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

JOURNAL OF NUMBER THEORY

卷:227

Article

Morton, Patrick¹ 

[1] Indiana Univ Purdue Univ Indianapolis IUPUI, Dept Math Sci, LD 270, Indianapolis, IN 46202 USA

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

DOI  :  10.1016/j.jnt.2021.03.006

来源: Elsevier

【 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.

【 授权许可】

Free