Fields generated by finite rank subgroups of ℚ¯∗

Let [Formula: see text] be a finite rank subgroup of [Formula: see text]. We prove that the multiplicative group of the field generated by all elements in the divisible hull of [Formula: see text] is free abelian modulo this divisible hull. This proves that a necessary condition for Rémond’s generalized Lehmer conjecture is satisfied.

A relation between Fourier coefficients of holomorphic cusp forms and exponential sums

We consider certain specific exponential sums related to holomorphic cusp forms, give a reformulation for the Lehmer conjecture and prove that certain exponential sums of Fourier coefficients of holomorphic cusp forms contain information on other similar non-overlapping exponential sums. Also, we prove an Omega result for short sums of Fourier coefficients.