Deligne-Lusztig reduction in the case of an affine Weyl group

The affine Deligne-Lusztig varieties were introduced for the first time in 2005 by M. Rapoport in "A guide to the reduction modulo p of Shimura varieties" (see [1]). Understanding the emptiness/nonemptiness and the dimension of these objects is fundamental to examine certain aspects of the reduction of Shimura varieties. In general, the affine Deligne-Lusztig varieties (abbreviated ADLV) are difficult to handle. However, Xuhua He in [6] managed to bring back the question of the dimension for arbitrary ADLV to some well-studied ADLV. His solution is based on the affine Deligne-Lusztig reduction and gives rise to a concrete algorithm for the calculation of the dimension. Our first goal was then to create a computer program for the dimensions of ADLV following his strategy. Indeed, such a program was not yet implemented. We decided to use the mathematics software SageMath (see [2]). Our program can be applied in several cases and many examples of computations are reported in the thesis. In particular, calculating the dimensions of a specific subset of ADLV it is possible to find the dimension of the supersingular locus of the moduli space of principally polarized abelian varieties of dimension g with Iwahori level structure at p over Fp