{"paper":{"title":"Higher R\\'edei reciprocity and integral points on conics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carlo Pagano, Peter Koymans","submitted_at":"2020-05-28T17:19:51Z","abstract_excerpt":"Fix an integer $l$ such that $|l|$ is a prime $3$ modulo $4$. Let $d > 0$ be a squarefree integer and let $N_d(x, y)$ be the principal binary quadratic form of $\\mathbb{Q}(\\sqrt{d})$. Building on a breakthrough of Alexander Smith, we give an asymptotic formula for the solubility of $N_d(x, y) = l$ in integers $x$ and $y$ as $d$ varies among squarefree integers divisible by $l$.\n  As a corollary we give, in case $l > 0$, an asymptotic formula for the event that the Hasse Unit Index of the field $\\mathbb{Q}(\\sqrt{-l}, \\sqrt{d})$ is $2$ as $d$ varies over all positive squarefree integers. We also"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2005.14157","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2005.14157/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}