On large Iwasawa $\lambda$-invariants of imaginary quadratic function fields

Abstract

Let $\ell$ be a prime number and $q$ be a power of $\ell$. Given an odd prime number $p$ and an imaginary quadratic extension $F$ of the rational function field ${\mathbb{F}}_q(T)$, let ${\lambda }_p(F)$ denote the Iwasawa $\lambda$-invariant of the constant ${\mathbb{Z}}_p$-extension of $F$. We show that for any number $r>0$ and all large enough values of $q\not\equiv 1\mathrm{mod}p$, there is a positive proportion of imaginary quadratic fields $F/{\mathbb{F}}_q(T)$ with the property that ${\lambda }_p(F)\ge r$. The main result is proved as a consequence of recent unconditional theorems of Ellenberg-Venkatesh-Westerland on the distribution of class groups of imaginary quadratic function fields.

8 pages; accepted for publication in the Ramanujan Journal

References in corpus (2)

• On Iwasawa $\lambda$-invariants for abelian number fields and random matrix heuristics
• A note on the distribution of Iwasawa invariants of imaginary quadratic fields