{"paper":{"title":"Congruence formulae for Legendre modular polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Adel Betina, Emmanuel Lecouturier","submitted_at":"2017-04-23T15:37:33Z","abstract_excerpt":"Let $p\\geq 5$ be a prime number. We generalize the results of E. de Shalit about supersingular $j$-invariants in characteristic $p$.\n  We consider supersingular elliptic curves with a basis of $2$-torsion over $\\overline{\\mathbf{F}}_p$, or equivalently supersingular Legendre $\\lambda$-invariants. Let $F_p(X,Y) \\in \\mathbf{Z}[X,Y]$ be the $p$-th modular polynomial for $\\lambda$-invariants. A simple generalization of Kronecker's classical congruence shows that $R(X):=\\frac{F_p(X,X^{p})}{p}$ is in $\\mathbf{Z}[X]$. We give a formula for $R(\\lambda)$ if $\\lambda$ is a supersingular. This formula is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06941","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"}