The theta multiplier for number fields via \(p\)-adic planes (Q1816551)

Given any number field and Dirichlet character for that field, one classically constructs a theta function associated to them as a function on a product of several hyperbolic spaces, one for each infinite prime. This work extends ideas of Stark and Schwartz to construct a theta function as a function on a restricted product of \(p\)-adic hyperbolic spaces (for \(p\) ranging over infinite and almost all finite primes). The ``theta group'' of transformations with respect to which the function has a nice functional equation is determined. An explicit formula for the theta multiplier appearing in the functional equations is obtained through local calculations.