Heuristics for anti-cyclotomic $\mathbb{Z}_p$-extensions

TL;DR

This paper introduces heuristics supported by computational evidence to predict the behavior of Iwasawa invariants in anti-cyclotomic $Z_p$-extensions of imaginary quadratic fields, focusing on intersection properties and invariants' vanishing.

Contribution

It proposes two novel heuristics—Intersection and Invariants heuristics—for understanding Iwasawa invariants in anti-cyclotomic towers, supported by computational data.

Findings

The Intersection Heuristics model the frequency of intersection of $p$-Hilbert class fields with anti-cyclotomic towers.

The Invariants Heuristics predict that $mbda$ and $mu$ invariants typically vanish when $p$ is non-split.

Computational evidence supports the validity of these heuristics.

Abstract

This paper studies Iwasawa invariants in anti-cyclotomic towers. We do this by proposing two heuristics supported by computations. First we propose the Intersection Heuristics: these model `how often' the $p$-Hilbert class field of an imaginary quadratic field intersects the anti-cyclotomic tower and to what extent. Second we propose the Invariants Heuristics: these predict that the Iwasawa invariants $\lambda$ and $\mu$ usually vanish for imaginary quadratic fields where $p$ is non-split.