Local-global principles for large rings of algebraic integers

Let $K$ be a global field. In other words, $K$ is either a number field, i.e. a finite extension of $\mathbf{Q}$, or a function field of one variable over a finite field. Denote the algebraic (respectively, separable) closure of $K$ by $\widetilde { K }$ (respectively, by $K _ { s }$; cf. also Extension of a field). A prime divisor of $K$ is an equivalence class $\text{p}$ of absolute values (cf. also Norm on a field). For each $\text{p}$, let $|.|_{\operatorname{p}}$ be a representative of $\text{p}$. Denote the completion of $K$ at $\text{p}$ by $\widehat { K } _ { \text{p} }$. Then $\widehat { K } _ { \text{p} }$ is either $\mathbf{R}$ or $\mathbf{C}$ ($\text{p}$ is metric), or $\widehat { K } _ { \text{p} }$ is a finite extension of $\widehat { \mathbf{Q} } _ { p }$ or a finite extension of $\mathbf{F} _ { p } ( ( t ) )$ ($\text{p}$ is ultra-metric).

There is a natural $\text{p}$-topology on $\widehat { K } _ { \text{p} }$ whose basic $\text{p}$-open subsets have the form $\left\{ x \in \widehat { K } _ { \operatorname {p} } : | x - a | _ { \operatorname {p} } \leq \epsilon \right\}$, for $a \in \widehat { K } _ { \text{p} }$ and $\epsilon \in \mathbf{R}$, $\epsilon > 0$. The $\text{p}$-topology has compatible extensions to all sets $V ( \widehat { K } _ { \operatorname {p} } )$, where $V$ is an affine algebraic set over $\widehat { K } _ { \text{p} }$. In each case, $V ( \widehat { K } _ { \operatorname {p} } )$ is locally compact.

Embed $\widetilde { K }$ into the algebraic closure of $\widehat { K } _ { \text{p} }$ and let $K _ { \text{p} } = K _ { s } \cap \widehat { K } _ { \text{p} }$. Then $K _ { \operatorname{p} }$ is a real (respectively, algebraic) closure of $K$ at $\text{p}$ if $\widehat { K } _ { \text{p} } = \bf R$ (respectively, $\widehat { K } _ { \operatorname{p} } = \mathbf{C}$), and is a Henselization of $K$ at $\text{p}$ if $\text{p}$ is ultra-metric (cf. also Henselization of a valued field). In the latter case, the valuation ring of $K _ { \operatorname{p} }$ is denoted by $O _ { K , \text{p} }$. In each case, $K _ { \operatorname{p} }$ is uniquely determined up to a $K$-isomorphism.

If $P$ is a set of prime divisors of $K$ and $L$ is an algebraic extension of $K$, then $P _ { L }$ denotes the set of all extensions to $L$ of all $\operatorname{p} \in P$.

In the sequel, let $P$ be a fixed set of prime divisors of $K$ which does not contain all prime divisors. For each algebraic extension $L$ of $K$ and each $\operatorname { p} \in P _ { L }$, let $O _ { \text{p} } = \{ x \in L : | x | _ { \text{p} } \leq 1 \}$. Thus, if $\text{p}$ is metric, then $O _ { \text{p} }$ is the $\text{p}$-unit ball and if $\text{p}$ is ultra-metric, then $O _ { \text{p} }$ is the valuation ring of $\text{p}$. Let $O_L = \cap_{\text{p}\in P_L} O_\text{p}$. If $P$ consists of ultra-metric primes only, then $O _ { K }$ is a Dedekind domain (cf. also Dedekind ring). For example, if $K = \mathbf{Q}$ and $P$ consists of all prime numbers, then $O _ { K } = \mathbf{Z}$.

Fix also a finite subset $S$ of $P$. Consider the field of totally $\text{p}$-adic numbers:

\begin{equation*} K _ { \operatorname{tot} S } = \bigcap _ { \operatorname{p} \in S } \bigcap _ { \sigma \in G ( K ) } K _ { \operatorname{p} } ^ { \sigma } \end{equation*}

This is the largest Galois extension of $K$ in which each $\text{p} \in S$ totally splits. Let $\mathbf{Z}_{\operatorname{tot} S} = O_{K_{\operatorname{tot} S}}$. If $K = \mathbf{Q}$, $O _ { K } = \mathbf{Z}$ and $S$ is empty, then $K_{\operatorname{tot} S} = \widetilde{\mathbf{Q}}$ and $\mathbf{Z}_{\operatorname{tot} S} = \widetilde{\mathbf{Z}}$ is the ring of all algebraic integers. The following two theorems, which can be found in [a10] and [a4], are therefore generalizations of Rumely's local-global principle and the density theorem (cf. also Local-global principles for the ring of algebraic integers).

The local-global principle: In the above notation, let $M = K_{\operatorname{tot} S}$. Consider an absolutely irreducible affine variety $V$ over $K$. Suppose that $V ( O _ { K , \text{p} } ) \neq \emptyset$ for each $\operatorname{p} \in P$. Suppose further that $V _ { \text{simp} } ( O _ { K , p } ) \neq \emptyset$ for each $\text{p} \in S$. Then $V ( O _ { M } ) \neq \emptyset$.

Here, $V _ { \text{simp} }$ is the Zariski-open subset of $V$ consisting of all non-singular points.

The density theorem: Let $M$ and $V$ be as in the local-global principle. Let $T$ be a finite subset of $P$ containing $S$. Suppose that $V ( O _ { K , \text{p} } ) \neq \emptyset$ for each $\operatorname{p} \in P$. For each $\text{p} \in S$, let ${\cal U} _ { \operatorname { p } }$ be a non-empty $\text{p}$-open subset of $V _ { \text {simp} } ( K _ { \text{p} } )$. For each $\text{p} \in T \backslash S$, let ${\cal U} _ { \operatorname { p } }$ be a non-empty $\text{p}$-open subset of $V ( K _ { \text{p} } )$. Then $V ( O _ { M } )$ contains a point which lies in ${\cal U} _ { \operatorname { p } }$ for each $\operatorname{p} \in T$.

Although the density theorem looks stronger than the local-global principle, one can actually use the weak approximation theorem and deduce the density theorem from the local-global principle.

Both the local-global principle and the density theorem are actually true for fields $M$ which are much smaller than $K _ { \operatorname{tot} S}$. To this end, call a field extension $M^{\prime}$ of $K$ PAC over $O _ { K }$ if for every dominating separable rational mapping $\phi : V \rightarrow \mathbf A ^ { r }$ of absolutely irreducible varieties of dimension $r$ over $M^{\prime}$, there exists an $x \in V ( M ^ { \prime } )$ such that $\phi ( x ) \in O^r_K$. If $K$ is a number field and $P$ consists of ultra-metric primes only, [a8], Thm. 1.4; 1.5, imply both the density theorem and the local-global principle for $M = M ^ { \prime } \cap K_{ \operatorname{tot} S}$. In the function field case, [a8] must replace $M$ by its maximal purely inseparable extension, which is denoted by $M _ { \operatorname {ins} }$. Accordingly, the fields $K _ { \operatorname{p} }$ in the assumption of the density theorem and the local-global principle must be replaced by $( K _ { p } ) _ { \text{ins} }$. However, using the methods of [a4] and [a5], it is plausible that even in this case one can restore the theorem for $M = M ^ { \prime } \cap K _ { \operatorname{tot} S }$.

By Hilbert's Nullstellensatz (cf. also Hilbert theorem), $K _ { s }$ is PAC over $O _ { K }$. Hence, [a8], Thms.1.4; 1.5, generalize the density theorem and the local-global principle above. Probability theory supplies an abundance of other algebraic extensions of $K$ which are PAC over $O _ { K }$. The measure space in question is the Cartesian product $G ( K ) ^ { e }$ of $e$ copies of the absolute Galois group of $K$ equipped with the Haar measure. For each $\overline { \sigma } = ( \sigma _ { 1 } , \ldots , \sigma _ { e } ) \in G ( K ) ^ { e }$, let $K _ { s } ( \overline { \sigma } )$ be the fixed field of $\sigma _ { 1 } , \ldots , \sigma _ { e }$ in $K _ { s }$. By [a7], Prop. 3.1, $K _ { s } ( \overline { \sigma } )$ is PAC over $O _ { K }$ for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. Together with the preceding paragraph, this yields the following result (the Jarden–Razon theorem): For every positive integer $e$ and for almost all $\overline { \sigma } \in G ( K ) ^ { e }$, the field $M = ( K _ { s } ( \overline { \sigma } ) \cap K _ { \operatorname{tot} S} )_{ \text{ins} }$ satisfies the conclusions of the local-global principle and the density theorem.

The local-global principle for rings implies a local-global principle for fields. An algebraic extension $M$ of $K$ is said to be P$S$C (pseudo $S$-adically closed) if each absolutely irreducible variety $V$ over $M$ which has a simple $M _ { \operatorname{p} }$-rational point for each $\operatorname{p} \in S _ { M}$, has an $M$-rational point. In particular, by the local-global principle and the Jarden–Razon theorem, the fields $K _ { \operatorname{tot} S }$ and $( K _ { s } ( \overline { \sigma } ) \cap K _ {\operatorname{tot} S}) _ {\text{ins}}$ are P$S$C for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. The main result of [a5] supplies P$S$C extensions of $K$ which are even smaller than the fields $K _ { s } ( \overline { \sigma } ) \cap K _ { \operatorname{tot}S }$ (the Geyer–Jarden theorem): For every positive integer $e$ and for almost all $\overline { \sigma } \in G ( K ) ^ { e }$, the field $K _ { s } [ \overline { \sigma } ] \cap K _ { \operatorname{tot}S }$ is P$S$C.

Here, $K _ { S } [ \overline { \sigma } ]$ is the maximal Galois extension of $K$ that is contained in $K _ { s } ( \overline { \sigma } )$. It is not known (1998) whether $O _ { K _ { s } [ \overline{\sigma} ] } $ satisfies the local-global principle. (So, the Geyer–Jarden theorem is not a consequence of the Jarden–Razon theorem.) Since a separable algebraic extension of a P$S$C field is P$S$C [a9], Lemma 7.2, the Geyer–Jarden theorem implies that $K _ { s } ( \overline { \sigma } ) \cap K _ { \operatorname{tot} S}$ is P$S$C for almost all $\overline { \sigma } \in G ( K ) ^ { e }$. Likewise, it reproves that $K _ { \operatorname{tot} S}$ is P$S$C.

A field $M$ which is P$S$C is also ample (i.e. if $V$ is an absolutely irreducible variety over $M$ and $V _ { \text { simp } } ( M ) \neq \emptyset$, then $V ( M )$ is Zariski-dense in $V$). Ample fields, in particular P$S$C fields, have the nice property that the inverse problem of Galois theory over $M ( t )$ has a positive solution (cf. also Galois theory, inverse problem of). That is, for every finite group $G$ there exists a Galois extension $F$ of $M ( t )$ such that $\operatorname{Gal}( F / M ( t ) ) \cong G$. Indeed, every finite split embedding problem over $M ( t )$ is solvable [a11], Main Thm. A, [a6], Thm. 2.

Another interesting consequence of the local-global principle describes the absolute Galois group of $K _ { \operatorname{tot} S }$: It is due to F. Pop [a11], Thm. 3, and may be considered as a local-global principle for the absolute Galois group of $K _ { \operatorname{tot} S }$ (Pop's theorem): The absolute Galois group of $K _ { \operatorname{tot} S }$ is the free pro-finite product

\begin{equation*} \prod _ { \text{p} ^ { \prime } \in S ^ { \prime } } G ( K _ { \text{p} ^ { \prime } } ), \end{equation*}

where $S ^ { \prime }$ is the set of all extensions to $K _ { \operatorname{tot}S }$ of all $\text{p} \in S$. This means that if $G$ is a finite group, then each continuous mapping $\alpha _ { 0 } : \cup _ { \text { p } ^ { \prime } \in S ^ { \prime } } G ( K _ { \text { p } ^ { \prime } } ) \rightarrow G$ whose restriction to each $G ( K _ { \operatorname{p} ^ { \prime } } )$ is a homomorphism, can be uniquely extended to a homomorphism $\alpha : G ( K _ { \operatorname {tot} S } ) \rightarrow G$.

As a consequence of the local-global principle, Yu.L. Ershov [a2], Thm. 3, has proved that the elementary theory of $\mathbf{Q}_{\operatorname{tot} S}$ is decidable. If $S$ does not contain $\infty$, this implies, by [a1], p. 86; Corol. 10, that the elementary theory of ${\bf Z} _ { \operatorname{tot} S }$ is decidable. In particular, Hilbert's tenth problem has an affirmative solution over ${\bf Z} _ { \operatorname{tot} S }$. If however, $S = \{ \infty \}$, then the elementary theory of $\mathbf{Q}_{\operatorname{tot} S}$ is decidable [a3] but the elementary theory of ${\bf Z} _ { \operatorname{tot} S }$ is undecidable [a12].

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