Special values of $p$-adic $L$-functions and Iwasawa $λ$-invariants of Dirichlet characters

Abstract

We investigate the Iwasawa $λ$-invariant $λ_p(χ)$ of the $p$-adic $L$-function associated with a Dirichlet character $χ$ for odd primes $p$. Using special values of the $p$-adic $L$-function and its derivative, we derive novel and computationally efficient criteria to distinguish between the cases $λ= 0$, $λ= 1$, $λ= 2$, and $λ\geq 3$. In particular, we study the case when the $p$-adic $L$-function vanishes at $s=0$. Building on the results of Ferrero-Greenberg and of Gross-Koblitz, we establish explicit conditions for $λ_p(χ) >1 $ and $λ_p(χ) > 2$. Furthermore, we extend the methods of Ernvall-Metsänkylä and of Dummit et al. to calculate the $λ$-invariant. This is achieved by twisting $χ$ by characters $ψ$ of $p$-power order and using the values of the $p$-adic $L$-function at $s=2-p, \dots, 0$. Additionally, we utilize the value at $s=1$ to compute $λ_p(χ)$. Finally, these results are applied to generate numerical data on the distribution of $λ$-invariants, when either the prime $p$ or the Dirichlet character $χ$ is fixed.