Euler systems are cohomological tools that play a crucial role in the study of special values of `$L$`

-functions; for instance, they have been used to prove cases of the Birch–Swinnerton-Dyer conjecture and have recently been used to prove cases of the more general Bloch–Kato conjecture. A fundamental technique in these recent advances is to show that Euler systems vary in `$p$`

-adic families. In this talk, we will first give a general introduction to the theme of `$p$`

-adic variation in number theory and introduce the necessary background from the theory of Euler systems; we will then explain the idea and importance of `$p$`

-adically varying Euler systems, and finally discuss current work in progress on `$p$`

-adically varying the Asai–Flach Euler system, which is an Euler system arising from quadratic Hilbert modular eigenforms.

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Last updated: 09 Sep 2024