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

Abstract

We investigate the Iwasawa $\lambda$-invariant $\lambda_p(\chi)$ of the $p$-adic $L$-function associated with a Dirichlet character $\chi$ 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 $\lambda = 0$, $\lambda = 1$, $\lambda = 2$, and $\lambda \geq 3$. 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 $\lambda_p(\chi) >1 $ and $\lambda_p(\chi) > 2$. Furthermore, we extend the methods of Ernvall-Mets\"ankyl\"a and of Dummit et al. to calculate the $\lambda$-invariant. This is achieved by twisting $\chi$ by characters $\psi$ 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 $\lambda_p(\chi)$. Finally, these results are applied to generate numerical data on the distribution of $\lambda$-invariants, when either the prime $p$ or the Dirichlet character $\chi$ is fixed.