The exact average size of the 2-Selmer set $\mathrm{Sel}_2(J^1)$ of locally soluble hyperelliptic curves over $\mathbb{Q}$

Bhargava, Gross, and Wang proved that when locally soluble hyperelliptic curves of genus $g$ over $\mathbb{Q}$ are ordered by height, the average size of the 2-Selmer set $\mathrm{Sel}_2(J^1)$ of the degree-one divisor class variety $J^1$ is at most 2, and stated the expectation that this average is in fact exactly 2, independent of $g$. This problem asks for the exact value of that average: prove or refute the expectation, and more generally determine the sharp constants left open by the source, namely the optimal integrality constant $\kappa$ in the orbit-integrality theorem (proved with $\kappa=4$ and expected to be improvable) and the exact values of the order-$k$ Selmer set averages.

Background

A hyperelliptic curve of genus $g\geq 1$ over $\mathbb{Q}$ is a smooth projective curve $C$ equipped with a degree-2 map to $\mathbb{P}^1$; when the map has no rational branch point, $C$ has a model $z^2=f(x,y)$ where $f$ is a binary form (a homogeneous polynomial in two variables) of even degree $n=2g+2$ with nonzero discriminant and integral coefficients. The curve is called locally soluble if it has a point over every completion of $\mathbb{Q}$, i.e., over $\mathbb{R}$ and over $\mathbb{Q}_p$ for every prime $p$. Curves in this family are ordered by height $H(f)=\max_i|f_i|$, the largest absolute value of the coefficients. Bhargava proved that, without the local-solubility hypothesis, most hyperelliptic curves over $\mathbb{Q}$ have no rational point at all; the locally soluble family, where any obstruction must be global rather than local, is the subtle one.

For a curve $C$ as above, let $J=\mathrm{Jac}(C)$ be its Jacobian — the abelian variety (projective algebraic group) parametrizing divisor classes of degree 0 on $C$, where a divisor on $C$ is a formal integral linear combination of points — and let $J^1$ denote the variety parametrizing degree-1 divisor classes on $C$. The variety $J^1$ is a principal homogeneous space (or torsor) for $J$ — a variety on which $J$ acts simply transitively — and $J^1(\mathbb{Q})\neq\emptyset$ if and only if $C$ has a $\mathbb{Q}$-rational divisor of degree 1. The tool for studying $J^1(\mathbb{Q})$ is descent. A two-cover of $J^1$ is an unramified cover $\pi:Y\to J^1$ by a torsor $Y$ for $J$ such that $\pi(y+a)=\pi(y)+2a$ for all points $y$ of $Y$ and $a$ of $J$; it is locally soluble if $Y$ has points over every completion. The 2-Selmer group $\mathrm{Sel}_2(J)$ is the finite group of isomorphism classes of locally soluble two-covers of $J$ itself (realized as a subgroup of the Galois cohomology group $H^1(\mathbb{Q},J[2])$, where $J[2]$ is the 2-torsion of $J$), and the 2-Selmer set $\mathrm{Sel}_2(J^1)$ is the finite set of isomorphism classes of locally soluble two-covers of $J^1$; concretely, it is the set of elements of the 4-Selmer group $\mathrm{Sel}_4(J)$ mapping to a distinguished class in $\mathrm{Sel}_2(J)$ induced by the hyperelliptic class (the divisor class of a fiber of the degree-2 map $C\to\mathbb{P}^1$). This set is either empty or a principal homogeneous space for the group $\mathrm{Sel}_2(J)$, and its emptiness is a strong global obstruction: if $\mathrm{Sel}_2(J^1)=\emptyset$, then $C$ has index 2 — no $\mathbb{Q}$-rational divisor of odd degree exists — so $C$ has no point over any extension of $\mathbb{Q}$ of odd degree.

A central program in arithmetic statistics determines the average size of Selmer groups in natural families of curves. Bhargava and Gross proved that for hyperelliptic curves of genus $g$ with a rational Weierstrass point (a rational fixed point of the degree-2 map to $\mathbb{P}^1$; equivalently, an odd-degree model), the average size of $\mathrm{Sel}_2(J)$ is exactly 3; combined with Chabauty's method (a $p$-adic technique for bounding rational points when the Mordell–Weil rank — the rank of the group of rational points of the Jacobian — is small compared to the genus), this led Poonen and Stoll to prove that most odd-degree hyperelliptic curves have only one rational point. Shankar and Wang proved the companion exact average 6 for monic even-degree curves, i.e., curves with a marked rational non-Weierstrass point (a rational point that is not a fixed point of the degree-2 map). Both values were predicted by the heuristic of Poonen and Rains, which models Selmer groups as random maximal isotropic subspaces of an infinite-dimensional quadratic space; for even-degree families their Conjecture 1.8 predicts the average size of $\mathrm{Sel}_2(J)$ to be 6, the inflation coming from a 'causal' Selmer element forced by the hyperelliptic class.

Bhargava, Gross, and Wang studied the family with no marked point at all: all locally soluble curves $z^2=f(x,y)$ of genus $g$ over $\mathbb{Q}$, ordered by height. Their Theorem 6 proves that the average size of $\mathrm{Sel}_2(J^1)$ over this family is at most 2; their Theorem 7 proves that for each fixed odd $k>0$, the average size of the order-$k$ Selmer set — the subset of $\mathrm{Sel}_2(J^1)$ whose elements locally, at each place $\nu$ of $\mathbb{Q}$, come from a $\mathbb{Q}_\nu$-rational point of $J^1$ of the form $e_\nu-\frac{k-1}{2}d$, where $e_\nu$ is an effective divisor (a divisor with nonnegative coefficients) of odd degree $k$ on the curve and $d$ is the hyperelliptic class — is strictly less than 2 when $k<g$ and tends to 0 as $g\to\infty$. Combining the bound $\leq 2$ with a parity result of Tim and Vladimir Dokchitser (their appendix to the paper: in the full family, and in any 'admissible' congruence subfamily, a positive proportion of the curves have even 2-Selmer rank), they deduced their title theorem: a positive proportion of locally soluble hyperelliptic curves have empty $\mathrm{Sel}_2(J^1)$, hence no point over any odd-degree extension of $\mathbb{Q}$, with the resulting failures of the Hasse principle (the expectation that a point over every completion implies a global rational point) explained by the Brauer–Manin obstruction (a cohomological mechanism accounting for failures of local-to-global principles).

The method parametrizes $\mathrm{Sel}_2(J^1)$ by arithmetic invariant theory (the study of orbits of linear algebraic groups on representation spaces, used to parametrize arithmetic objects). Its elements correspond to locally soluble orbits of $(\mathrm{SL}_n/\mu_2)(\mathbb{Q})$ (the group of rational points of $\mathrm{SL}_n$, the special linear group, modulo its order-2 scalar center) on the space $\mathbb{Q}^2\otimes\mathrm{Sym}^2\mathbb{Q}^n$ of pairs $(A,B)$ of symmetric bilinear forms in $n=2g+2$ variables whose discriminant form $\mathrm{disc}(Ax-By):=(-1)^{g+1}\det(Ax-By)$ equals $f$; the two-covers arise as Fano varieties (varieties parametrizing families of linear subspaces) of maximal linear subspaces common to the two quadrics defined by $A$ and $B$, which form a pencil of quadrics (a one-parameter family $\{Ax-By=0\}$). To count these orbits by geometry of numbers (the theory of counting lattice points in regions via volume estimates) one needs integral representatives: the paper's Theorem 15 asserts that there is a positive integer $\kappa$, depending only on $n$, such that every locally soluble orbit with invariant $\kappa^2 f$ contains an element of $\mathbb{Z}^2\otimes\mathrm{Sym}^2\mathbb{Z}^n$, and this is proved for $\kappa=4$. The upper bound on the average then comes from counting integral orbits of bounded height, sieving out forms with squarefull discriminant (a discriminant divisible by $p^2$ for every prime $p$ dividing it), and applying a local mass formula (a weighted count of local orbits, weighted by inverse stabilizer sizes). Both sources of slack — the integrality scaling by $\kappa^2$ and the sieve — are what separate the proved bound $\leq 2$ from the authors' stated expectation that the average is exactly 2.

Problem Statement

Fix $g\geq 1$, and order all locally soluble hyperelliptic curves $C:z^2=f(x,y)$ of genus $g$ over $\mathbb{Q}$ (with $f$ a binary form of degree $2g+2$ having nonzero discriminant and height $H(f)=\max_i|f_i|$) by height. Determine the exact average size of the 2-Selmer set $\mathrm{Sel}_2(J^1)$ over this family. In particular, prove or refute the expectation stated by Bhargava–Gross–Wang that this average equals exactly 2 for every $g\geq 1$ — and is therefore independent of $g$ — which would sharpen their Theorem 6 upper bound of 2 to an equality. The problem comprises two named companion refinements of the same counting problem that the source explicitly leaves open: (i) determine the optimal integrality constant $\kappa$ in the source's Theorem 15 — the minimal positive integer, depending only on $n=2g+2$, such that for every integral binary form $f$ of degree $n$ with $z^2=f$ locally soluble over $\mathbb{Q}$, every locally soluble $(\mathrm{SL}_n/\mu_2)(\mathbb{Q})$-orbit of pairs $(A,B)$ of symmetric bilinear forms in $n$ variables with $\mathrm{disc}(Ax-By)=\kappa^2 f$ contains an integral representative in $\mathbb{Z}^2\otimes\mathrm{Sym}^2\mathbb{Z}^n$ (currently proved with $\kappa=4$ and expected to be improvable); and (ii) determine the exact values of the averages of the order-$k$ Selmer sets of the source's Theorem 7 for all odd $k>0$ (currently known only to be strictly less than 2 when $k<g$, and to tend to 0 as $g\to\infty$). A resolution consists of the exact average for every $g\geq 1$ — either a proof that the average equals 2 for all $g$, or a proof that the expectation fails for some $g\geq 1$ — together with the optimal integrality constant $\kappa$ and the exact averages of the order-$k$ Selmer sets for all odd $k>0$.

References

1. Manjul Bhargava, Benedict H. Gross, and Xiaoheng Wang, with an appendix by Tim Dokchitser and Vladimir Dokchitser, 'A positive proportion of locally soluble hyperelliptic curves over Q have no point over any odd degree extension', Journal of the American Mathematical Society 30 (2017), no. 2, 451-493, DOI: 10.1090/jams/863, https://doi.org/10.1090/jams/863 (preprint arXiv:1310.7692, https://arxiv.org/abs/1310.7692)
2. Manjul Bhargava, Arul Shankar, and Xiaoheng Wang, 'Geometry-of-numbers methods over global fields II: Coregular representations', preprint (2026), arXiv:2604.16978, https://arxiv.org/abs/2604.16978
3. Jef Laga and Jack A. Thorne, 'Lower bounds on heights of odd degree points of hyperelliptic curves', preprint (2025), arXiv:2507.08652, https://arxiv.org/abs/2507.08652
4. Ashvin A. Swaminathan, '2-Selmer groups, 2-class groups, and the arithmetic of binary forms', PhD thesis, Princeton University (2022), http://arks.princeton.edu/ark:/88435/dsp012v23vx55w, also at https://dataspace.princeton.edu/handle/88435/dsp012v23vx55w (Appendix 4A of the thesis contains the correction of the argument for Theorem 15 of the source)
5. Manjul Bhargava and Benedict H. Gross, 'The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point', in Automorphic Representations and L-functions, TIFR Studies in Mathematics 22, Hindustan Book Agency, New Delhi (2013), 23-91, arXiv:1208.1007, https://arxiv.org/abs/1208.1007
6. Arul Shankar and Xiaoheng Wang, 'Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves' (published as 'Rational points on hyperelliptic curves having a marked non-Weierstrass point'), Compositio Mathematica 154 (2018), no. 1, 188-222, DOI: 10.1112/S0010437X17007515, https://doi.org/10.1112/S0010437X17007515, arXiv:1307.3531, https://arxiv.org/abs/1307.3531
7. Bjorn Poonen and Eric Rains, 'Random maximal isotropic subspaces and Selmer groups', Journal of the American Mathematical Society 25 (2012), no. 1, 245-269, DOI: 10.1090/S0894-0347-2011-00710-8, https://doi.org/10.1090/S0894-0347-2011-00710-8, arXiv:1009.0287, https://arxiv.org/abs/1009.0287
8. Adam Morgan, '2-Selmer parity for hyperelliptic curves in quadratic extensions', Proceedings of the London Mathematical Society (6) 127 (2023), no. 5, 1507-1576, DOI: 10.1112/plms.12565, https://doi.org/10.1112/plms.12565, arXiv:1504.01960, https://arxiv.org/abs/1504.01960
9. Adam Morgan, 'Quadratic twists of abelian varieties and disparity in Selmer ranks', Algebra & Number Theory 13 (2019), 839-899, DOI: 10.2140/ant.2019.13.839, https://doi.org/10.2140/ant.2019.13.839, arXiv:1706.06063, https://arxiv.org/abs/1706.06063
10. Brendan Creutz, 'Most binary forms come from a pencil of quadrics', Proceedings of the American Mathematical Society, Series B 3 (2016), 18-27, DOI: 10.1090/bproc/24, https://doi.org/10.1090/bproc/24, arXiv:1601.03451, https://arxiv.org/abs/1601.03451
11. Brendan Creutz, 'Generalized Jacobians and explicit descents', Mathematics of Computation 89 (2020), no. 323, 1365-1394, DOI: 10.1090/mcom/3491, https://doi.org/10.1090/mcom/3491, arXiv:1601.06445, https://arxiv.org/abs/1601.06445
12. Brendan Creutz, 'Improved rank bounds from 2-descent on hyperelliptic Jacobians', International Journal of Number Theory 14 (2018), no. 6, 1709-1713, DOI: 10.1142/S179304211850104X, https://doi.org/10.1142/S179304211850104X, arXiv:1707.05877, https://arxiv.org/abs/1707.05877
13. Brendan Creutz and Duttatrey Nath Srivastava, 'Brauer-Manin obstructions on hyperelliptic curves', Advances in Mathematics 431 (2023), article 109238, DOI: 10.1016/j.aim.2023.109238, https://doi.org/10.1016/j.aim.2023.109238, arXiv:2112.00230, https://arxiv.org/abs/2112.00230
14. Joseph Gunther and Jackson S. Morrow, 'Irrational points on random hyperelliptic curves', preprint, arXiv:1709.02041, https://arxiv.org/abs/1709.02041
15. Jef Laga and Ashvin A. Swaminathan, 'A positive proportion of monic odd-degree hyperelliptic curves of genus g at least 4 have no unexpected quadratic points', International Mathematics Research Notices 2024 (2024), no. 19, 12857-12866, DOI: 10.1093/imrn/rnae184, https://doi.org/10.1093/imrn/rnae184, arXiv:2405.09421, https://arxiv.org/abs/2405.09421
16. Kai Huang and Yongqi Liang, 'Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4', Journal of Algebra 713 (2027), 106-140, DOI: 10.1016/j.jalgebra.2026.07.022, https://doi.org/10.1016/j.jalgebra.2026.07.022, arXiv:2312.11204, https://arxiv.org/abs/2312.11204
17. Daniel Barrera Salazar, Ariel Pacetti, and Gonzalo Tornaría, 'On the 2-Selmer group of Jacobians of hyperelliptic curves', preprint, arXiv:2308.08663, https://arxiv.org/abs/2308.08663
18. Dao Van Thinh, '2-Selmer groups of even hyperelliptic curves over function fields', Transactions of the American Mathematical Society 376 (2023), no. 7, 4679-4712, DOI: 10.1090/tran/8878, https://doi.org/10.1090/tran/8878, arXiv:1811.10539, https://arxiv.org/abs/1811.10539
19. Ananth N. Shankar, '2-Selmer groups of hyperelliptic curves with two marked points' (published as '2-Selmer groups of hyperelliptic curves with marked points'), Transactions of the American Mathematical Society 372 (2019), 267-304, DOI: 10.1090/tran/7546, https://doi.org/10.1090/tran/7546, arXiv:1611.03172, https://arxiv.org/abs/1611.03172
20. Bjorn Poonen and Michael Stoll, 'Most odd degree hyperelliptic curves have only one rational point', Annals of Mathematics (2) 180 (2014), no. 3, 1137-1166, DOI: 10.4007/annals.2014.180.3.7, https://doi.org/10.4007/annals.2014.180.3.7, arXiv:1302.0061, https://arxiv.org/abs/1302.0061
21. Manjul Bhargava, 'Most hyperelliptic curves over Q have no rational points', preprint (2013), arXiv:1308.0395, https://arxiv.org/abs/1308.0395