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

TL;DR

This paper investigates the discriminants of minimal polynomials of Ramanujan $t_n$ class invariants, revealing their divisibility properties, sign determination, and prime factorization characteristics related to class group structures.

Contribution

It explicitly relates the discriminants of Ramanujan $t_n$ invariants to Hilbert class polynomial discriminants and class group structures, providing new divisibility and sign results.

Findings

Discriminant of $ ext{min poly}$ divides Hilbert class polynomial discriminant with a perfect square quotient.

Sign of discriminant determined by class group structure of order $-n$.

3 does not divide the discriminant for all squarefree $n ot ext{ mod } 24$.

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 $n\equiv11\pmod{24}$. 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 $n\equiv11\pmod{24}$.