Universal quadratic forms over real quadratic fields

A positive integral quadratic form is called universal if it represents all positive integers, such as sums of four squares. This definition makes sense for totally positive quadratic forms over the ring of integers of totally real number fields. For instance, Hans Maa{\ss} showed that sums of three squares are universal over $\mathbb{Q}(\sqrt{5})$. We will investigate universal quadratic forms over real quadratic fields and see that there is a connection between the minimal rank of universal quadratic forms, the class number, properties of class group $L$-functions (subconvexity!), and certain continued fraction expansions. In particular, many number fields do not possess any universal quadratic form with a small number of variables. (Joint work with Vítězslav Kala.)