Motivating the multiplicative spectrum.- Modular Mahler measures I.- Diophantine problems in many variables: The role of additive number theory.- Equations F(x, y, ?x) = 0.- A fundamental invariant in the theory of partitions.- Congruence properties of values of L-functions and applications.- On Tschirnhaus transformations.- Hecke operators and the nonvanishing of L-functions.- A new minor-arcs estimate for number fields.- Modular functions, Maple and Andrews’ tenth problem.- Hecke characters and formal group characters.- On certain Gauss periods with common decomposition field.- Theorems and conjectures involving rook polynomials with only real zeros.- An example of an elliptic curve with a positive density of prime quadratic twists which have rank zero.- Exponents of class groups of quadratic fields.- A local-global principle for densities.- Divisibility of the specialization map for twists of abelian varieties.