Open AccessExplicit descent for Jacobians of prime power cyclic covers of the projective line
Author(s) -
Edward F. Schaefer
Publication year - 2017
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/7060
Subject(s) - mathematics , subvariety , abelian group , divisor (algebraic geometry) , prime (order theory) , prime power , degree (music) , cover (algebra) , rank (graph theory) , jacobian matrix and determinant , pure mathematics , line (geometry) , discrete mathematics , combinatorics , variety (cybernetics) , geometry , mechanical engineering , statistics , physics , acoustics , engineering
The Jacobian of a cyclic cover of the projective line is isogenous to a product of abelian subvarieties, one for each positive divisor of the degree of the cover. In this article, we show how to compute a Selmer group that bounds the Mordell-Weil rank for each abelian subvariety corresponding to a non-trivial prime power divisor of the degree. In the case that the Chabauty condition holds for that abelian subvariety, we show how to bound the number of rational points on the curve.