### Abstract

We construct a parametric family {E^{(±)} (s, t, u)} of minimal Q-curves of degree 5 over the quadratic fields Q(√s^{2} + st - t^{2}), and the family {C(s, t, u)} of genus two curves over Q covering E^{(+)} (s, t, u) whose jacobians are abelian surfaces of GL_{2}-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 language | English |
---|---|

Pages (from-to) | 165-182 |

Number of pages | 18 |

Journal | Manuscripta Mathematica |

Volume | 98 |

Issue number | 2 |

Publication status | Published - 1999 Feb |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

_{2}-type.

*Manuscripta Mathematica*,

*98*(2), 165-182.

**Q-curves of degree 5 and jacobian surfaces of GL _{2}-type.** / Hashimoto, Kiichiro.

Research output: Contribution to journal › Article

_{2}-type',

*Manuscripta Mathematica*, vol. 98, no. 2, pp. 165-182.

_{2}-type. Manuscripta Mathematica. 1999 Feb;98(2):165-182.

}

TY - JOUR

T1 - Q-curves of degree 5 and jacobian surfaces of GL2-type

AU - Hashimoto, Kiichiro

PY - 1999/2

Y1 - 1999/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0033485203&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033485203&partnerID=8YFLogxK

M3 - Article

VL - 98

SP - 165

EP - 182

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 2

ER -