There are two families of conjectures, formulated for general classes of
-functions (the very general setting being for -functions associated to
Chow motives over
number fields), the division into two reflecting the questions of: 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 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 n for which a 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==