The Gauss conjecture states the existence of infinitely many real quadratic number fields with class number one. The weak Gauss conjecture states the existence of infinitely many number fields having class number one.
In the paper under review, the authors generalize the Gauss conjecture as follows. Consider pairs \((K, S)\) with \(K\) a global field and \(S\) a finite nonempty set of places of \(K\) containing the archimedean ones in the number field case. Fix a pair \((K_0, S_0)\) with the ring of \(S_0\)-integers principal. Then the main conjecture is that there are infinitely many pairs \((K,S)\) such that: (1) all the places of \(S_0\) split completely in \(K\), (2) \(K/K_0\) is of degree two and (3) the ring of \(S\)-integers is principal. The Gauss conjecture is just the case where \(K_0= {\mathbb Q}\) and \(S=\{P_\infty\}\) where \(P_\infty\) is the usual absolute value of the field of rational numbers.
The authors consider several cases of the main conjecture. In particular the case of complex quadratic fields, the case of hyperelliptic function fields and function fields which are Galois extensions of a given one or with certain ramification conditions. Among several other results, they prove that for \(q=4, 9, 25, 49\) or \(169\) there are infinitely many extensions \((K,S)\) of \(({\mathbb F}_q (T), \{P\})\) such that \(P\) splits completely in \(K\), \(K/{\mathbb F}_q(T)\) is a Galois extension and the ring of \(S\)-integers is principal.
From the geometric point of view, they show that if \(X\) is a curve of genus \(g_X\geq 2\) over \({\mathbb F}_q\) such that \({|S|\over g_X - 1} > \sqrt{q} -1\), then the class field tower of \((X, S)\) is finite. A similar result is obtained for Drinfeld modular curves. Finally, they apply classical and Drinfeld modular curves to solve cases of the main problem discussed at the beginning of the paper.