Representations of binary quadratic forms by quaternary quadratic forms
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918459596603392 |
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| author | Kim, Wooyeon Wieser, Andreas Yang, Pengyu |
| author_facet | Kim, Wooyeon Wieser, Andreas Yang, Pengyu |
| contents | We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. Our method is a variant of Linnik's ergodic method showing density for certain homogenous toral sets. The central ingredient is a measure classification result of Einsiedler and Lindenstrauss for actions of rank two diagonalizable groups on quotients of products of $\mathrm{SL}_2$. This rigidity result together with an application of the Siegel mass formula reduces the density problem to a counting problem on a certain affine variety. We solve that counting problem using the determinant method of Bombieri-Pila and Heath-Brown. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19437 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Representations of binary quadratic forms by quaternary quadratic forms Kim, Wooyeon Wieser, Andreas Yang, Pengyu Number Theory Dynamical Systems We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. Our method is a variant of Linnik's ergodic method showing density for certain homogenous toral sets. The central ingredient is a measure classification result of Einsiedler and Lindenstrauss for actions of rank two diagonalizable groups on quotients of products of $\mathrm{SL}_2$. This rigidity result together with an application of the Siegel mass formula reduces the density problem to a counting problem on a certain affine variety. We solve that counting problem using the determinant method of Bombieri-Pila and Heath-Brown. |
| title | Representations of binary quadratic forms by quaternary quadratic forms |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2604.19437 |