ON DISCRIMINANTS OF MINIMAL POLYNOMIALS OF THE RAMANUJAN CLASS INVARIANTS

Abstract

AbstractWe study the discriminants of the minimal polynomials $\mathcal {P}_n$ of the Ramanujan $t_n$ class invariants, which are defined for positive $n\equiv 11\pmod {24}$ . We show that $\Delta (\mathcal {P}_n)$ divides $\Delta (H_n)$ , where $H_n$ is the ring class polynomial, with quotient a perfect square and determine the sign of $\Delta (\mathcal {P}_n)$ based on the ideal class group structure of the order of discriminant $-n$ . We also show that the discriminant of the number field generated by $j({(-1+\sqrt {-n})}/{2})$ , where j is the j-invariant, divides $\Delta (\mathcal {P}_n)$ . Moreover, using Ye’s computation of $\log|\Delta(H_n)|$ [‘Revisiting the Gross–Zagier discriminant formula’, Math. Nachr. 293 (2020), 1801–1826], we show that 3 never divides $\Delta(H_n)$ , and thus $\Delta(\mathcal{P}_n)$ , for all squarefree $n\equiv11\pmod{24}$ .