Stark's question on special values of \(L\)-functions (Q2459337)

Let \(K/k\) be an abelian extension of global fields with Galois group \(G\). For a field \(F\) of characteristic zero denote by \(\hat{G}(F)\) the set of characters associated to the irreducible \(F\)-representations of \(G\). Let \(\mu_K\) denote the group of roots of unity in \(K\) and \(w_K\) its order. Let \(S\) be a finite nonempty set of primes in \(k\), containing at least all the infinite primes and all the primes which ramify in \(K/k\), and let \(U_S\) be the \({\mathbb Z}[G]\)-module of \(S\)-units in \(K\). One constructs a \({\mathbb C}[G]\)-module isomorphism (a so called \(G\)-equivariant \(S\)-regulator map) \[ {\mathcal R}_S:{\det}_{{\mathbb C}[G]}({\mathbb C}U_S)\to {\mathbb C}[G]. \] Let \[ \Theta_S:{\mathbb C}\setminus\{1\}\to {\mathbb C}[G], \] \[ \Theta_S(s)=\sum_{\chi\in\hat{G}({\mathbb C})}L_S(\chi,s)\cdot e_{\chi^{-1}}, \] where \(L_S(\chi,s)\) denotes the complex valued Dirichlet \(L\)-function with Euler factors at primes in \(S\) removed and \( e_{\chi}\in{\mathbb C}[G] \) is the usual idempotent, for all \(\chi\in\hat{G}({\mathbb C}).\) Let \(r_{S,\chi}:=\text{ord}_{s=0}L_S(\chi,s)\), and for all \(n\geq 0\) let \(e_{S,n}:=\sum_{\chi,r_{S,\chi}=n}e_{\chi}.\) What is the preimage \( {\mathcal L}_S^{(n)}:={\mathcal R}_S^{-1}(e_{S,n}\cdot{\mathbb Z}[G])\)? Assume that \(\text{card}(S)\geq 3\) and that \(S\) contains two distinct primes which split completely in \(K/k\). Let \(\Theta_S^{(2)}(0):=\lim_{s\to 0}\frac{\Theta_S(s)}{s^2}\). Stark's question: Is it true that \[ \Theta_S^{(2)}(0)\cdot {\mathcal L}_S^{(2)}\subseteq {\mathbb Z}[1/w_K]\cdot {\mathbb Q}(\wedge^2U_S)? \] The author proved in [Compos. Math. 140, No. 3, 631--646 (2004; Zbl 1059.11069)] that the answer is negative in the case of positive characteristic. In the present paper he shows that even in the case of number fields the answer is negative.