We start by introducing Brauer-Manin obstructions to local-global principles over varieties. We then move to some of the known literature on Brauer-Manin obstructions for Kummer surfaces of products of elliptic curves. We finally present our work on some of the special cases where we calculate the Brauer group of a Kummer surface $X=Kum(E \times E')$ of a product of CM elliptic curves $E$ and $E'$, where $End(E)=End(E')=\mathbb{Z}[\zeta_3]$, and show that a non-trivial 5-torsion element of the transcendental Brauer group gives rise to Brauer Manin obstruction to weak approximation for $X$.

Speaker: Mohamed Tawfik, 14:00-14:20

Title: Brauer-Manin obstructions on Kummer Surfaces of CM elliptic curves.

Abstract: We discuss Brauer-Manin obstructions on Kummer surfaces of products of certain CM elliptic curves. We start by putting necessary and sufficient conditions on these surfaces to get a non-trivial transcendental Brauer group, then we find a generator of this group. Further, using a theorem by Harpaz and Skorobogatov, we show that a non-trivial element of order 5 of the transcendental Brauer group always gives rise to Brauer-Manin obstruction to weak approximation on these surfaces. Finally, we show that for most cases there is no obstruction coming from the algebraic part.

Speaker: Lorenzo La Porta, 14:30-14:50

Title: Generalised theta operators on some unitary Shimura varieties

Abstract: The theory of the classical theta operator was instrumental in Edixhoven's proof of the weight part of Serre's modularity conjecture. Because of this, much work has been devoted to extending the construction of this operator to other Shimura varieties, with an eye towards generalisations of Serre's conjecture, or to gain insight in the Langlands programme (mod p) in a broader sense. My goal is to present the construction of a new ``generalised'' theta operator that seems to produce exactly the weight shifts that one would expect from a representation-theoretic viewpoint and ties in neatly with the theory of generalised Hasse invariants of Boxer and Goldring-Koskivirta. I will sketch the classical construction in the modular case (i.e. on modular curves), review the ``ordinary'' theta operator (inspired by the work Eischen, Mantovan et al. and Goren-de Shalit) in the Picard case (i.e. on a unitary Shimura surface) and show how one can obtain a similar ``generalised'' operator in the same setting, but on the closure of the 1-dimensional Ekedahl-Oort stratum.