A question of Stark (Q1382067)

The author studies a problem related to \textit{H. M. Stark}'s conjectures about values of Artin's \(L\)-functions and their derivatives. For these conjectures, see Stark's article [Adv. Math. 35, No. 3, 197-235 (1980; Zbl 0475.12018)]. Let \(K\) be a totally real abelian extension of a real quadratic field \(k\). For a finite set \(S\) of primes in \(k\), denote by \(L_S(s,\chi)\) the Artin \(L\)-function attached to a character \(\chi\) of \(\text{Gal} (K/k)\), with the Euler factors corresponding to the primes in \(S\) removed. The set \(S\) is assumed to contain at least the infinite primes plus the ramifying finite primes. Let \(S_K\) consist of all primes in \(K\) extending the primes in \(S\). The problem in question asks about the existence of two particular \(S_K\)-units (i.e., elements of \(K\) that are locally units outside \(S_K\)) \(\varepsilon_1\), \(\varepsilon_2\) with the following properties: (i) For every character \(\chi\), the second derivative of \(L_S(s,\chi)\) at \(s=0\) can be expressed as a certain linear combination of regulator-type \(2 \times 2\) determinants defined by \(\varepsilon_1\) and \(\varepsilon_2\); (ii) the fields \(K(\sqrt{\varepsilon_1})\) and \(K(\sqrt{\varepsilon_2})\) are abelian over \(k\) and coincide; (iii) the principal ideals generated by \(\varepsilon_1\) and by its conjugates (over \(k\)) are equal, and the same is true of \(\varepsilon_2\). The author shows that such \(S_K\)-units do exist in the case of a quadratic extension \(K/k\), provided \(S\) contains at least four primes. It seems that the hardest problem here is to have the condition \(K(\sqrt{\varepsilon_1})=K(\sqrt{\varepsilon_2})\) satisfied when \(| S| \) equals 4.