Bloch–Kato Conjecture (L-functions)
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In
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 (1777â ...
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 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 Â¼.


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 fields), the division into two reflecting the questions of:
  1. 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
  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 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" (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 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 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, 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

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External links


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