{"paper":{"title":"CM Elliptic Curves: Volcanoes, Reality and Applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Pete L. Clark","submitted_at":"2022-12-26T22:52:10Z","abstract_excerpt":"For positive integers $M \\mid N$ and an order of discriminant $\\Delta$ in an imaginary quadratic field $K$ with discriminant $\\Delta_K < -4$, we determine the fiber of the morphism $X_0(M,N) \\rightarrow X(1)$ over the closed point $J_{\\Delta}$ corresponding to $\\Delta$. We also show that the fiber of the natural map $X_1(M,N) \\rightarrow X_0(M,N)$ over $J_{\\Delta}$ is connected. Putting this together we deduce the number of points in the fiber of $X_1(M,N) \\rightarrow X(1)$ over $J_{\\Delta}$ and their residual degrees. In the continuation of this work with F. Saia, these results will be extend"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.13316","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2212.13316/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"}