Stark's question and Popescu's conjecture for abelian \(L\)-functions. (Q2710133)

Let \(K/k\) be an abelian extension of number fields and let \(S\) be a finite set of primes of \(k\) containing the infinite primes plus the finite primes that ramify in \(K/k\). Fix two primes \(v_1,v_2\in S\) which split completely in \(K\), and assume that \(\text{card} S>2\). Let \(w_j\) be a prime of \(K\) above \(v_j\) (\(j=1,2\)). For a character \(\chi\) of the group \(G=\text{Gal}(K/k)\), define the Artin \(L\)-function \(L_S(s,\chi)\) by the usual Euler product, with the primes \(\in S\) removed. A question posed by \textit{H.M. Stark} in 1980 asks about the existence of two \(S\)-units \(\varepsilon_1\) and \(\varepsilon_2\) having some special properties and giving the value at \(s=0\) of \(L^{(2)}(s,\chi)\) (the second derivative of \(L_S(s,\chi)\) divided by 2!) in terms of certain \(2\times 2\) matrices depending upon \(| \varepsilon_i^\sigma| _{w_j}\) \((i,j=1,2)\). Here \(\sigma\) runs through \(G\). The author proves that an affirmative answer to this question implies the truth of a conjecture by \textit{C. Popescu} [J. Reine Angew. Math. 542, 85--111 (2002; Zbl 1074.11062)] for this particular situation. More exactly, the unit \(\varepsilon=(1/w^2)\varepsilon_1\wedge\varepsilon_2\) (external product) satisfies the main conditions in this conjecture, notably \(\overline{\chi}(R(\varepsilon))=L_S^{(2)}(0,\chi)\), where \(w\) is the number of roots of 1 in \(K\), the bar denotes complex conjugation and \(R(\varepsilon)\) is the regulator in \(\mathbb R[G]\).NEWLINENEWLINEThere is also a detailed discussion of an example in which \(k={\mathbb Q}(\sqrt{229})\) and \(K\) is its Hilbert class field, \([K:k]=3\). Using PARI, the author shows that the units \(\epsilon_1,\varepsilon_2\) do exist. From his main result it then follows that Popescu's conjecture holds in this case. Another example, in which a prime ramifies in \(K/k\), is discussed briefly.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].