Beilinson regulator explained

In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology:

K _{n} (X) → ⊕ _{p} H ^{2 p - n} _{D} (X, Q(p)).

Here, X is a complex smooth projective variety, for example. It is named after Alexander Beilinson. The Beilinson regulator features in Beilinson's conjecture on special values of L-functions.

l O _{F} of a number field Fl O ^{x} _{F} → R ^{r _{1} + r _{2}} , x \mapsto ( log |\sigma (x)|) _{\sigma}

is a particular case of the Beilinson regulator. (As usual,

\sigma: F \subset C runs over all complex embeddings of F, where conjugate embeddings are considered equivalent.) Up to a factor 2, the Beilinson regulator is also generalization of the Borel regulator.

References

• Book: Beilinson's conjectures on special values of L-functions. 1988. Academic Press. 0-12-581120-9. M. Rapoport, N. Schappacher and P. Schneider.