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.

Modular forms continue to attract attention for decades with many different application areas. To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to be able to compute a large number of Fourier coefficients. In this talk, firstly, we will show that this can be achieved in level 4 for a large range of half-integral weights by making use of one of three explicit bases, the elements of which can be calculated via fast power series operations. After having "many" Fourier coefficients, it is time to ask the following question: Can the dis- tribution of normalised Fourier coefficients of half-integral weight level 4 Hecke eigenforms with bounded indices be approximated by a distribution? We will suggest that they follow the generalised Gaussian distribution and give some numerical evidence for that. Finally, we will see that the appar- ent symmetry around zero of the data lends strong evidence to the Bruinier- Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute values are distributed independently. This is joint work with Gabor Wiese (Luxembourg), Zeynep Demirkol Ozkaya (Van) and Elif Tercan (Bilecik).

Diophantine equations involving specific number sequences have attracted considerable attention. For instance, studying when a Tribonacci number can be expressed as the product of two Fibonacci numbers is an interesting problem. In this case, the corresponding Diophantine equation has only two nontrivial integer solutions. While finding these solutions is relatively straightforward, proving that no further solutions exist requires a rigorous argument-this is where Baker’s method plays a crucial role. After conducting a comprehensive literature review on the topic, we present our recent results on Diophantine equations involving Fibonacci, Tribonacci, Jacobsthal, and Perrin numbers. Furthermore, as an application of Baker’s method, we will briefly demonstrate how linear forms in logarithms can be effectively applied to Diophantine equations involving Fibonacci-like sequences. This is joint work with Zeynep Demirkol Ozkaya (Van), Zekiye Pinar Cihan (Bilecik) and Meltem Senadim (Bilecik).