Shin-ichiro Seki, On the extension of the Green-Tao theorem to number fields

In 2006, Tao established the Gaussian counterpart of the celebrated Green-Tao theorem on arithmetic progressions of primes. In this talk, I will explain the extension of Tao’s theorem and the Green-Tao theorem to the case of general number fields. Our combinatorial tool is the relative hypergraph removal lemma by Conlon-Fox-Zhao. I will discuss the difficulties that arise in the case of general number fields and an application of our results to prime representations by binary quadratic forms. This is based on joint work with Wataru Kai, Masato Mimura, Akihiro Munemasa, and Kiyoto Yoshino.