Ravenel's Global Conjecture is true

Abstract

I prove Ravenel's 1983 "Global Conjecture" on $\Ext^1$ over the classifying Hopf algebroid of formal $A$-modules, equivalently, the first flat cohomology group $H_{fl}^1$ of the moduli stack ${\mathcal{M}}_{fmA}$ of formal $A$-modules. I then show that the Hecke $L$-functions of certain Großencharakters of Galois extensions $K/\mathbb{Q}$ can be computed from $H_{fl}^1({\mathcal{M}}_{fmA})$, and vice versa; as a consequence I show that, for a large class of Galois extensions of $\mathbb{Q}$, two extensions $K,L$ are arithmetically equivalent (i.e., they have the same Dedekind zeta-function) if and only if the flat cohomology groups $H_{fl}^1({\mathcal{M}}_{fm{\mathcal{O}}_K})$ and $H_{fl}^1({\mathcal{M}}_{fm{\mathcal{O}}_L})$ agree.

Updated with a little more detail on K. Johnson's work on the Local Conjecture

References in corpus (1)

• On the moduli stack of commutative, 1-parameter formal Lie groups