Rank growths of elliptic curves over some Galois extensions and Markov operators

I would like to give an overview of the insights from the work by Swinnerton-Dyer and Klagsbrun--Mazur--Rubin, who established relations between Markov operators over countable state spaces and changes in Mordell-Weil rank of elliptic curves with respect to base change to cyclic Galois extensions over number fields. If time allows, we will use this insight to generalize the technique to understand changes in Mordell-Weil ranks of elliptic curves with respect to families of S3-cubic extensions over number fields with fixed quadratic resolvents. My hope is to make the talk be accessible and explicit for non-specialists as well. This talk is based on joint works in progress with Daniel Keliher.