## STNB 2016 (30è any)

### Twists of the Klein quartic: a complete classification and a modular interpretation

Elisa Lorenzo

#### Resum

The Klein quartic is, up to isomorphism, the genus 3 curve with biggest automorphism group. We will show a classification of its twists over number fields, which provides a complete classification of twists of non-hyperelliptic genus 3 curves defined over number fields, already started by the speaker in her thesis. The Klein quartic is isomorphic to the modular curve $X(7)$, which endows the twists with a modular interpretation. We use this interpretation to provide counterexamples to the Hasse principle.

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