On Ramanujan's cubic continued fraction
We study a certain finitely generated multiplicative subgroup of the Hilbert class field of a quartic CM field. It consists of special values of certain theta functions of genus 2 and is analogous to the group of Siegel units. Questions of integrality of these specials values are related to the arithmetic of the Siegel moduli space.
We have for is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue to which the Hecke series is attached. This result yields the new bound
We address the question of when an integer in a totally real number field can be written as the sum of three squared integers from the field and more generally whether it can be represented by a positive definite integral ternary quadratic form over the field. In recent work with Piatetski-Shapiro and Sarnak we have shown that every sufficiently large totally positive square free integer is globally integrally represented if and only if it is so locally at all places, thus essentially resolving...