# Colloquium abstracts

**Carlo Pagano**

University of Glasgow

December 17, 2021

**The negative Pell conjecture and related problems: reciprocity laws in higher nilpotency class**: In this talk I will overview an upcoming joint work with Peter Koymans, settling a conjecture made by Nagell (made around 1930) on the solvability of the negative Pell equation, in the refined form proposed by Stevenhagen in 1995. We achieved this by discovering certain reciprocity laws for (so-called) governing expansions: such reciprocity laws can be thought as higher-nilpotency generalizations of a reciprocity law established by Re'dei around the 30's (which corresponds to nilpotency class 2). Governing expansions were introduced by Smith in 2017 in his groundbreaking work on Goldfeld's and Cohen--Lenstra's conjectures, where he used them to establish reflection-principles to compare 2-power Selmer groups of different twists of a Galois module: we used these objects in previous work to establish a simplicial generalization of Gauss' genus theory from quadratic to multi-quadratic fields, a result whose proof is based on the control of the Lie-algebra of certain Galois groups over the