Pro-nilpotent arithmetic statistics by Carlo Pagano
The pro-nilpotent closure of a global field, obtained as the compositum of all sequences of successive central extensions, is explicit enough, in terms of the arithmetic of the base field, but yet complex enough to act as a natural environment for several problems in arithmetic statistics, ranging from the solvability of conics over the integers (Stevenhagen's conjecture) to the vanishing of L-functions at 1/2 over function fields (Chowla's conjecture). I will overview work, past and in progress, that profits from this contrast.
I begin with a parametrization of nilpotent extensions, introduced in joint work with Peter Koymans where we made progress on the strong form of Malle's conjecture. I will use the parametrization to explain how the Malle's predicted counting function naturally arises, in a sequence of successive central extensions, as a consequence of a certain "Inertia versus Frobenius" mechanism (IvF later), a mechanism already exploited in the Scholz-Reichardt method resolving the Galois inverse problem for odd nilpotents.
Next I will explain how certain counts of nilpotent extensions with prescribed ramification, aside from being the key to do further progress on many remaining cases of Malle's conjecture itself, are also the key to the classical problems in arithmetic statistics mentioned above. More specifically here the IvF-mechanism naturally leads to a hierarchy of pairings, which one can approach with ideas coming from Smith's breakthrough. I will discuss the innovations in the Smith's method, introduced in joint work with Peter Koymans, that allowed us to establish Stevenhagen's conjecture on the negative Pell equation, and are currently used in ongoing work in progress with Peter Koymans and Mark Shusterman whose aim is at to establish that the L-function of 100% of imaginary quadratic characters of Fq(T) does not vanish at 1/2, as soon as q=3 mod 4 (Chowla's conjecture).
Finally the examined features of pro-nilpotent arithmetic statistics (versus pro-solvable arithmetic statistics) are compared with the different status of anabelianity of the respective Galois groups: I will conclude explaining how one can use statistical ideas to establish the lack of anabelianity for maximal 2-nilpotent Galois groups, a negative results inspired by (and to be compared with) recent work of Saidi and Tamagawa on the maximal 2 and 3-solvable Galois groups (in the latter case refining a classical theorem of Neukirch and Uchida), this is joint work in progress with Peter Koymans.