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Degrees of Polarizations on an Abelian Surface with Real Multiplication
Author(s) -
Wilson John
Publication year - 2001
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1017/s0024609301007974
Subject(s) - mathematics , isogeny , abelian group , order (exchange) , multiplication (music) , degree (music) , cyclotomic field , conductor , coprime integers , field (mathematics) , complex multiplication , algebraic number field , ring (chemistry) , surface (topology) , pure mathematics , class (philosophy) , combinatorics , discrete mathematics , elliptic curve , geometry , physics , acoustics , economics , artificial intelligence , computer science , chemistry , organic chemistry , finance
Let F be a real quadratic field, and let R be an order in F . Suppose given, a polarized abelian surface ( A ; λ), defined over a number field k , with a symmetric action of R defined over k . This paper considers varying A within the k ‐isogeny class of A to reduce the degree of λ and the conductor of R . It is proved, in particular, that there is a k ‐isogenous principally polarized abelian surface with an action of the full ring of integers of F , when F has class number 1 and the degree of λ and the conductor of R are odd and coprime.