{"paper":{"title":"The average of the first invariant factor for reductions of CM elliptic curves mod $p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Paul Pollack, Tristan Freiberg","submitted_at":"2014-10-27T12:53:37Z","abstract_excerpt":"Let $E/\\mathbb{Q}$ be a fixed elliptic curve. For each prime $p$ of good reduction, write $E(\\mathbb{F}_p) \\cong \\mathbb{Z}/d_p \\mathbb{Z} \\oplus \\mathbb{Z}/e_p \\mathbb{Z}$, where $d_p \\mid e_p$. Kowalski proposed investigating the average value of $d_p$ as $p$ runs over the rational primes. For CM curves, he showed that $x\\log\\log{x}/\\log{x} \\ll \\sum_{p \\le x} d_p \\ll x\\sqrt{\\log{x}}$. It was shown recently by Felix and Murty that in fact $\\sum_{p \\le x} d_p$ exceeds any constant multiple of $x\\log\\log{x}/\\log{x}$, once $x$ is sufficiently large. In the opposite direction, Kim has shown that "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.7212","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}