Today I wanted to share with you a write-up of a talk I have given to other PROMYS Europe 2018 counsellors about two weeks ago. The topic of the talk was *higher reciprocity laws*, which is a general term used to describe a great variety of results which, in one way or another, can be seen as generalizations of the classical law of quadratic reciprocity. I have tried to give a somewhat motivated account of how each and every one of those generalizes the previous ones and, ultimately, how all of them are just special cases of one theorem (or of a single, even more general conjecture!)

This is a topic I have been fascinated by for years now and I have been seeking an opportunity/excuse to read more into it. Around a year ago I have given a 30-minute talk on this topic during a student conference I was attending, but due to time constraints I couldn’t get far into the topic nor did I get deep into it myself. My talk during PROMYS Europe was different – I would like to thank all counsellors for having patience to sit through my 2.5 hour long (!) talk, despite my promises I will keep it reasonably short. I have greatly enjoyed reading up on the topic and then preparing to present it to others.

I must admit I am quite proud of the write-up for this talk, which is why I would like to share it with everyone. Except for some minor fixes, this is a version which ended up in our yearbook, but I hope some people find some benefit in this, perhaps even find this topic as interesting as I do!