Stark conjectures

Let $K/k$ be a finite Galois extension of number fields with Galois group $G$, and let $S$ be a finite set of places of $k$ containing all archimedean places and all places ramified in $K/k$. Let $S_K$ be the set of places of $K$ above $S$, let $Y_S(K)$ be the free abelian group on $S_K$, and put

$$X_S(K)=\ker\!\left(Y_S(K)\xrightarrow{\sum n_w w\mapsto\sum n_w}\mathbf Z\right).$$

Let $E_S(K)=\mathcal O_{K,S}^{\times}$ be the group of $S$-units. With absolute values normalized so that the product formula holds, the logarithmic map is

$$\lambda_S:E_S(K)\longrightarrow \mathbf R\otimes_{\mathbf Z}X_S(K),\qquad u\longmapsto-\sum_{w\in S_K}\log|u|_w\,w.$$

After tensoring with $\mathbf R$, this map is an isomorphism. Choose a $\mathbf Q[G]$-module isomorphism

$$f:\mathbf Q\otimes_{\mathbf Z}X_S(K)\xrightarrow{\sim}\mathbf Q\otimes_{\mathbf Z}E_S(K).$$

For an irreducible complex character $\chi$ of $G$, let $V_\chi$ be a representation affording $\chi$, and set

$$r_S(\chi)=\dim_{\mathbf C}\operatorname{Hom}_{\mathbf C[G]} \!\left(V_\chi,\mathbf C\otimes_{\mathbf Z}X_S(K)\right).$$

The $G$-equivariant automorphism $\lambda_S\circ f$ induces an automorphism of this multiplicity space; define the Stark regulator by

R_S(\chi,f)=\det\!\left((\lambda_S\circ f)_*\,igg|\, \operatorname{Hom}_{\mathbf C[G]}\!\left(V_\chi, \mathbf C\otimes_{\mathbf Z}X_S(K)\right) \right).

Let $L_S(s,\chi,K/k)$ be the Artin $L$-function with the Euler factors at places in $S$ omitted, and let

$$L_S^*(0,\chi,K/k)=\lim_{s\to0}s^{-r_S(\chi)}L_S(s,\chi,K/k)$$

be its leading coefficient at $s=0$. Then, for every $\chi$ and every such $f$,

$$\frac{L_S^*(0,\chi,K/k)}{R_S(\chi,f)}\in\overline{\mathbf Q}.$$

Equivalently, there is an algebraic number $A_S(\chi,f)$ such that

$$L_S^*(0,\chi,K/k)=A_S(\chi,f)R_S(\chi,f).$$