Optimal dependence on the ambient ternary form of the count of embeddings of a binary quadratic form

Counting embeddings of one integral quadratic form into another is a classical problem in the arithmetic of quadratic forms and the key arithmetic input to Linnik-type ergodic proofs of equidistribution. Einsiedler, Lindenstrauss, Michel and Venkatesh (Enseign. Math. 2012) proved that the number $N(q;Q)$ of embeddings of a non-degenerate integral binary quadratic form $q$ into a non-degenerate integral ternary quadratic form $Q$, counted modulo $\mathrm{SO}_Q(\mathbb{Z})$ (the group of integer matrices of determinant $1$ preserving $Q$), satisfies $N(q;Q)\ll_{Q,\varepsilon}f\max(|a_1|,|a_2|,|a_3|)^{\varepsilon}$, where $f^2$ is the greatest square dividing $\gcd(a_1,a_2,a_3)$; but the implicit constant depends on $Q$ in an uncontrolled way. Their Remark A.1 asks what the optimal dependence on $Q$ is, records that the shape $\ll_\varepsilon|\mathrm{disc}(Q)|^{1/2+\varepsilon}$ can be shown when $Q$ is a multiple of a quaternionic norm form with an Eichler order, and states it as plausible that this shape holds in general - a uniformity that is precisely what equidistribution applications with varying $Q$ require. This problem asks for a proof or refutation of that uniform bound for all non-degenerate integral ternary forms over $\mathbb{Z}$ (with the analogous question over a fixed number field). The question is open: the closest later works obtain uniformity in different formulations - a uniform Linnik basic lemma over varying base fields by Wieser-Yang, and a local-global principle for representations of binary forms by quaternary forms by Kim-Wieser-Yang - but neither settles the ternary uniformity question.

Background

The modular surface is the homogeneous space $X=\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$, the space of unimodular lattices in the plane (lattices of covolume one), equipped with the geodesic flow (the action of the diagonal subgroup of $\mathrm{SL}_2(\mathbb{R})$). Every primitive closed geodesic on $X$ determines a real indefinite binary quadratic form, and the discriminant of that form is called the discriminant of the geodesic. A theorem of W. Duke (1988) states that closed geodesics of discriminant $d>0$ become equidistributed on $X$ with respect to the hyperbolic volume measure as $d\to\infty$ through fundamental discriminants (discriminants of maximal quadratic orders). Duke's proof used the spectral theory of half-integral-weight Maass forms (automorphic eigenfunctions of the hyperbolic Laplacian whose weight is an odd multiple of $1/2$). Linnik's ergodic method - an approach to equidistribution of arithmetic points via dynamics on homogeneous spaces - had earlier yielded the same conclusion only under an additional splitting condition on $d$ (that a fixed prime $p$ split in the quadratic field $\mathbb{Q}(\sqrt{d})$, equivalently that $d$ be a square modulo $p$); obtaining it without that condition by ergodic means was a long-standing goal. Einsiedler, Lindenstrauss, Michel and Venkatesh (2012) achieved this with a proof combining two ingredients: a finitary uniqueness statement for the measure of maximal entropy (the invariant measure with the largest dynamical entropy, i.e. the asymptotic complexity of orbits), and an arithmetic spacing statement for pairs of points in the orbit packets (the finite families of arithmetic points attached to a given discriminant), known as Linnik's Basic Lemma.

The Basic Lemma reduces to a purely arithmetic counting problem. An embedding of the integral binary quadratic form $q(x,y)=a_1x^2+a_2xy+a_3y^2$ into the integral ternary quadratic form $Q$ on $\mathbb{Z}^3$ is a linear map $\iota\colon\mathbb{Z}^2\to\mathbb{Z}^3$ with $Q(\iota(x))=q(x)$ for all $x$; the integral special orthogonal group $\mathrm{SO}_Q(\mathbb{Z})$ (the integer $3\times3$ matrices of determinant $1$ preserving $Q$) acts on embeddings, and $N(q;Q)$ denotes the (finite) number of orbits. Bounds for fixed $Q$ are classical and can be deduced from Siegel's mass formula, which expresses representation counts averaged over a genus - the set of forms locally equivalent to $Q$ at every prime - as products of local densities (the contributions of the individual completions of $\mathbb{Q}$). Einsiedler-Lindenstrauss-Michel-Venkatesh give a self-contained elementary proof, in the spirit of the adelic proof of the mass formula, of $N(q;Q)\ll_{Q,\varepsilon}f\max(|a_1|,|a_2|,|a_3|)^{\varepsilon},$ where $f^2$ is the greatest square dividing $\gcd(a_1,a_2,a_3)$, and $\ll_{Q,\varepsilon}$ means that the implicit constant is allowed to depend on $Q$ and $\varepsilon$ but on nothing else.

The dependence of that constant on $Q$ is not tracked anywhere in the argument, and Remark A.1 of the paper asks for it. Here the natural height of $Q$ is $|\mathrm{disc}(Q)|$, where the discriminant of a quadratic form is, up to a conventional normalizing factor and sign, the determinant of the symmetric matrix of its coefficients. The remark records: (i) the argument in fact bounds only the sum of the embedding counts over a set of representatives of the genus classes of $Q$, so control for individual forms requires understanding how the mass distributes across the genus; (ii) when $Q$ is a multiple of the norm form on a lattice in the space of trace-zero elements of a quaternion algebra (a central simple algebra of dimension $4$ over $\mathbb{Q}$, whose trace-zero elements form a rational three-dimensional space carrying a natural ternary norm form) whose associated order is an Eichler order (an intersection of two maximal orders, the standard well-understood class of orders in such an algebra), the dependency $\ll_\varepsilon|\mathrm{disc}(Q)|^{1/2+\varepsilon}$ can be shown; and (iii) the argument carries over without significant changes to quadratic forms over a general number field. The authors then state that it 'seem[s] plausible that this holds in general' - i.e. that the ternary-form bound with $Q$-dependence $\ll_\varepsilon|\mathrm{disc}(Q)|^{1/2+\varepsilon}$ is valid for all non-degenerate integral ternary forms. That uniform claim is the problem posed here.

What is at stake is a methodological capability. Linnik-type equidistribution results in which the ambient quadratic structure varies - the source cites, as an instance of what such uniformity would buy, a theorem of Ellenberg-Michel-Venkatesh on the distribution of integer points on spheres - require the arithmetic input to be uniform in the ambient form, and the available bound is not. Modern work needing this uniformity has routed around the ternary-form estimate rather than proving it: Wieser-Yang prove a uniform Linnik basic lemma over varying base fields, but with an arithmetic input (theta-series, i.e. modular generating functions whose Fourier coefficients encode representation counts of a quadratic form, combined with geometric invariant theory, the study of quotient spaces of algebraic varieties under group actions, and adelic orbital integrals, i.e. integrals over orbits of points in the product of all completions of the base field) that replaces the ternary-form count instead of bounding it uniformly. Whether the original ternary-form claim is true - and if not, what the correct shape of the dependence is - is unknown.

Problem Statement

Let $Q$ be a non-degenerate integral ternary quadratic form on $\mathbb{Z}^3$ (of either signature; the indefinite case of signature $(2,1)$ is the one arising in Linnik-type equidistribution) and let $q(x,y)=a_1x^2+a_2xy+a_3y^2$ be a non-degenerate integral binary quadratic form. Let $f^2$ be the greatest square dividing $\gcd(a_1,a_2,a_3)$, let $\mathrm{disc}(Q)$ denote the discriminant of $Q$ (the determinant of its coefficient matrix, up to the conventional normalizing factor and sign), and let $N(q;Q)$ be the number of embeddings - linear maps $\iota\colon\mathbb{Z}^2\to\mathbb{Z}^3$ with $Q(\iota(x))=q(x)$ for all $x$ - modulo the action of $\mathrm{SO}_Q(\mathbb{Z})$ (the integer $3\times3$ matrices of determinant $1$ preserving $Q$). Einsiedler-Lindenstrauss-Michel-Venkatesh prove that $N(q;Q)\ll_{Q,\varepsilon}f\max(|a_1|,|a_2|,|a_3|)^{\varepsilon}$ with an implicit constant whose dependence on $Q$ is not controlled, and state in Remark A.1 that it seems plausible that the dependence on $Q$ can be taken of the shape $|\mathrm{disc}(Q)|^{1/2+\varepsilon}$. Prove or disprove: for every $\varepsilon>0$, $N(q;Q)\ll_\varepsilon|\mathrm{disc}(Q)|^{1/2+\varepsilon}\cdot f\cdot\max(|a_1|,|a_2|,|a_3|)^{\varepsilon}$ for all non-degenerate integral ternary $Q$ and all non-degenerate integral binary $q$, the implicit constant depending only on $\varepsilon$. If the conjectured shape is false, determine the correct shape of the optimal dependence on $Q$. The same question over the ring of integers of any fixed number field in place of $\mathbb{Z}$, with the analogous definitions, is part of the same problem (the source records that its argument carries over to that setting).

References

1. Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, Akshay Venkatesh, The distribution of closed geodesics on the modular surface, and Duke's theorem, L'Enseignement Mathematique (2) 58 (2012), no. 3-4, 249-313. DOI: 10.4171/LEM/58-3-2, https://doi.org/10.4171/lem/58-3-2 (preprint: arXiv:1109.0413, https://arxiv.org/abs/1109.0413)
2. W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Inventiones Mathematicae 92 (1988), no. 1, 73-90. DOI: 10.1007/bf01393993, https://doi.org/10.1007/bf01393993
3. Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, Akshay Venkatesh, Distribution of periodic torus orbits and Duke's theorem for cubic fields, Annals of Mathematics (2) 173 (2011), no. 2, 815-885. DOI: 10.4007/annals.2011.173.2.5, https://doi.org/10.4007/annals.2011.173.2.5 (preprint: arXiv:0708.1113, https://arxiv.org/abs/0708.1113)
4. Jordan S. Ellenberg, Philippe Michel, Akshay Venkatesh, Linnik's ergodic method and the distribution of integer points on spheres, preprint (2010). arXiv:1001.0897, https://arxiv.org/abs/1001.0897
5. Andreas Wieser, Linnik's problems and maximal entropy methods, Monatshefte fur Mathematik 190 (2019), no. 1, 153-208. DOI: 10.1007/s00605-019-01320-7, https://doi.org/10.1007/s00605-019-01320-7 (preprint: arXiv:1801.09012, https://arxiv.org/abs/1801.09012)
6. Andreas Wieser, Pengyu Yang, A uniform Linnik basic lemma and entropy bounds, Commentarii Mathematici Helvetici (published online 2026-01-22). DOI: 10.4171/cmh/613, https://doi.org/10.4171/cmh/613 (preprint: arXiv:2201.05380, https://arxiv.org/abs/2201.05380)
7. Wooyeon Kim, Andreas Wieser, Pengyu Yang, Representations of binary forms by quaternary quadratic forms, preprint (2025). arXiv:2511.22877, https://arxiv.org/abs/2511.22877
8. Shirali Kadyrov, Positive entropy invariant measures on the space of lattices with escape of mass, Ergodic Theory and Dynamical Systems 32 (2012), no. 1, 141-157. DOI: 10.1017/s0143385710000726, https://doi.org/10.1017/s0143385710000726 (preprint: arXiv:0912.0487, https://arxiv.org/abs/0912.0487)
9. Taehyeong Kim, High entropy measures on the space of lattices with escape of mass, Nonlinearity 38 (2025), no. 12, 125002. DOI: 10.1088/1361-6544/ae240f, https://doi.org/10.1088/1361-6544/ae240f (preprint: arXiv:2403.16840, https://arxiv.org/abs/2403.16840)
10. S. Kadyrov, D. Kleinbock, E. Lindenstrauss, G. A. Margulis, Singular systems of linear forms and non-escape of mass in the space of lattices, Journal d'Analyse Mathematique 133 (2017), no. 1, 253-277. DOI: 10.1007/s11854-017-0033-4, https://doi.org/10.1007/s11854-017-0033-4
11. Manfred Einsiedler, Shirali Kadyrov, Entropy and escape of mass for SL3(Z)\SL3(R), Israel Journal of Mathematics 190 (2012), no. 1, 253-288. DOI: 10.1007/s11856-011-0185-3, https://doi.org/10.1007/s11856-011-0185-3 (preprint: arXiv:0912.0475, https://arxiv.org/abs/0912.0475)
12. Manfred Einsiedler, Shirali Kadyrov, Anke Pohl, Escape of mass and entropy for diagonal flows in real rank one situations, Israel Journal of Mathematics 210 (2015), no. 1, 245-295. DOI: 10.1007/s11856-015-1252-y, https://doi.org/10.1007/s11856-015-1252-y (preprint: arXiv:1110.0910, https://arxiv.org/abs/1110.0910)
13. Manfred Einsiedler, Elon Lindenstrauss, Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations, preprint. arXiv:2307.04163, https://arxiv.org/abs/2307.04163
14. Ron Mor, Bounding entropy for one-parameter diagonal flows on $\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$ using linear functionals, Journal of the European Mathematical Society (published online 2025-02-13). DOI: 10.4171/jems/1592, https://doi.org/10.4171/jems/1592 (preprint: Ron Mor, Upper bounds for the entropy in the cusp for one-parameter diagonal flows on $\mathrm{SL}_d(\mathbb{R})/\mathrm{SL}_d(\mathbb{Z})$, arXiv:2309.01665, https://arxiv.org/abs/2309.01665)