On the Shafarevich Group of Restricted Ramification Extensions of Number Fields in the Tame Case

TL;DR

This paper investigates the structure of the Shafarevich group in tame ramification extensions of number fields, revealing that it can increase with larger sets of ramified places, contrasting with the wild case.

Contribution

It establishes a new relationship between the Shafarevich group and certain dual groups in the tame case, showing that the group can grow with larger ramification sets, unlike in the wild case.

Findings

In the tame case, the Shafarevich group can increase as the set of ramified places grows.

The paper provides a construction of sets where the Shafarevich group enlarges.

Contrasts the behavior of the Shafarevich group in tame versus wild ramification cases.

Abstract

Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S} H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$, into the dual $\CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $\Sha_{S\cup X} \stackrel{\simeq}{\hookrightarrow} \CyB_{S\cup X} \stackrel{\simeq}{\twoheadleftarrow} \CyB_X \hookleftarrow \Sha_X$. In particular, we show that in the tame case $\Sha_Y$ can {\it increase} with increasing $Y$. This is in contrast with the wild case where $\Sha_Y$ is nonincreasing in size with increasing $Y$.