Stacky Modifications and Operations in the Étale Cohomology of Number Fields

Abstract: This thesis consists of 4 papers. In Paper A we define stacky building data for stacky covers in the spirit of Pardini and give an equivalence of (2,1)- categories between the category of stacky covers and the category of stacky building data. We show that every stacky cover is a flat root stack in the sense of Olsson and Borne–Vistoli and give an intrinsic description of it as a root stack using stacky building data. When the base scheme S is defined over a field, we give a criterion for when a stacky building datum comes from a ramified cover for a finite abelian group scheme over k, generalizing a result of Biswas–Borne.In Paper B we compute the étale cohomology ring H'(X,Z/nZ) for X the spectrum of the ring of integers of a number field K. As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim. We also give examples of two distinct number fields whose rings of integers have isomorphic cohomology groups but distinct cohomology ring structures.In Paper C we generalize the results of Paper B to include the case when X is replaced by an open subset U ⊆ X, where we have removed a finite number of closed points from X. We show that when U is the complement of two odd primes p and q which are congruent to 1 (mod 4), the Legendre symbol of p over q may be interpreted as a cup product in H'(U,Z/2Z).In Paper D we find formulas for Massey products in étale cohomology of the ring of integers of a number field. Then we use these formulas to, with the help of a computer, find the first ever known examples of imaginary quadratic fields with p-class group of rank 2 for odd p and infinite class field tower. We also compute examples disproving McLeman’s (3, 3)-conjecture.