Positive definite binary hermitian forms with finitely many exceptions

Let E/F be a CM extension of number fields, and L be a positive definite binary hermitian lattice over the ring of integers of E. An element in F is called an exception of L if it is represented by every localization of L but not by L itself. We show that if E/F and a positive integer k are given, then there are only finitely many similarity classes of positive definite binary hermitian lattices with at most k exceptions. This generalizes the corresponding finiteness result by Earnest and Khosravani [A.G. Earnest, A. Khosravani, Representation of integers by positive definite binary hermitian lattices over imaginary quadratic fields, J. Number Theory 62 (1997) 368–374, Theorem 2.2] for the case F=Q. We also prove that for a fixed totally real field F of odd degree over Q, there are only finitely many CM extensions E/F for which there exists a positive definite regular normal binary hermitian lattice over the ring of integers of E.