论文标题
在Iwasawa上
On Iwasawa $λ$-invariants for abelian number fields and random matrix heuristics
论文作者
论文摘要
在Ernvall-Metsänkylä和Ellenberg-Jain-Venkatesh之后，我们研究了$ p $ p $ adic-adic zeta-unctuction twist twist twist的零命令$ cnection $ p $ cnuction $ p $ c $χ$χ$χ$χ$χ$χ$χ$。我们对两种情况感兴趣：（i）字符$χ$是固定且Prime $ p $的不同，（ii）$ \ text {ord}（χ）$和Prime $ p $均固定，但允许$χ$变化。我们使用$ p $ -Adic随机矩阵理论预测这些$λ$ invariants的分布，并为这些预测提供数值证据。我们还研究了$χ$定期的素数的比例，这取决于$ \ mathbb {q}（χ）$中的$ p $分裂。最后，在广泛的附录中，我们将每个字符$χ$的导体$ \ leq 1000 $的$λ$ invariant的值和小尺寸的奇数$ p $。
Following both Ernvall-Metsänkylä and Ellenberg-Jain-Venkatesh, we study the density of the number of zeroes (i.e. the cyclotomic $λ$-invariant) for the $p$-adic zeta-function twisted by a Dirichlet character $χ$ of any order. We are interested in two cases: (i) the character $χ$ is fixed and the prime $p$ varies, and (ii) $\text{ord}(χ)$ and the prime $p$ are both fixed but $χ$ is allowed to vary. We predict distributions for these $λ$-invariants using $p$-adic random matrix theory and provide numerical evidence for these predictions. We also study the proportion of $χ$-regular primes, which depends on how $p$ splits inside $\mathbb{Q}(χ)$. Finally in an extensive Appendix, we tabulate the values of the $λ$-invariant for every character $χ$ of conductor $\leq 1000$ and for odd primes $p$ of small size.
