The negative Pell conjecture and related problems: reciprocity laws in higher nilpotency class

Abstract: 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 rational numbers. Smith's reflection principles broke down in the study of the negative Pell equation, and our reciprocity laws provide new supplementary reflection principles for this problem. A number of seemingly different problems are affected by these same difficulties: ranging from Chowla's conjecture over function fields (non-vanishing of L-functions at 1/2 for 100% of characters), open cases of Goldfeld's conjecture, statistics of Iwasawa's modules, to the proof that the Brauer--Manin obstruction is the only obstruction to the Hasse principle for Kummer surfaces over number fields (under finiteness of Tate--Shafarevich groups of abelian surfaces). I will conclude explaining what we expect to achieve with these new tools on those other problems.