Eta products of weight one, and decomposition of primes in non-abelian extensions

Abstract

This article is a slightly extended and revised version of a conference talk at “Arithmetik an der A7” in Wurzburg, June 23rd, 2017. We present a conjecture on the coincidence of Hecke theta series of weight 1 on three distinct quadratic fields. Then we discuss a special instance of the Deligne–Serre Theorem, implying that the decomposition of prime numbers in a certain extension of the rationals is governed by the coefficients of the eta product $$\eta^{2}(z)$$ .