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arXiv · 2207.13902

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

Anwesh Ray
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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.

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Anwesh Ray. 2022-07-28. On large Iwasawa $\lambda$-invariants of imaginary quadratic function fields. https://doi.org/10.1007/s11139-023-00717-1

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