Explicit identities for invariants of elliptic curves

✍ Scribed by Patrick Morton

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 318 KB
• Volume: 120
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2005.12.008

✦ Synopsis

New explicit formulas are given for the supersingular polynomial ss p (t) and the Hasse invariant Ĥp (E) of an elliptic curve E in characteristic p. These formulas are used to derive identities for the Hasse invariants of elliptic curves E n in Tate normal form with distinguished points of order n. This yields a proof that Ĥ (E 4 ) and Ĥ (E 5 ) are projective invariants (mod p) for the octahedral group and the icosahedral group, respectively; and that the set of fourth roots λ 1/4 of supersingular parameters of the Legendre normal form Y 2 = X(X -1)(X -λ) in characteristic p has octahedral symmetry. For general n 4, the field of definition of a supersingular E n is determined, along with the field of definition of the points of order n on E n .