The affine DeligneLusztig 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 DeligneLusztig 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 wellstudied ADLV. His solution is based on the affine DeligneLusztig 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.
DeligneLusztig reduction in the case of an affine Weyl group
Bonan, Elena
2018/2019
Abstract
The affine DeligneLusztig 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 DeligneLusztig 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 wellstudied ADLV. His solution is based on the affine DeligneLusztig 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.File  Dimensione  Formato  

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https://hdl.handle.net/20.500.12608/23605