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

TL;DR

This paper demonstrates that for large enough finite fields, a positive proportion of imaginary quadratic function fields have arbitrarily large Iwasawa $\lambda$-invariants, based on recent class group distribution results.

Contribution

It establishes the existence of many imaginary quadratic function fields with large Iwasawa $\lambda$-invariants, extending understanding of their distribution over finite fields.

Findings

Positive proportion of fields with $\lambda_p(F) o ext{large}$

Dependence on the size of the finite field $q$

Application of recent class group distribution theorems

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 $q\not\equiv 1\mod{p}$, there is a positive proportion of imaginary quadratic fields $F/\mathbb{F}_q(T)$ with the property that $\lambda_p(F)\geq r$. 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.