Representations of binary forms by quaternary quadratic forms
Wooyeon Kim, Andreas Wieser, Pengyu Yang
TL;DR
The paper proves a local-global principle for primitively representing binary quadratic forms by quaternary positive-definite forms in the codimension-two setting (m=2, n=4) under a splitting condition at two odd primes. The authors fuse Adelic toral-period methods with Einsiedler–Lindenstrauss measure rigidity to obtain a positive entropy gain for limits of toral measures, then reduce the core step to a Linnik-type counting problem. The counting is handled by combining Siegel mass formula reductions with determinant-method bounds (Bombieri–Pila, Heath–Brown), yielding entropy improvements and, consequently, representations from local spin-genus data. This approach yields both a local-global principle and quantitative lower bounds on the number of primitive representations, advancing the understanding of representations of binary forms by quaternary forms in low codimension via dynamics and arithmetic geometry.
Abstract
We prove a local-global principle for representations of binary by quaternary quadratic forms. One of the main ingredients is a recent measure rigidity result of Einsiedler and Lindenstrauss for diagonalizable actions on quotients of products of $\mathrm{SL}_2$'s. Based on this, it suffices to show that limits of the uniform measures on the associated rank one adelic toral packets have more entropy than one half of the maximal entropy. The latter is proved using the Siegel mass formula and the determinant method as developed by Bombieri and Pila as well as Heath-Brown.