Positive definite binary Hermitian forms with finitely many exceptions
Let \(E/F\) be a CM extension of algebraic number fields and let \(L\) be a positive definite Hermitian lattice over the ring of integers \(\mathfrak O\) of \(E\). An element \(a\) of \(F\) is said to be exceptional for \(L\) if \(a\) is represented by all localizations of \(L\) but not by \(L\) itself. When the exceptional set is empty (finite), the lattice is said to be regular (almost regular, respectively). If the rank of \(L\) exceeds 2, it is known that every such lattice is almost regular [\textit{J. S. Hsia} and \textit{J. P. Prieto-Cox}, J. Number Theory 47, 175--189 (1994; Zbl 0797.11040)], but this is no longer the case for lattices of rank 2. Another Hermitian \(\mathfrak O\)-lattice \(M\) is said to be similar to \(L\) if \(L\) is isometric to \(M^{\alpha}\) for some nonzero \(\alpha \in F\). Similarity is an equivalence relation on the set of \(\mathfrak O\)-lattices, and similar lattices have exceptional sets of equal size. In this paper, it is shown that if a CM extension \(E/F\) and a non-negative integer \(k\) are fixed, then there are at most finitely many similarity classes of positive definite binary Hermitian \(\mathfrak O\)-lattices for which the exceptional set has size at most \(k\). This generalizes a finiteness result for the case \(F=\mathbb Q\) proved by the reviewer and \textit{A. Khosravani} [J. Number Theory 62, 368--374 (1997; Zbl 0871.11028)]. It is also proven here that for a fixed totally real field \(F\) of odd degree over \(\mathbb Q\), there are at most finitely many CM extensions \(E/F\) for which there exists a positive definite regular normal (i.e., norm and scale ideals are equal) binary Hermitian lattice over the ring of integers of \(E\).
scientific article; zbMATH DE number 5149377
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