Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

✍ Scribed by John Brillhart; Patrick Morton

• Publisher: Elsevier Science
• Year: 2004
• Tongue: English
• Weight: 389 KB
• Volume: 106
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.01.006

✦ Synopsis

Using the theory of elliptic curves, we show that the class number hðÀpÞ of the field Qð ffiffiffiffiffiffi ffi Àp p Þ appears in the count of certain factors of the Legendre polynomials P m ðxÞ ðmod pÞ; where p is a prime 43 and m has the form ðp À eÞ=k; with k ¼ 2; 3 or 4 and p e ðmod kÞ: As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y 2 þ axy þ y ¼ x 3 and find an elementary expression for the supersingular polynomial ss p ðxÞ whose roots are the supersingular j-invariants of elliptic curves in characteristic p: As a corollary we show that the class number hðÀpÞ also shows up in the factorization ðmod pÞ of certain Jacobi polynomials.