论文标题
关于Ramanujan $ t_n $ class不变的最小多项式的判别因素
On discriminants of minimal polynomials of the Ramanujan $t_n$ class invariants
论文作者
论文摘要
我们研究了Ramanujan $ T_N $类不变的最小多项式$ \ Mathcal {P} _n $的判别因子，这些n $是为正整数定义的$ n \ equiv11 \ pmod {24} $。这样做的历史先例来自Gross和Zagier，该公司以计算Hilbert类多项式$ H_N $的某些结果和判别因素的主要因素化而闻名。我们表明，$δ\ left（\ Mathcal {p} _n \ right）$划分$Δ\ left（h_n \右）$，商标是完美的正方形，因此，我们明确确定了$Δ\ left的符号（\ nathcal {p} _n \ right）$基于类别$的$ - 我们还表明，由$ j \ left生成的数字字段的歧视（\ frac {-1+ \ sqrt {-n}}} {2}} {2} \ right）$，其中$ j $是$ j $ -invariant，$ j $ invariant，$ j $ tivides $δ\ weft（\ nathcal {p} p} _n \ right）$。此外，我们表明3永远不会除以$δ\ left（\ Mathcal {p} _n \ right）$ for All SquareFore Pastic Integers $ n \ equiv11 \ pmod {24} $。
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 $Δ\left(\mathcal{P}_n\right)$ divides $Δ\left(H_n\right)$ with quotient a perfect square, and as a consequence, we explicitly determine the sign of $Δ\left(\mathcal{P}_n\right)$ 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 $Δ\left(\mathcal{P}_n\right)$. Moreover, we show that 3 never divides $Δ\left(\mathcal{P}_n\right)$ for all squarefree positive integers $n\equiv11\pmod{24}$.
