The course will cover four major subjects of number theory with applications in cryptography. First, it will cover Global Fields, in particular number fields and function fields (especially cyclotomic fields and function fields of curves), rings of integers, Dedekind rings and orders (“ideal lattices”). Secondly, the course will deal with the geometry of curves, in particular morphisms and function field extensions, as well as isogenies of elliptic curves. Moreover, we will cover the geometry of numbers (a.k.a. Minkowski theory), focusing on class groups, units, Dirichlet’s unit theorem, and the finiteness of the class group. Finally, we will discuss some analytic aspects. Upon successful completion of the module, the participants will know the basics of global fields, both in the number field and the function field settings. He or she masters basics of algebraic curves, especially aspects related to their function fields, as well as standard results in the geometry of numbers. Moreover, he or she will aware of analytic aspects, such as Dirichlet series, L-functions and Chebotarev’s density theorem. Duration: 4 days. ECTS: 0.5.
The course took place in hybrid mode at the University of Luxembourg (uni.lu) Campus Belval. Remote participation was allowed.
02.11.2020 – 11.11.2020, MSA.