Yesterday, I watched a talk by Manjul Bhargava called What is the Birch and Swinnerton-Dyer Conjecture?. I think he did a really good job in explaining the conjecture on a level that can be understood by a non-expert yet mathematically inclined person.
In the caption, there is a link to these lecture notes (not written by Bhargava himself). I thought it might be interesting to go through these lecture notes and explore the territory of topics mentioned in the talk (especially in the introductory part of the talk). Therefore, I will walk through the lecture notes in this blogpost (and one or two following posts) and try to look into the mentioned theorems in a bit more detail.