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ON DISCRIMINANTS OF MINIMAL POLYNOMIALS OF THE RAMANUJAN
$t_n$ CLASS INVARIANTS
Published online by Cambridge University Press: 11 April 2023
Abstract
We 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
$n\equiv11\pmod{24}$.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 264 - 275
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
The author was supported by The 2022 Spirit of Ramanujan Fellowship and The 2022 Mehta Fellowship.
Dedicated to all my Rickoid friends who turned into a family
References
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