Special values ofL-functions

Inmathematics,the study ofspecial values ofL-functionsis a subfield ofnumber theorydevoted to generalising formulae such as theLeibniz formula forπ,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 also$L(1)$where$L(s)$is theDirichletL-functionfor the field ofGaussian rationalnumbers. This formula is a special case of theanalytic class number formula,and in those terms reads that the Gaussian field hasclass number 1.The factor${\tfrac {1}{4}}$on the right hand side of the formula corresponds to the fact that this field contains fourroots of unity.

Conjectures

There are two families of conjectures, formulated for general classes ofL-functions(the very general setting being forL-functions associated toChow motivesovernumber fields), the division into two reflecting the questions of:

1. how to replace$\pi$in theLeibniz formulaby some other "transcendental" number (regardless of whether it is currently possible fortranscendental number theoryto 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 theL-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 calledBeilinson's conjectures,forAlexander Beilinson.The idea is to abstract from theregulator of a number fieldto some "higher regulator" (theBeilinson regulator), a determinant constructed on a real vector space that comes fromalgebraic K-theory.

The conjectures for (b) are called theBloch–Kato conjectures for special values(forSpencer BlochandKazuya Kato;this circle of ideas is distinct from theBloch–Kato conjectureof K-theory, extending theMilnor conjecture,a proof of which was announced in 2009). They are also called theTamagawa number conjecture,a name arising via theBirch–Swinnerton-Dyer conjectureand its formulation as anelliptic curveanalogue of theTamagawa numberproblem forlinear algebraic groups.In a further extension, the equivariant Tamagawa number conjecture (ETNC) has been formulated, to consolidate the connection of these ideas withIwasawa theory,and its so-calledMain Conjecture.

Current status

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

See also

• Brumer–Stark conjecture

References

• Kings, Guido (2003),"The Bloch–Kato conjecture on special values ofL-functions. A survey of known results ",Journal de théorie des nombres de Bordeaux,15(1): 179–198,doi:10.5802/jtnb.396,ISSN1246-7405,MR2019010
• "Beilinson conjectures",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• "K-functor in algebraic geometry",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Mathar, Richard J. (2010), "Table of Dirichlet L-Series and Prime Zeta Modulo Functions for small moduli",arXiv:1008.2547[math.NT]

External links

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