Click here for the program of the summer school

Exercises for the course of Christophe Ritzenthaler

The mathematical school will focus on effective and explicit methods in the theory of algebraic curves and applications to coding theory and cryptography. An important aspect of the school will be tutorials accompanying each of the lecture series. In these tutorials participants will have the opportunity to put into practice the theory from the lectures and experiment with the introduced object using various computer algebra systems. Depending on the background of the students, they might be given small research problems on which they can work together during the week. Active participation and interaction will be a key aspect of the mathematical school. This school also serves the purpose of preparing students for a sequence of conferences and workshops to be organized in the near future.

Nurdagül Anbar - Johannes Kepler University, Linz

*Algebraic function fields and applications to cryptography and coding
theory*

Anne Frühbis-Krüger - Leibniz Universität Hannover

*Algorithmic Aspects of Desingularization of 2-dimensional
schemes*

Christophe Ritzenthaler - University Rennes 1

*Models of curves of low genus and number of points over finite
fields*

**Nurdagül Anbar**

Algebraic function fields and applications to cryptography and coding theory

- Algebraic function fields and connection with algebraic curves

- The Hasse-Weil bound, maximal curves, and some examples of maximal function fields.

- Galois extension of function fields (Kummer and Artin-Schreier Extensions)

- Algebraic Geometric Codes

- Application of function fields to arcs and permutation polynomials

**Anne Frühbis-Krüger**

Algorithmic Aspects of Desingularization of 2-dimensional schemes

Resolution of singularities is a well-known theoretical and practical tool in algebraic geometry over fields of characteristic zero and for surfaces also in arbitrary characteristic. For surfaces in mixed characteristic, it is by far less used, as there are a number of additional algorithmic issues to consider. In this mini-course we shall first explore the tasks and challenges of algorithmic desingularization in general and explicitly provide basic algorithmic tools like blow-ups. In the second lecture we will focus on the desingulariziation invariant which controls the whole resolution process and on algorithmically determining the locus of maximal order. The last of the three lectures will be devoted to a variant of the algorithm of Cossart-Jannsen-Saito for resolving 2-dimensional schemes in arbitrary and mixed characteristic.

The mini-course is designed for PhD-students and advanced Master-students, so there will be an emphasis on instructive examples and explicit algorithmic techniques.

**Christophe Ritzenthaler**

Models of curves of low genus and number of points over finite fields

- Review of some basic properties: function fields, divisors, Riemann-Roch, Riemann-Hurwitz

- The canonical map and hyperelliptic curves/non hyperelliptic curves

- Equation of curves of genus up to 5

- Weil conjectures for curves with the proof of Stepanov/Bombieri

- Jacobians: application to cryptography and construction of curves with many points

We have financial support available especially for young participants to partially/totally cover accommodation and travel expenses. People from developing countries are especially invited to apply.

In order to attend the summer school, please send a short CV (containing
mathematical background and name of one/two reference(s)). Indicate if you
request financial support for travel and/or accommodation. Deadline for
applications will be **15th of May**.

alp.bassa (a) gmail.com

ritzenthalerchristophe (a) gmail.com

Female mathematicians are particularly encouraged to apply.

Alp Bassa, Ekin Özman and Christophe Ritzenthaler