Symmetries of quadratic form classes and of quadratic surd continued fractions. Part II: Classification of the periods’ palindromes

Abstract

According to a theorem by Lagrange, the continued fractions of quadratic surds are periodic. Their periods may have different types of symmetries. This work relates these types of symmetries to the symmetries of the classes of the corresponding indefinite quadratic forms. This allows classifying the periods of quadratic surds and simultaneously finding the symmetry type of the class of an arbitrary indefinite quadratic form and the number of its integer points contained in each domain of the Poincaré tiling of the de Sitter world, introduced in Part I of this paper. Moreover, we obtain the same result for every class of forms representing zero, i.e., when the quadratic surds are replaced by rational, using the finite continued fraction obtained from a special representative of that class. Finally, we show the relation between the reduction procedure for indefinite quadratic forms defined by continued fractions and the classical reduction theory, which acquires a geometric description by the results in Part I.