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 study of special values of L-functions is a subfield of

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� ...

devoted to generalising formulae such as the

Leibniz formula for pi Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...

, namely :

1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \;=\; \frac,\!

by the recognition that expression on the left-hand side is also ''L''(1) where ''L''(''s'') is the

Dirichlet L-function In mathematics, a Dirichlet ''L''-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and ''s'' a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By a ...

for the Gaussian field. This formula is a special case of 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. ...

, and in those terms reads that the Gaussian field has class number 1, and also contains four

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 in ...

, so accounting for the factor ¼.

Conjectures

There are two families of conjectures, formulated for general classes of ''L''-functions (the very general setting being for ''L''-functions ''L''(''s'') associated to

Chow motive In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham coho ...

s over

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 division into two reflecting the questions of:

how to replace π in the Leibniz formula by some other "transcendental" number (whether or not it is yet possible for
transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendence ...
to provide a proof of the transcendence); and
how to generalise the rational factor in the formula (class number divided by number of roots of unity) by some algebraic construction of a rational number that will represent the ratio of the L-function value to the "transcendental" factor.

Subsidiary explanations are given for the integer values of ''n'' for which such formulae ''L''(''n'') can be expected to hold. The conjectures for (a) are called ''Beilinson's conjectures'', for

Alexander Beilinson Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1 ...

. The idea is to abstract from the

regulator of a number field In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a posi ...

to some "higher regulator" (the

Beilinson regulator In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology: :K_n (X) \rightarrow \oplus_ H_D^ (X, \mathbf Q(p)). Here, ''X'' is a complex smooth projective variety ...

), a determinant constructed on a real vector space that comes from

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense o ...

. The conjectures for (b) are called the ''Bloch–Kato conjectures for special values ''(for

Spencer Bloch Spencer Janney Bloch (born May 22, 1944; New York City) is an American mathematician known for his contributions to algebraic geometry and algebraic ''K''-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Departm ...

and

Kazuya Kato is a Japanese mathematician. He grew up in the prefecture of Wakayama in Japan. He attended college at the University of Tokyo, from which he also obtained his master's degree in 1975, and his PhD in 1980. He was a professor at Tokyo University ...

– NB this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the

Milnor conjecture In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' wi ...

, a proof of which was announced in 2009). For the sake of greater clarity, they are also called the ''Tamagawa number conjecture'', a name arising via the

Birch–Swinnerton-Dyer conjecture In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory a ...

and its formulation as an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...

analogue of 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 ...

problem for

linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...

s.Matthias Flach, ''The Tamagawa Number Conjecture'' (PDF)
/ref> In a further extension, the equivariant Tamagawa number conjecture (ETNC) has been formulated, to consolidate the connection of these ideas with

Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the ea ...

, and its so-called Main Conjecture.

Current status

All of these conjectures are known to be true only in special cases.

Notes

References

* * * *

External links

L-funktionen und die Vermutingen von Deligne und Beilinson (L-functions and the conjectures of Deligne and Beilsnson)
__notoc__ {{DEFAULTSORT:Special Values Of L-Functions Zeta and L-functions

Conjectures

Current status

See also

Notes

References

External links