unproved for number fields. 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 g ...

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

. 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.

Variations

In 1980, Benedict Gross formulated the Gross–Stark conjecture, a ''p''-adic analogue of the Stark conjectures relating derivatives of Deligne–Ribet ''p''-adic ''L''-functions (for totally even characters of

totally real number field In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...

s) to ''p''-units. This was proved conditionally by

Henri Darmon Henri Rene Darmon (born 22 October 1965) is a French-Canadian mathematician. He is a number theorist who works on Hilbert's 12th problem and its relation with the Birch–Swinnerton-Dyer conjecture. He is currently a professor of mathematics at ...

, Samit Dasgupta, and Robert Pollack in 2011. The proof was completed and made unconditional by Dasgupta, Mahesh Kakde, and Kevin Ventullo in 2018. A further refinement of the ''p''-adic conjecture was proposed by Gross in 1988. In 1984, John Tate formulated the

Brumer–Stark conjecture The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's theorem about the factorization of Gauss sums ...

, which gives a refinement of the abelian rank-one Stark conjecture at totally split finite primes (for totally complex extensions of totally real base fields). The function field analogue of the Brumer–Stark conjecture was proved by John Tate and

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 Crafoor ...

in 1984. In 2023, Dasgupta and Kakde proved the

away from the prime 2. In 1996, Karl Rubin proposed an

integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...

refinement of the Stark conjecture in the abelian case. In 1999,

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 ca ...

proposed a function field analogue of Rubin's conjecture and proved it in some cases.

Notes

References

* * * * * * * * * *

External links

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