Special Values of Class Group L-Functions for CM Fields

Masri, Riad

Canadian journal of mathematics, Tome 62 (2010) no. 1, pp. 157-181

Résumé

Abstract. Let $H$ be the Hilbert class field of a $\text{CM}$ 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 ${{\Theta }_{{{S}_{\infty }}\,}}\left( s \right)$ associated to the unramified abelian characters of $\text{Gal}\left( H/K \right)$ . This is an identity in the group ring $\mathbb{C}\left[ \text{Gal}\left( H/K \right) \right]$ expressing $\Theta _{{{S}_{\infty }}}^{(n)}\,\left( 0 \right)$ as essentially a linear combination of logarithms of special values $\left\{ \Psi ({{z}_{\sigma }}) \right\}$ , where $\Psi :\,{{\mathbb{H}}^{n}}\,\to \,\mathbb{R}$ is a Hilbert modular function for a congruence subgroup of $S{{L}_{2}}\left( {{\mathcal{O}}_{F}} \right)$ and $\left\{ {{z}_{\sigma }}\,:\,\sigma \,\in \,\text{Gal}\left( H/K \right) \right\}$ are $\text{CM}$ 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 $\left( {{h}_{K}}\,-\,1 \right)\,\times \,\left( {{h}_{K}}\,-\,1 \right)$ 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.

Détail

DOI : 10.4153/CJM-2010-009-5

Mots-clés : Artin L-function, CM point, Hilbert modular function, Rubin–Stark conjecture