Distribution of 2-primary class groups in the symmetric-pairing setting

Determine the distribution of \(2\cdot\operatorname{Cl}(K)[2^{\infty}]\) for random quadratic extensions \(K/\mathbb Q(i)\), a case in which the relevant first pairing is symmetric.

Background

In comparing their method with Smith’s approach to Goldfeld’s conjecture, the authors explain that symmetric pairings create a fundamental obstruction. They identify the distribution of the 2-primary part of the class group for random quadratic extensions of Q(i)\mathbb Q(i) as a particularly fundamental open case excluded by Smith’s methods. This is presented as an unresolved distribution problem rather than as a result proved in the paper.

References

This is for instance the case for the distribution of $2 \cdot \text{Cl}(K)[2{\infty}]$ when $K$ is a random quadratic extension of $Q(i)$, which is one of the most fundamental open cases left out by Smith's methods.

— Chowla's non-vanishing conjecture over $\mathbb{F}_q(T)$  (2609.11855 - Koymans et al., 10 Sep 2026) in Section 1, subsection Method of proof