Ravenel's Global Conjecture is true

TL;DR

This paper proves Ravenel's 1983 Global Conjecture relating Ext groups over formal A-modules to flat cohomology of moduli stacks, and connects these cohomology groups to Hecke L-functions and arithmetic equivalence of Galois extensions.

Contribution

It confirms Ravenel's conjecture and establishes a novel link between flat cohomology of formal A-modules and number-theoretic invariants like L-functions and zeta-functions.

Findings

Proof of Ravenel's Global Conjecture on Ext groups.

Equivalence of flat cohomology groups and Hecke L-functions.

Characterization of arithmetic equivalence via flat cohomology.

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 $H^1_{fl}$ of the moduli stack $\mathcal{M}_{fmA}$ of formal $A$-modules. I then show that the Hecke $L$-functions of certain Gro{\ss}encharakters of Galois extensions $K/\mathbb{Q}$ can be computed from $H^1_{fl} (\mathcal{M}_{fmA})$, 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 $H^1_{fl}(\mathcal{M}_{fm\mathcal{O}_K})$ and $H^1_{fl}(\mathcal{M}_{fm\mathcal{O}_L})$ agree.