Q-curves of degree 5 and jacobian surfaces of GL2-type

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    Abstract

    We construct a parametric family {E(±) (s, t, u)} of minimal Q-curves of degree 5 over the quadratic fields Q(√s2 + st - t2), and the family {C(s, t, u)} of genus two curves over Q covering E(+) (s, t, u) whose jacobians are abelian surfaces of GL2-type. We also discuss the modularity for them and the sign change between E(+) (s, t, u) and its twist E(-) (s. t, u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C (s, t, u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A f attached to cusp forms of Neben type character of level N = 29, 229, 349, 461, and 509.

    Original languageEnglish
    Pages (from-to)165-182
    Number of pages18
    JournalManuscripta Mathematica
    Volume98
    Issue number2
    Publication statusPublished - 1999 Feb

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    ASJC Scopus subject areas

    • Mathematics(all)

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