Special values of L-functions

Template:More citations needed In mathematics, the study of special values of Template:Mvar-functions is a subfield of number theory devoted to generalising formulae such as the [[Leibniz formula for pi|Leibniz formula for Template:Pi]], namely $$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots =\frac{\pi }{4},$$

by the recognition that expression on the left-hand side is equal to $L(1)$ where $L(s)$ is the [[Dirichlet L-function|Dirichlet Template:Mvar-function]] for the field of Gaussian rational numbers. The right-hand side then follows from the analytic class number formula, and the factor $\frac{1}{4}$ appears because the Gaussian field has class number 1 and contains four roots of unity.

Conjectures

There are two families of conjectures, formulated for general classes of [[L-function|Template:Mvar-functions]] (the very general setting being for Template:Mvar-functions associated to Chow motives over number fields), the division into two reflecting the questions of:

1. how to replace $\pi$ in the Leibniz formula by some other "transcendental" number (regardless of whether it is currently possible for transcendental number theory to provide a proof of the transcendence); and
2. 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 Template:Mvar-function value to the "transcendental" factor.

Subsidiary explanations are given for the integer values of $n$ for which formulae of this sort involving $L(n)$ 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" (the Beilinson 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; 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 and its formulation as an elliptic curve analogue of the Tamagawa number problem for linear algebraic groups. 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.

Current status

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

See also

• Brumer–Stark conjecture

References

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External links

• L-funktionen und die Vermutingen von Deligne und Beilinson (L-functions and the conjectures of Deligne and Beilsnson)