Special values of L-functions
Encyclopedia
In mathematics
, the study of special values of L-functions is a subfield of number theory
devoted to generalising formulae such as the Leibniz formula for pi, namely
by the recognition that expression on the left-hand side is also L(1) where L(s) is the Dirichlet L-function for the Gaussian field
. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor ¼.
There are two families of conjectures, formulated for general classes of L-function
s (the very general setting being for L-functions L(s) associated to Chow motives over number fields), 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 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
. The idea is to abstract from the regulator of a number field to some "higher regulator", a determinant constructed on a real vector space that comes from algebraic K-theory
.
The conjectures for (b) are called the Bloch–Kato conjectures for special values (for Spencer Bloch
and Kazuya Kato
– NB 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). For the sake of greater clarity they are also called the Tamagawa number conjecture, a name arising via the Birch–Swinnerton-Dyer conjecture and its formulation as an elliptic curve
analogue of the Tamagawa number problem for linear algebraic group
s. In a further extension, the equivariant Tamagawa number conjecture (ETNC) has been formulated, to consolidate the connection of these ideas with Iwasawa theory
, and its so-called Main Conjecture; it is mathematical folklore that the ETNC and Main Conjecture should be equivalent.
All these conjectures are known only in special cases.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the study of special values of L-functions is a subfield of number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
devoted to generalising formulae such as the Leibniz formula for pi, namely
by the recognition that expression on the left-hand side is also L(1) where L(s) is the Dirichlet L-function for the Gaussian field
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....
. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor ¼.
There are two families of conjectures, formulated for general classes of L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s (the very general setting being for L-functions L(s) associated to Chow motives over number fields), 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 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 Beilinson
Alexander A. Beilinson 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...
. The idea is to abstract from the regulator of a number field to some "higher regulator", a determinant constructed on a real vector space that comes from algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
.
The conjectures for (b) are called the Bloch–Kato conjectures for special values (for Spencer Bloch
Spencer Bloch
Spencer Janney Bloch 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 Department of Mathematics of the University of Chicago. He is a member of the U.S...
and Kazuya Kato
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, Tokyo Institute of Technology and Kyoto University...
– NB this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the Milnor conjecture
Milnor conjecture
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory of a general field F with characteristic different from 2, by means of the Galois cohomology of F with coefficients in Z/2Z. It was proved by .-Statement of the theorem:Let F be a field of...
, 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 and its formulation as an elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
analogue of the Tamagawa number problem for linear algebraic group
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
s. In a further extension, the equivariant Tamagawa number conjecture (ETNC) has been formulated, to consolidate the connection of these ideas with Iwasawa theory
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 Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...
, and its so-called Main Conjecture; it is mathematical folklore that the ETNC and Main Conjecture should be equivalent.
All these conjectures are known only in special cases.