In this talk I shall describe how the components of J_b-orbits of an affine Deligne-Lusztig varieties contribute to the components of the basic locus of a Shimura variety. Then we shall present results for the cases of PEL-type A and C and describe the mass formula part for type C. This is based on joint works with Jiangwei Xue and Yasuhiro Terakado.

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This is also a KAI-X invited lecture. There will be coffee right before the seminar.

I will discuss how to study the mod p non-vanishing problem for Dirichlet L-values with characters whose kernels are unbounded as the conductors grow. Main ingredients are a dynamical reformulation of the problem using a variant of trace defined by Poisson kernel and a study on the spectral properties of transfer operators related to p-Bernoulli map on the interval. This is ongoing research joint with A. Burungale.