The local-global principle for representations of quadratic forms in codimension two

Ellenberg and Venkatesh (2008) proved, via the ergodic theory of $p$-adic groups, that a positive definite integral quadratic form $Q$ on $\mathbb{Z}^n$ represents every locally representable form of squarefree discriminant in $m\leq n-5$ variables with minimum above an ineffective constant $c(Q)$, and every such form in $m\leq n-3$ variables whose discriminant is prime to a fixed auxiliary odd prime $p$, with minimum above an ineffective constant $c(Q,p)$. They posed four follow-up problems: effectivity of the constants, removal of the auxiliary prime, weakening of the squarefree-discriminant hypothesis, and the codimension-two case $n=m+2$. The first three have since been resolved in the range $n-m\geq 3$: Schulze-Pillot (2009) replaced the squarefree-discriminant hypothesis by bounded imprimitivity of the local representations, and Einsiedler, Lindenstrauss, Mohammadi and Wieser (2025) proved an effective local-global principle for primitive representations for all $n\geq m+3$ with no splitting condition, together with effective asymptotics for representation numbers — the Fourier coefficients of Siegel theta series. The surviving open core is codimension two, where the stabilizer of an embedding is a $p$-adic torus and unipotent dynamics is unavailable. The conjecture, formalized by Kim, Wieser and Yang (2025): if $n-m=2$ and a form $q$ is primitively represented by the spin genus of $Q$, with minimum large enough in terms of $Q$, then $q$ is primitively represented by $Q$. The first case, primitive binary forms by quaternary forms, is a theorem of Kim, Wieser and Yang — but under a splitting condition at two auxiliary primes that the authors call an artifact of their method, and they report no viable analytic approach without it. The problem is to prove the codimension-two local-global principle in general, or refute it.

Background

An integral quadratic lattice is a free $\mathbb{Z}$-module $L\simeq\mathbb{Z}^n$ equipped with an integer-valued positive definite quadratic form $Q$. A form $q$ on $\mathbb{Z}^m$ with $m<n$ is represented by $Q$ if there is a linear isometry $(\mathbb{Z}^m,q)\to(\mathbb{Z}^n,Q)$, i.e. an integer matrix $X$ with $X^{\mathsf T}A_QX=A_q$ for Gram matrices $A_Q,A_q$; the representation is primitive if the image is a primitive (saturated) sublattice of $\mathbb{Z}^n$. The form $q$ is locally representable if such an isometry exists over the $p$-adic integers $\mathbb{Z}_p$ for every prime $p$ and over $\mathbb{R}$. The genus of $Q$ is the set of integral forms locally isometric to $Q$, so by the Hasse-Minkowski theorem (two rational quadratic forms are isomorphic over $\mathbb{Q}$ if and only if they are isomorphic over $\mathbb{R}$ and over every $\mathbb{Q}_p$) local representability of $q$ is equivalent to representation of $q$ by some form in the genus of $Q$. The local-global principle asks when representation by the genus descends to the individual class of $Q$.

Between a single class and its genus lies the spin genus: the orbit of the class of $Q$ under the adelic points of the spin group (the universal cover of the special orthogonal group), i.e. under local transformations of trivial spinor norm (the spinor norm of a product of reflections in vectors $v_i$ is the product of the values $Q(v_i)$, taken modulo squares, so trivial spinor norm picks out a subgroup of index at most $2$ of the local orthogonal group). This is the natural intermediate notion because, when the codimension $n-m$ is at least $3$, strong approximation for spin groups (for a simply connected group such as the spin group, adelic points decompose into a single orbit of the integral adelic points away from one place) upgrades local representability to representation by some form in the spin genus. The classical thresholds for the positive definite local-global principle are: high codimension, $n\geq 2m+3$ (Hsia, Kitaoka and Kneser 1978, with refinements by Jöchner and Kitaoka), and the first low-codimension case $m=1$, $n=3$ — representation of integers by ternary forms — settled by Duke and Schulze-Pillot (1990), necessarily formulated with the spin genus in the hypothesis (the circle method of the 1920s–30s had earlier handled $m=1$ with $n\geq 4$). Ellenberg and Venkatesh (2008) introduced homogeneous dynamics: the embeddings of $q$ into the genus of $Q$ form an orbit of the special orthogonal group $H=\operatorname{SO}_{n-m}$ on an adelic quotient associated to $Q$, and for $n-m\geq 3$ the group $H$ is semisimple, so measure-classification theorems for unipotent flows (flows generated by nilpotent elements of the group) apply — in the $p$-adic setting, Ratner's theorem on products of real and $p$-adic groups. Their published theorem: $Q$ represents every locally representable $q$ of rank $m\leq n-5$ and squarefree discriminant with minimum (least nonzero value) at least an ineffective constant $c(Q)$; and, for a fixed auxiliary odd prime $p$, every such $q$ of rank $m\leq n-3$ whose discriminant is prime to $p$, with minimum at least an ineffective $c(Q,p)$ — a Linnik-type splitting condition analogous to the auxiliary prime in Linnik's ergodic method (which exploits the dynamics of a $p$-adic rotation group acting on the solutions, and therefore needs a prime at which the form splits; see Ellenberg-Michel-Venkatesh 2010 for a modern account of that method).

In codimension two the mechanism breaks down: the stabilizer of an embedding is $\operatorname{SO}_2(\mathbb{Q}_p)$, an abelian torus, on which unipotent dynamics says nothing. Ellenberg and Venkatesh proposed replacing Ratner theory by rigidity of higher-rank diagonalizable (torus) actions: passing from $\mathbb{Q}_p$ to a product of two completions $\mathbb{Q}_p\times\mathbb{Q}_q$, the group $\operatorname{SO}_2(\mathbb{Q}_p)\times\operatorname{SO}_2(\mathbb{Q}_q)\simeq\mathbb{Q}_p^\times\times\mathbb{Q}_q^\times$ is a rank-$2$ torus, the setting of the Einsiedler-Lindenstrauss measure-rigidity program (realized for maximal tori in $\operatorname{PGL}_2$-type groups in the Einsiedler-Lindenstrauss-Michel-Venkatesh proof of Duke's theorem for cubic fields). They flagged three obstacles: the needed rigidity statements were then unavailable, positive entropy of the limiting measures must be proven, and concentration of limiting measures on intermediate subgroups must be excluded. The rigidity tool now exists (Einsiedler-Lindenstrauss 2023: classification for higher-rank diagonalizable actions on irreducible quotients of forms of $\operatorname{SL}_2$, with an adelic version requiring no entropy assumption), and Kim, Wieser and Yang (2025) used it — together with the Siegel mass formula (which computes the average number of representations of $q$ over a genus as an explicit product of local factors) and determinant-method counting (the technique of Bombieri-Pila and Heath-Brown bounding the number of integral points on algebraic varieties) — to prove the first codimension-two case: primitive binary forms ($m=2$) by quaternary forms ($n=4$), primitively represented by the spin genus, with minimum at least $C(p_1,p_2,Q)$ and $-\operatorname{disc}(q)\operatorname{disc}(Q)$ a nonzero square modulo two auxiliary odd primes $p_1,p_2$. Meanwhile the other three limitations of the 2008 paper were resolved in codimension $\geq 3$: Schulze-Pillot (2009) replaced squarefree discriminant by bounded imprimitivity, and Einsiedler, Lindenstrauss, Mohammadi and Wieser (2025) — building on the effective equidistribution program of Einsiedler-Margulis-Venkatesh (2009), Einsiedler-Margulis-Mohammadi-Venkatesh (2020) and Wieser (2024) — proved an effective local-global principle for primitive representations for all $n\geq m+3$, with effective constants depending only on the dimension and no splitting condition. What remains open is the codimension-two principle itself, without auxiliary primes, in general rank — the direct generalization of the Duke-Schulze-Pillot theorem from $m=1$ to $m\geq 2$.

Problem Statement

Throughout, $Q$ is a positive definite integral quadratic form on $\mathbb{Z}^n$ and $q$ a positive definite integral quadratic form on $\mathbb{Z}^m$ with $n=m+2$ and $m\geq 2$; $\min(q)$ denotes the least nonzero value of $q$ on $\mathbb{Z}^m\setminus\{0\}$, and $\operatorname{disc}(\cdot)$ the discriminant (determinant of the Gram matrix of the associated bilinear form, up to the standard factor-of-$2$ convention). A representation of $q$ by $Q$ is an integer matrix $X$ with $X^{\mathsf T}A_QX=A_q$, i.e. a linear isometry $(\mathbb{Z}^m,q)\to(\mathbb{Z}^n,Q)$; it is primitive if the image sublattice is primitive (saturated) in $\mathbb{Z}^n$. The genus of $Q$ is the set of integral forms locally isometric to $Q$ (over $\mathbb{Z}_p$ for every prime $p$, and over $\mathbb{R}$); the spin genus of $Q$ is the orbit of the class of $Q$ under local transformations of trivial spinor norm (the spinor norm of a product of reflections in vectors $v_i$ is the product of the values $Q(v_i)$, modulo squares), an intermediate collection between the single class of $Q$ and its full genus; and $q$ is primitively represented by the spin genus of $Q$ if some form in that collection primitively represents $q$.

Main question (the codimension-two local-global principle; Conjecture 1.2 of Kim-Wieser-Yang 2025, formalizing the codimension-two direction left open by Ellenberg-Venkatesh 2008). Prove, or refute: there is a constant $c(Q)>0$ such that every $q$ as above that is primitively represented by the spin genus of $Q$ and satisfies $\min(q)\geq c(Q)$ is primitively represented by $Q$ itself.

(a) The conjecture has the following variants, each for all $m\geq 2$ (all pairs with $n-m=2$): (i) with $q$ in addition primitive (has coprime coefficients — the setting of Kim-Wieser-Yang's Theorem 1.3); (ii) with the local representations non-primitive, but only by a uniformly bounded index; (iii) the general case, without the splitting condition. For $m=1$, $n=3$ the analogous statement is the theorem of Duke-Schulze-Pillot.

(b) The codimension-two principle for quaternary $Q$, with no condition on $-\operatorname{disc}(q)\operatorname{disc}(Q)$ modulo auxiliary primes: prove that for quaternary $Q$ there is $C(Q)$ such that every binary $q$ primitively represented by the spin genus of $Q$ with $\min(q)\geq C(Q)$ is primitively represented by $Q$. This is the codimension-two analogue of the removal of the Linnik-type splitting condition that Einsiedler-Lindenstrauss-Mohammadi-Wieser (2025) achieved in codimension $\geq 3$.

A refutation of the main question consists of an explicit $Q$ together with an infinite family $q_i$ of forms primitively represented by the spin genus of $Q$, with $\min(q_i)\to\infty$, such that for all sufficiently large $i$ the form $q_i$ is not primitively represented by $Q$.

References

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