Special Values of Class Group L-Functions for CM Fields Academic Article uri icon

abstract

  • Let H be the Hilbert class field of a CM number field K with maximal totally real subfield F of degree n over Q. We evaluate the second term in the Taylor expansion at s = 0 of the Galois-equivariant L-function θs∞ (s) associated to the unramified abelian characters of Gal(H/K). This is an identity in the group ring C[Gal(H/K)] expressing θs∞(n) (0) as essentially a linear combination of logarithms of special values {Ψ(zσ)}, where Ψ: Hn -R is a Hilbert modular function for a congruence subgroup of SL2(OF) and {zσ : σ ∈ Gal(H/K)} are CM points on a universal Hilbert modular variety. We apply this result to express the relative class number hH/hK as a rational multiple of the determinant of an (hK -1) x (hK -1) matrix of logarithms of ratios of special values Ψ(zσ), thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for Ψ(zσ) in terms of exponentials of special values of L-functions. © 2010 Canadian Mathematical Society.

author list (cited authors)

  • Masri, R.

citation count

  • 0

publication date

  • February 2010