Theta series and number fields: theorems and experiments

Abstract

Let d and n be positive integers and let K be a totally real number field of discriminant d and degree n. We construct a theta series $\theta _{K} \in {\mathcal {M}}_{d,n}$, where ${\mathcal {M}}_{d,n}$ is a space of modular forms defined in terms of n and d. Moreover, if d is square free and n is at most 4 then $\theta _{K}$ is a complete invariant for K. We also investigate whether or not the collection of $\theta$-series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant d, is a linearly independent subset of ${\mathcal {M}}_{d,4}$. This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well.