Periodic points of algebraic functions and Deuring’s class number formula
dc.contributor.author | Morton, Patrick | |
dc.contributor.department | Mathematical Sciences, School of Science | en_US |
dc.date.accessioned | 2020-06-26T18:13:47Z | |
dc.date.available | 2020-06-26T18:13:47Z | |
dc.date.issued | 2019 | |
dc.description.abstract | The exact set of periodic points in Q of the algebraic function ˆ F(z) = (−1±p1 − z4)/z2 is shown to consist of the coordinates of certain solutions (x, y) = ( , ) of the Fermat equation x4+y4 = 1 in ring class fields f over imaginary quadratic fields K = Q(p−d) of odd conductor f, where −d = dKf2 1 (mod 8). This is shown to result from the fact that the 2-adic function F(z) = (−1 + p1 − z4)/z2 is a lift of the Frobenius automorphism on the coordinates for which | |2 < 1, for any d 7 (mod 8), when considered as elements of the maximal unramified extension K2 of the 2-adic field Q2. This gives an interpretation of the case p = 2 of a class number formula of Deuring. An algebraic method of computing these periodic points and the corresponding class equations H−d(x) is given that is applicable for small periods. The pre-periodic points of ˆ F(z) in Q are also determined. | en_US |
dc.eprint.version | Author's manuscript | en_US |
dc.identifier.citation | Morton, P. (2019). Periodic points of algebraic functions and Deuring’s class number formula. The Ramanujan Journal, 50(2), 323–354. https://doi.org/10.1007/s11139-018-0120-x | en_US |
dc.identifier.uri | https://hdl.handle.net/1805/23121 | |
dc.language.iso | en | en_US |
dc.publisher | Springer | en_US |
dc.relation.isversionof | 10.1007/s11139-018-0120-x | en_US |
dc.relation.journal | The Ramanujan Journal | en_US |
dc.rights | Publisher Policy | en_US |
dc.source | ArXiv | en_US |
dc.subject | periodic points | en_US |
dc.subject | algebraic function | en_US |
dc.subject | class number formula | en_US |
dc.title | Periodic points of algebraic functions and Deuring’s class number formula | en_US |
dc.type | Article | en_US |