Supersingular j-invariants as singular moduli mod p
Let \({\mathfrak O}_ D={\mathbb{Z}}[{1\over2}D+\sqrt{-D})]\) for \(D\equiv 0\) or 3 mod 4, and let \(P_ D(X)\) be the minimal monic polynomial whose roots are the distinct j-invariants of elliptic curves over \({\bar {\mathbb{Q}}}\) with complex multiplication by \({\mathfrak O}_ D\). The author gives several results concerning the reduction \(mod\quad p\) of the polynomials \(P_ D(X).\) Let E be an elliptic curve over \({\mathbb{Q}}\), and let J be its j-invariant. Then every prime p of good, supersingular reduction for E is a prime factor of the numerator of \(P_ D(J)\) for some D such that p does not split in \({\mathbb{Q}}(\sqrt{-D})\). \textit{N. D. Elkies} [Invent. Math. 89, 561- 567 (1987; Zbl 0631.14024)] noted that D can be chosen to be \(<2p^{2/3}.\) The author shows that this bound can be improved to \(D<(4/\sqrt{3})\sqrt{p}\) as a consequence of his result that every supersingular j-invariant contained in \(F_ p\) is a root of some \(P_ D(X)\quad mod\quad p\) with \(D<(4/\sqrt{3})\sqrt{p}.\) The author also gives bounds on the prime divisors of the resultant of \(P_{D_ 1}(X)\) and \(P_{D_ 2}(X)\) for different discriminants \(- D_ 1\) and \(-D_ 2\). When \(D_ 1\) and \(D_ 2\) are relatively prime fundamental discriminants this is a result due to \textit{B. H. Gross} and \textit{D. B. Zagier} [J. Reine Angew. Math. 355, 191-220 (1985; Zbl 0545.10015)]. Finally, the author gives a new proof of a result, first due to Elkies, that if \(p\equiv 3\quad mod\quad 4\) then \(P_ p(X)\equiv (X- 12^ 3)(R(X))^ 2mod\quad p\) and \(P_{4p}(X)\equiv (X-12^ 3)(S(X))^ 2mod\quad p\) with R(X), S(X)\(\in {\mathbb{Z}}[X]\).
scientific article; zbMATH DE number 4212220