The refined Swan conductors in mixed characteristics

Thời gian: 16:00 - 17h15, thứ năm, 06/07/2023

Hình thức: Offline tại phòng 612 A6 và online qua google meet, cụ thể https://meet.google.com/yep-kbzk-eao?pli=1&authuser=1

Tóm tắt: The Swan conductor is an invariant that measures the ramification of a Galois representation of a curve over a field $K$. Initially introduced by Serre in the 1960s for degree one representations, it was later extended by Kato in 1989 to encompass characters of arbitrary degree. Notably, the Swan conductor has been applied in the study of the $L$-function of elliptic surfaces.

In situations where the field $K$ has equal characteristics, Matsuda has developed a highly practical algorithm for computing the Swan conductor. In this presentation, we will delve into the details of this algorithm and explore its efficiency in such scenarios. Moreover, we will introduce a conjecture, jointly developed with Tossici, proposing an analogous algorithm for fields with mixed characteristics. By presenting this conjecture, we hope to contribute to the broader understanding and application of the Swan conductor in various mathematical contexts.