$p$-adic $L$-函数的二阶导数与秩二CM椭圆曲线的Shafarevich--Tate群

中文摘要 AI 辅助

对于一条具有复乘的秩二椭圆曲线 $E/\mathbb{Q}$，Coates、Liang和Sujatha给出了在好的普通素数 $p$ 处 $Sha(E/\mathbb{Q})[p^\infty]$ 消失的一个判据，并将其应用于 $p<30{,}000$ 时的五条此类曲线。我们证明了同一类型的分圆判据：在一个显式排除的素数集合之外，Mazur-Tate-Teitelbaum $p$-adic $L$-函数在中心点的归一化二阶Taylor系数是 $p$-adic 单位，当且仅当分圆 $p$-adic 调节子是单位且 $Sha(E/\mathbb{Q})[p^\infty]=0$；并且，根据Bannai和Kobayashi的理论，这当且仅当三个权为 $2p-1$ 的临界Hecke $L$-值的显式组合的赋值为恰好二。遵循Stein和Wuthrich的算法，我们对同样的五条曲线在低于 $30{,}000$ 的每个好的普通素数处计算了调节子：在排除集合之外的 $8{,}050$ 个素数中，除三个外，调节子均为单位。Coates、Liang和Sujatha的判据遗留的一个情形，即 $y^2=x^3+34x$ 在 $p=577$ 处，被新判据解决。我们提供了第一个等价性的Lean 4形式化证明，其中假设了文献中陈述的结果。

英文摘要

For an elliptic curve $E/\mathbb{Q}$ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of $Sha(E/\mathbb{Q})[p^\infty]$ at a good ordinary prime $p$ and applied it to five such curves for $p < 30{,}000$. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum $p$-adic $L$-function at the central point is a $p$-adic unit if and only if the cyclotomic $p$-adic regulator is a unit and $Sha(E/\mathbb{Q})[p^\infty] = 0$, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke $L$-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below $30{,}000$: it is a unit at all but three of the $8{,}050$ primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, $p = 577$ for $y^2 = x^3 + 34x$, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.

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48 pages, comments welcome