These advanced courses will take place over both weeks. The program will consist of three courses:
- Introduction to p-adic Galois representations by Ahmed Abbes, CNRS & IHÉS, France [second week].
Summary. I will present some of the fundamental results of Tate and Sen on p-adic Galois representations of p-adic local fields. Let K be a complete discrete valuation field of characteristic 0, with perfect residue field of characteristic p>0, K an algebraic closure of K, C the p-adic completion of K, and GK=Gal(K/K) the Galois group of K over K. I will first present the results of Tate and Sen on the continuous cohomology of C.
I will then explain Sen's theory for continuous C-representations of GK, that is, continuous semi-linear representations of GK on finite dimensional C-vector spaces. Let K∞ be the cyclotomic Zp-extension of K contained in K. The ultimate goal of this theory is to associate to such a C-representation W, a finite dimensional K∞-vector space V equipped with a K∞-linear endomorphism σ, called Sen's endomorphism, which encodes many properties of the representation. Indeed, the functor W ↦ (V,σ) is exact and faithful and (V,σ) determines W. If time permits, I will conclude the lectures with a brief overview of the p-adic Simpson theory, which can be naturally viewed as the geometric generalization of Sen theory.
On the first week there will be advanced preparatory lectures for this course.
For instance: Hermann Soré's lecture notes on profinite group cohomology. - Lattices and codes : arithmetic for communication systems by Frédérique Oggier, Birmingham University, UK [first week]. This course is supported by CIMPA as a CIMPA Course.
Summary. In this course we will show how lattices and codes, both independently and jointly, are used in the context of communication systems. The course is structured as follows :
- Introduction Lecture: Number Theory, Algebra and Applications (slides)
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Introduction to lattices and geometry of numbers (slides)
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Introduction to linear codes and lattices from codes (slides + exercises)
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Introduction to number fields and lattices from number fields (slides + exercises)
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Introduction to quaternion algebras and codes from quaternion algebras (slides)
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Practical aspects : channel modeling and decoding
- Introduction to PARI/GP by Marine Rougnant, University Marie et Louis Pasteur , Besançon, France [first week].
Summary. Numerical experiments play a crucial role in mathematical research, whether for computing examples, reinforcing intuition, or formulating conjectures through an exploratory approach. The PARI/GP software is designed for fast computations in arithmetic and number theory. Its collaborative development provides a wide range of efficient tools across various areas of number theory, including algebraic number theory, elliptic curves, modular forms, and L-functions.
After a brief introduction to the software’s syntax, this course will cover the fundamental functions necessary for using the software at the Master’s level. Participants will be invited to practice with exercise sheets and will have the opportunity, upon request, to explore topics more closely related to their research.
➡️ All related documents, course material and exercises, are available here:
https://pari.math.u-bordeaux.fr/Events/PARI2025Libreville/
These courses will be accompanied by exercises sessions.
