The Hasse invariant of the Tate normal form E5 and the class number of Q(√-5l)

[摘要] It is shown that the number of irreducible quartic factors of the form g(x) = x(4) + ax(3) + (11a+2)x(2) - ax + 1 which divide the Hasse invariant of the Tate normal form E-5 in characteristic lis a simple linear function of the class number h(-5l) of the field Q(root-5l), when l equivalent to 2, 3 modulo 5. A similar result holds for irreducible quadratic factors of g(x), when l equivalent to 1, 4 modulo 5. This implies a formula for the number of linear factors over F-p of the supersingular polynomial ss(p)((5)*()) (x) corresponding to the Fricke group Gamma(0)*(5). (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-10-01

[关键词] Tate normal form;Hasse invariant;Class number;Class equation;Class field theory;Fricke group