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Special Values of Class Group $L$-Functions for CM Fields

  Published:2009-12-04
 Printed: Feb 2010
  • Riad Masri,
    University of Wisconsin, USA
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Abstract

Let $H$ be the Hilbert class field of a CM number field $K$ with maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$. We evaluate the second term in the Taylor expansion at $s=0$ of the Galois-equivariant $L$-function $\Theta_{S_{\infty}}(s)$ associated to the unramified abelian characters of $\operatorname{Gal}(H/K)$. This is an identity in the group ring $\mathbb{C}[\operatorname{Gal}(H/K)]$ expressing $\Theta^{(n)}_{S_{\infty}}(0)$ as essentially a linear combination of logarithms of special values $\{\Psi(z_{\sigma})\}$, where $\Psi\colon \mathbb{H}^{n} \rightarrow \mathbb{R}$ is a Hilbert modular function for a congruence subgroup of $SL_{2}(\mathcal{O}_{F})$ and $\{z_{\sigma}: \sigma \in \operatorname{Gal}(H/K)\}$ are CM points on a universal Hilbert modular variety. We apply this result to express the relative class number $h_{H}/h_{K}$ as a rational multiple of the determinant of an $(h_{K}-1) \times (h_{K}-1)$ matrix of logarithms of ratios of special values $\Psi(z_{\sigma})$, thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for $\Psi(z_{\sigma})$ in terms of exponentials of special values of $L$-functions.
Keywords: Artin $L$-function, CM point, Hilbert modular function, Rubin-Stark conjecture Artin $L$-function, CM point, Hilbert modular function, Rubin-Stark conjecture
MSC Classifications: 11R42, 11F30 show english descriptions Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]
Fourier coefficients of automorphic forms
11R42 - Zeta functions and $L$-functions of number fields [See also 11M41, 19F27]
11F30 - Fourier coefficients of automorphic forms
 

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