# EPSRC DTP PhD studentship: Anabelian geometry of curves over finite fields

**University of Exeter**

Qualification type: | PhD |

Location: | Exeter |

Funding for: | UK Students, EU Students |

Funding amount: | £14,296 |

Hours: | Full Time |

Placed on: | 31st October 2016 |

Closes: | 11th January 2017 |

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**Main supervisor:** Prof. Mohamed Saidi (University of Exeter)

This project is in Number Theory and Arithmetic Geometry.

This project is within the research area of anabelian geometry, a newly developed theory which has recently grown very successful by achieving very impressive results in Number theory and arithmetic geometry. This theory centres around the so-called anabelian conjectures of Grothendieck whose ultimate aim is to establish an equivalence of categories between certain algebraic varieties, the so-called anabelian varieties, and certain categories of profinite groups, thus establishing a bridge between algebraic geometry and the theory of (profinite group). These conjectures have been investigated by some leading research groups in arithmetic geometry worldwide especially at the Research Institute for Mathematical Sciences (RIMS) of Kyoto university.

Together with my collaborator Prof. Akio Tamagawa from RIMS we proved a refined version of these conjectures for algebraic curves over finite fields whereby we prove that the isomorphism type of such a curve is entirely determined by the isomorphism type of its fundamental group. More precisely, we prove that there exists a group-theoretic algorithm within the category of profinite groups which allows to reconstruct such a curve from its geometrically pro-Sigma etale geometric fundamental group where Sigma is a large set of primes.

The aim of this PhD project is to investigate a certain Hom-form of the above result where one aims to show that (non constant) morphisms (and not only isomorphism) between curves over finite fields can be reconstructed from open homomorphisms between their (geometrically pro-Sigma) arithmetic fundamental groups. In order to prove such result some improvement of the techniques used with Tamagawa is required.

**Funding Minimum**3.5 year studentship: UK/EU tuition fees and an annual maintenance allowance at current Research Council rate. Current rate of £14,296 per year.

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