Computation of $p$-units in ray class fields of real quadratic number fields

Let $K$ be a real quadratic field, let $p$ be a prime number which is inert in $K$ and let $K_p$ be the completion of $K$ at $p$. As part of a Ph.D. thesis, we constructed a certain $p$-adic invariant $u\in K_p^{\times}$, and conjectured that $u$ is, in fact, a $p$-unit in a suitable narrow ray class field of $K$. In this paper we give numerical evidence in support of that conjecture. Our method of computation is similar to the one developed by Dasgupta and relies on partial modular symbols attached to Eisenstein series.