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Summer school on effective geometry and arithmetic of curves (Istanbul June 4-8 2018).

This 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.

• Practical information

The school will take place on Boğaziçi university campus in Istanbul from 4th to 8th of June 2018.

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

In order to attend the summer school, please send to us a short CV (containing mathematical background and name of one/two reference(s)). Indicate as well if you request financial support for travel and/or accommodation. Deadline for applications will be 15 of May. Female mathematicians are particularly encouraged to apply.

• Lecturers and courses

Nurdagül Anbar RICAM, 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

• Description of each course

Nurdagül Anbar

-Algebraic function fields and their extensions -Congruence function fields, rational places, The Hasse-Weil bound, maximal curves, Serre-bound, Ihara bound, Drinfeld-Vladut bound -Towers of function fields -Examples of exceptional function fields (Hermitian, Ree, Suzuki, GK, etc.) -Goppa codes, almost perfect sequences, low-discrepancy sequences

Anne Frühbis-Krüger

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

• Review of some basic properties: 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

The organization committee, Alp Bassa and Christophe Ritzenthaler