In
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 study of special values of -functions is a subfield of
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 ...
devoted to generalising formulae such as the
Leibniz formula for , namely
by the recognition that expression on the left-hand side is also
where
is the
Dirichlet -function for the field of
Gaussian rational
In mathematics, a Gaussian rational number is a complex number of the form ''p'' + ''qi'', where ''p'' and ''q'' are both rational numbers.
The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(''i''), obtained b ...
numbers. This formula is a special case of 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 ...
, and in those terms reads that the Gaussian field has
class number 1. The factor
on the right hand side of the formula corresponds to the fact that this field contains four
roots of unity
In mathematics, a root of unity 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 number theory, the theory of group char ...
.
Conjectures
There are two families of conjectures, formulated for general classes of
-functions (the very general setting being for -functions 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 cohom ...
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 ...
s), the division into two reflecting the questions of:
- how to replace in the Leibniz formula by some other "transcendental" number (regardless of whether it is currently 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.
Transcendenc ...
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 -function value to the "transcendental" factor.
Subsidiary explanations are given for the integer values of
for which a formulae of this sort involving
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 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 vari ...
), 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 sens ...
.
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 who works at the University of Chicago and specializes in number theory and arithmetic geometry.
Early life and education
Kazuya Kato grew up in the prefecture of Wakayama in Japan. He attended college at the Univer ...
; this circle of ideas is distinct from the
Bloch–Kato conjecture of K-theory, extending the
Milnor conjecture, a proof of which was announced in 2009). 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 the ...
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 ...
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 ...
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 Tower of fields, towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic ...
, and its so-called Main Conjecture.
Current status
All of these conjectures are known to be true only in special cases.
See also
*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 ...
Notes
References
*
*
*
*
External links
L-funktionen und die Vermutingen von Deligne und Beilinson (L-functions and the conjectures of Deligne and Beilsnson)
{{DEFAULTSORT:Special Values Of L-Functions
Zeta and L-functions