关于椭圆曲线二次扭转的尖点/平坦2 - 进\(L\)函数的岩泽不变量
Iwasawa invariants of sharp/flat $2$-adic $L$-functions for quadratic twists of elliptic curves
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中文总结 AI 辅助
研究\(\mathbb{Q}\)上在\(2\)处具有良好超奇异约化的椭圆曲线的Sprung尖点/平坦2 - 进\(L\)函数解析岩泽不变量在二次扭转下的变化，\(\mu\) - 不变量为零时给出\(\lambda\) - 不变量的显式差分公式及应用。
中文摘要 AI 辅助
本文旨在研究\(\mathbb{Q}\)上在\(2\)处具有良好超奇异约化的椭圆曲线的Sprung尖点/平坦2 - 进\(L\)函数的解析岩泽不变量在二次扭转下的变化。在\(\mu\) - 不变量消失的假设下，我们得到了尖点/平坦\(\lambda\) - 不变量的显式差分公式。该公式给出了在良好普通情形下松野公式的超奇异类似物。作为应用，按照哈特利 - 雷的方法，我们得到了具有规定尖点/平坦2 - 进岩泽\(\lambda\) - 不变量的二次扭转数的渐近下界。
英文摘要
The aim of this paper is to study the variation under quadratic twists of the analytic Iwasawa invariants of Sprung's sharp/flat 2-adic $L$-functions for elliptic curves over $\mathbb{Q}$ with good supersingular reduction at $2$. Under the hypothesis that the $μ$-invariant vanishes, we obtain an explicit formula for the sharp/flat $λ$-invariants. This formula gives a supersingular analogue of Matsuno's formula in the good ordinary case. As an application, we show that the sharp/flat $λ$-invariants can be made arbitrarily large even among quadratic twists by single primes. Moreover, using the method of Hatley-Ray, we obtain an asymptotic lower bound for the number of quadratic twists with a prescribed sharp/flat 2-adic Iwasawa $λ$-invariant.
