Summary: We study realization fields and integrality of characters of discrete and finite subgroups of \(SL_2(\mathbf{C})\) and related lattices with a focus on on the integrality of characters of finite groups \(G\). Theory of characters of finite and infinite groups plays the central role in the group theory and the theory of representations of finite groups and associative algebras. The classical results are related to some arithmetic problems: the description of integral representations are essential for finite groups over rings of integers in number fields, local fields, or, more generally, for Dedekind rings. A substantial part of this paper is devoted to the following question, coming back to W. Burnside: whether every representation over a number field can be made integral. Given a linear representation \(\rho: G\to GL_n(K)\) of finite group \(G\) over a number field \(K/\mathbf{Q}\), is it conjugate in \(GL_n(K)\) to a representation \(\rho: G\to GL_n(O_K)\) over the ring of integers \(O_K\)? To study this question, it is possible to translate integrality into the setting of lattices.
This question is closely related to globally irreducible representations; the concept introduced by J. G. Thompson and B. Gross, was developed by Pham Huu Tiep and generalized by F. Van Oystaeyen and A. E. Zalesskii, and there are still many open questions. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, splitting fields, and, in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras.