On discriminants of minimal polynomials of the Ramanujan $t_n$ class invariants

Abstract

We study the discriminants of the minimal polynomials ${\mathcal{P}}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive integers $n\equiv 11(\mathrm{mod}24)$. The historical precedent for doing so comes from Gross and Zagier, which is known for computing the prime factorizations of certain resultants and discriminants of the Hilbert class polynomials $H_n$. We show that $\Delta \left({\mathcal{P}}_n\right)$ divides $\Delta \left(H_n\right)$ with quotient a perfect square, and as a consequence, we explicitly determine the sign of $\Delta \left({\mathcal{P}}_n\right)$ based on the class group structure of the order of discriminant $-n$. We also show that the discriminant of the number field generated by $j\left(\frac{-1+\sqrt{-n}}{2}\right)$, where $j$ is the $j$-invariant, divides $\Delta \left({\mathcal{P}}_n\right)$. Moreover, we show that 3 never divides $\Delta \left({\mathcal{P}}_n\right)$ for all squarefree positive integers $n\equiv 11(\mathrm{mod}24)$.

11 pages