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 \(\uplambda _p(F)\) denote the Iwasawa \(\uplambda \)-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 \(\uplambda _p(F)\ge r\). 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.
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Ray, A. On large Iwasawa \(\uplambda \)-invariants of imaginary quadratic function fields. Ramanujan J 62, 853–861 (2023). https://doi.org/10.1007/s11139-023-00717-1
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DOI: https://doi.org/10.1007/s11139-023-00717-1