The arithmetic of Prym varieties in genus 3

Details The arithmetic of Prym varieties in genus 3 The arithmetic of Prym varieties in genus 3

2004-08-04

arxiv.org/abs/math/0408069v2

Compos. Math. 144 (2008), no. 2, 317--338.

Mathematics

Number Theory

21 pages

Scientific paper

10.1112/S0010437X07003314

Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym-variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to do Chabauty- and Brauer-Manin type calculations for curves of genus 5 with an unramified involution. As an application, we determine the rational points on a smooth plane quartic with no particular geometric properties and give examples of curves of genus 3 and 5 violating the Hasse-principle. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over Q(t).

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