Planes in quadratic 4-space and associated shapes of lattices

Abstract: Let $Q=-x_1<sup>1-x_2<sup>2-x_3<sup>2+x_4<sup>2$ be the standard signature $(1,3)$ quadratic form. To each non-degenerate rational plane $L$ in the four-dimensional quadratic space $(\mathbb{Q}<sup>4,Q)$ we can naturally attach a periodic geodesic on the Bianchi orbifold $$\mathrm{SL}<em>2(\mathbb{Z}[i])\backslash \mathbb{H}<sup>3$$ which records the position of $L$ in the Grassmannian up to integer rotations. Moreover, each such plane $L$ defines a CM point and a periodic geodesic on the modular curve through restriction of $Q$ to $L$ and its orthogonal complement. Lastly, the local isomorphism between $\mathrm{SO}</em>{1,3}(\mathbb{R})$ and $\mathrm{SL}_2(\mathbb{C})$ gives rise to a further periodic geodesic on the Bianchi orbifold. In this article, we exhibit a natural coupling of all the above objects and prove simultaneous equidistribution under a Linnik-type splitting condition. The main ingredient is the classification of joinings of higher-rank diagonalizable actions on homogeneous spaces due to Einsiedler and Lindenstrauss.

• The paper proves simultaneous equidistribution of four arithmetic objects associated with rational planes—Bianchi geodesics, CM points, and modular geodesics—as square-free discriminants grow under a splitting condition.
• The authors construct a general Clifford-algebraic Klein map for non-degenerate planes in any quaternary quadratic space, while establishing integral parametrization and distribution results for the form of signature (1,3).
• The proof combines Duke-type torus-orbit estimates, class-group bounds, and Einsiedler–Lindenstrauss joining rigidity to show that the four coupled objects become asymptotically independent, with removing the auxiliary prime condition remaining open.

Setting and motivation

The paper studies oriented two-dimensional rational subspaces ("planes") $L$ of the quadratic space $(\mathbb{Q}^4, Q)$ with $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$, a form of signature $(1,3)$. To each non-degenerate such plane one can attach several classical arithmetic objects: a periodic geodesic on the Bianchi orbifold $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$ recording the "position" of $L$ in the Grassmannian up to integral rotations; a CM point on the modular surface $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$ given by the shape of the lattice $L \cap \mathbb{Z}^4$ (i.e., the restriction of $Q$ to it); a periodic geodesic on the unit tangent bundle of the modular surface given by the indefinite binary form obtained by restricting $Q$ to $(\mathbb{Q}^4, Q)$0; and — via a new construction generalizing the Klein vectors of Aka–Einsiedler–Wieser (2606.03472) — a second periodic geodesic on the Bianchi orbifold attached to the orthogonal complement of the Klein vector.

The central result is a simultaneous equidistribution theorem for these four objects coupled along their natural torus orbits. This extends the precursor work on the positive-definite form $(\mathbb{Q}^4, Q)$1, where an accidental local isomorphism $(\mathbb{Q}^4, Q)$2 produced four CM points per plane. The signature-$(\mathbb{Q}^4, Q)$3 setting is structurally different: $(\mathbb{Q}^4, Q)$4 is isogenous to $(\mathbb{Q}^4, Q)$5, an outer form of $(\mathbb{Q}^4, Q)$6, and the stabilizer of a plane in $(\mathbb{Q}^4, Q)$7 is non-compact, so the position of $(\mathbb{Q}^4, Q)$8 must be captured by dualizing to a periodic geodesic rather than a CM point.

The Klein map via Clifford algebras

A key technical contribution is a Clifford-algebraic construction of the Klein map valid for any non-degenerate quaternary quadratic space $(\mathbb{Q}^4, Q)$9 over $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$0. For a pure 2-wedge $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$1, the commutator $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$2 in the even Clifford algebra $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$3 is traceless, satisfies $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$4, and the resulting map from non-degenerate pure wedges to traceless norm-invertible elements of $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$5 is an equivariant bijection. Orthogonal planes have Klein vectors differing by multiplication by a traceless element of the center $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$6, which is a separable quadratic algebra generated by the volume element $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$7. The stabilizer $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$8 of a plane in $Q = -x_1^2 - x_2^2 - x_3^2 + x_4^2$9 is shown to be isogenous to $(1,3)$0, a maximal $(1,3)$1-torus of absolute rank two.

For the specific form $(1,3)$2, the even Clifford algebra identifies with $(1,3)$3 and $(1,3)$4 with a conjugate of $(1,3)$5; the paper computes this conjugation explicitly and verifies that the normalizer of the relevant order coincides exactly with $(1,3)$6. The Klein vector of a plane becomes a primitive vector in a free rank-three $(1,3)$7-lattice $(1,3)$8 equipped with the sum-of-three-squares form, and its discriminant equals $(1,3)$9 for the Klein orthogonal lattice. The authors note plainly that freeness of $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$0 over $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$1 fails for general forms (e.g., $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$2), which is why the distribution results are restricted to this particular form.

The coupled objects and lengths

For a plane $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$3 of square-free discriminant $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$4, the four projections of the adelic torus orbit yield: (i) a compact periodic geodesic $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$5 on the frame bundle of the Bianchi orbifold, of length $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$6 depending only on $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$7; (ii) a CM point $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$8 corresponding to the definite form $\mathrm{SL}_2(\mathbb{Z}[i])\backslash\mathbb{H}^3$9; (iii) a periodic geodesic $L$0 for the indefinite form $L$1; and (iv) a compact periodic geodesic $L$2 on the Bianchi orbifold for the binary form over $L$3 on the Klein orthogonal lattice. A lemma relating these lengths shows $L$4 is either $L$5 or $L$6, while $L$7; all are logarithms of fundamental units in $L$8 or $L$9 with relative norm one. The joint measure $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$0 averages length measures over the finite set $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$1 of $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$2-orbits, which has size $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$3.

Main results

Two equidistribution statements are proved. The first is a Duke-type theorem for the position alone:

Theorem (position equidistribution). The measure $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$4 on $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$5 converges to the uniform measure on the unit tangent bundle of the Bianchi orbifold as $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$6 through square-free integers.

The main theorem couples all four factors:

Theorem (simultaneous equidistribution). For any odd prime $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$7, as $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$8 with $\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}^2$9 square-free and $L \cap \mathbb{Z}^4$0 a non-zero square modulo $L \cap \mathbb{Z}^4$1, the measures $L \cap \mathbb{Z}^4$2 converge to Haar measure on $L \cap \mathbb{Z}^4$3.

The congruence condition at $L \cap \mathbb{Z}^4$4 is a Linnik-type splitting hypothesis. The authors state explicitly that conjecturally it should be removable, but doing so would require bypassing the Einsiedler–Lindenstrauss joining rigidity, which needs congruence conditions at two distinct places; they instead exploit the prime at infinity (the geodesic flow) via an elementary disjointness argument to reduce to a single auxiliary prime.

Proof architecture

The dynamical core concerns the adelic group $L \cap \mathbb{Z}^4$5 and shifted orbits $L \cap \mathbb{Z}^4$6 of two-dimensional anisotropic tori $L \cap \mathbb{Z}^4$7 embedded diagonally across the four factors. Three ingredients combine:

Reduction to packets. Each adelic orbit decomposes into finitely many pieces indexed by representatives of the class number of $L \cap \mathbb{Z}^4$8; projecting to the arithmetic quotient produces precisely the collections $L \cap \mathbb{Z}^4$9 of planes, with disjointness of the projected pieces verified via the real stabilizer structure. The pushforward of the adelic Haar measure is then the average of the $Q$0.

Equidistribution in each factor. The class group of the biquadratic field $Q$1 controls the first and fourth factors: the relevant quotient factors onto the squares in $Q$2 with absolutely bounded kernel. For the middle factors, the images of the morphisms $Q$3 contain the squares, with cokernel of size $Q$4 (bounded via 2- and 4-torsion estimates for class groups of imaginary quadratic and biquadratic fields). A Duke-type theorem for torus orbits on adelic quotients of forms of $Q$5 — proved here via Jacquet–Langlands transfer, Waldspurger's formula, and subconvex bounds of Michel–Venkatesh under a volume-versus-discriminant condition — then gives equidistribution of each factor, since the missing squares contribute only $Q$6 mass.

Joining classification. The limit measure is invariant under both the diagonal flow $Q$7 and, using the splitting condition at $Q$8 (which forces the stabilizer torus to be split over $Q$9 by Hensel's lemma), an independent $Q$0-action of class-$Q$1 in the remaining three factors. After decoupling the CM-point factor by ergodicity of the geodesic flow (Howe–Moore plus a trivial-joining lemma), the Einsiedler–Lindenstrauss classification of joinings of higher-rank actions forces every ergodic component to be algebraic; pairwise joining analysis eliminates all proper subgroups — the case of two $Q$2 factors is excluded because the eigenvalue mismatch of $Q$3 prevents an isomorphism between the factors from containing the diagonal flow. Hence the limit is Haar.

Limitations and open questions

Several restrictions are acknowledged explicitly. The congruence condition $Q$4 a non-zero square mod $Q$5 is assumed for technical reasons and conjecturally unnecessary; removing it requires avoiding the Einsiedler–Lindenstrauss rigidity machinery. The distribution results are established only for the specific form $Q$6; the Klein map itself works for arbitrary quaternary forms, but integrality difficulties concerning the Steinitz class of the complement of the Klein vector obstruct the general equidistribution statement. The discriminants are required to be square-free, ensuring primitivity of the Klein vectors. The authors also note that analytic methods (Blomer–Brumley and successors) have not yet addressed multi-factor equidistribution problems of this type, and that no analogue of Khayutin's marked-point refinement for sphere problems is known in this setting. Whether the auxiliary prime can be dropped, whether the theorem extends to general quaternary forms, and whether analytic techniques can handle the coupled problem remain open.

Conclusion

The paper establishes simultaneous equidistribution of planes in quadratic 4-space together with their associated shapes, CM points, and periodic geodesics, under a single Linnik-type splitting condition. Its methodological contribution is twofold: a Clifford-algebraic Klein map that extends the parametrization of rational planes to arbitrary quaternary quadratic spaces, and a demonstration that the higher-rank joining classification of Einsiedler–Lindenstrauss, combined with a Waldspurger-based Duke theorem for torus packets, suffices to force full independence of the four coupled arithmetic objects as the discriminant grows.