Homological stability for generalized Hurwitz spaces and Selmer groups in quadratic twist families over function fields

TL;DR

The paper establishes function-field analogues of the BKLPR heuristics for ν-Selmer groups in quadratic twist families of abelian varieties by linking Selmer data to Hurwitz stacks and proving a new homological stability theorem for generalized Hurwitz-type spaces on higher-genus punctured curves. It develops a robust topological-to-arithmetic bridge via arc-complex spectral sequences, Frobenius-equivariant stabilization, and monodromy analysis of symplectically self-dual sheaves, enabling the computation of moments and the extraction of distributional limits. The main results show that, up to an error decaying with the size of the constant field q, the ν-Selmer distributions and their moments agree with BKLPR predictions; in particular, a minimalist-type conjecture for ℓ-Selmer ranks holds in the large-q regime. The work also demonstrates how parity phenomena and rank-doubling mechanisms govern equidistribution and component counts, and it outlines potential extensions to ℓ∞-Selmer ranks and broader twist-settings. This provides a powerful geometric-analytic framework for understanding Selmer-rank statistics in function-field arithmetic with potential far-reaching consequences for the study of ranks and Tate–Shafarevich groups.

Abstract

We prove a version of the Bhargava-Kane-Lenstra-Poonen-Rains heuristics for Selmer groups of quadratic twist families of abelian varieties over global function fields. As a consequence, we derive a result towards the "minimalist conjecture" on Selmer ranks of abelian varieties in such families. More precisely, we show that the probabilities predicted in these two conjectures are correct to within an error term in the size of the constant field, $q$, which goes to $0$ as $q$ grows. Two key inputs are a new homological stability theorem for a generalized version of Hurwitz spaces parameterizing covers of punctured Riemann surfaces of arbitrary genus, and an expression of average sizes of Selmer groups in terms of the number of rational points on these Hurwitz spaces over finite fields.

Figures (7)

• Figure 1: A diagram depicting the structure of the proof of the main result, \ref{['theorem:main-finite-field']}.
• Figure 2: Some notation introduced in the paper.
• Figure 3: The blue surface with green punctures is a picture of $A_{2,3} \simeq \Sigma^{1}_{2,3}$ and the black surface is $X \simeq \Sigma^2_{0,0}$. The yellow circles correspond to the point $x$, the red rectangles are the subsurface $Y$ with $x \in Y \subset X$. We also depict $X^{\oplus 3}$ and $X^{\oplus 3} \oplus A_{2,3}$.
• Figure 4: We depict the spectral sequence coming from the arc complex. To avoid clutter in the picture, we write $\mathcal{K}(M_i)$ where we should write $\{\mathcal{K}(M_i^{V,F})\}_n$. The entries here start on the $E^2$ page, so $E^2_{q,p} = H_q(\{\mathcal{K}(M_p^{V,F})\}_n)$. The blue arrows depict the differentials on the $E^2$ page, the red arrows depict the differentials on the $E^3$ page, and the green arrow depicts a differential on the $E^4$ page.
• Figure 5: This picture depicts a cell in the configuration space $\mathop{\mathrm{Conf}}\nolimits^{12}{\Sigma^1_{1,2}}$. The boundary component consists of the union of the upper, left, and lower edges. The arrows indicate the orientations of the segments of the edges. Note that the segments of the same color are glued to each other with the orientations indicated, so there are only $4$ distinct points represented by the yellow dots on the right boundary despite the fact that there are $8$ yellow dots on the right boundary in the picture. The two black dots indicate the two punctures comprising $W$. The yellow dots indicate the $12$ points in configuration space. The cell is labeled by the $12$-tuple $\mathfrak t = ((3,1,2,2), (2,1),(0,1))$ with $b = 4$.

Theorems & Definitions (221)

• Theorem 1.1.1
• Theorem 1.1.2
• Remark 1.1.1
• Remark 1.1.2
• Remark 1.1.3
• Theorem 1.1.3
• Remark 1.1.4
• Remark 1.1.5
• Remark 1.1.6
• Theorem 1.1.4