1 results
The arithmetic of Prym varieties in genus 3
- Part of
-
- Journal:
- Compositio Mathematica / Volume 144 / Issue 2 / March 2008
- Published online by Cambridge University Press:
- 01 March 2008, pp. 317-338
- Print publication:
- March 2008
-
- Article
-
- You have access
- Export citation
-
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 perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The methods are, in principle, applicable to any genus 3 curve with a double cover. 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
. By specialization, this also gives examples over 
.
. By specialization, this also gives examples over 
.