Linear theories of global fields with absolute values

1. Problem Statement and Motivation

The paper investigates the decidability of the first-order theory of global fields (finite extensions of $\mathbb{Q}$ or $\mathbb{F}_p(t)$) when viewed not as rings, but as vector spaces over themselves, augmented with predicates for absolute values.

• Context: The standard first-order theory of global fields in the language of rings ($+, \cdot$) is known to be undecidable (Robinson, Rumely). The decidability of the existential theory (Hilbert's 10th problem for global fields) remains a major open problem.
• The "Linear" Approach: The authors restrict the language to scalar multiplication ($\cdot_\lambda$ for $\lambda \in k$) rather than full field multiplication. This creates a "linear theory."
• The Question: What happens to decidability when we add arithmetic structure in the form of absolute values (places)? Specifically, does the theory become decidable or undecidable depending on the type of place (finite/ultrametric, real archimedean, or complex archimedean)?

2. Methodology

The authors employ a combination of model-theoretic techniques and number-theoretic properties:

1. Valued Vector Spaces: They develop a theory of vector spaces over a field $k$ equipped with a valuation. They define a two-sorted language (vector space sort and value sort) and prove Quantifier Elimination (QE) for discretely valued vector spaces.
2. Bi-interpretation: They establish bi-interpretations between:
   • The two-sorted language of valued vector spaces.
   • The one-sorted language of vector spaces with absolute value predicates ($L_p, M_p$).
   • This allows them to transfer model-theoretic properties (completeness, QE, decidability) from the valued vector space theory to the absolute value theory.
3. Weak Approximation: They utilize the Artin-Whaples Approximation Theorem to axiomatize the interaction between multiple distinct absolute values on a global field. This allows them to handle families of places simultaneously.
4. Definability of Multiplication: A crucial part of their methodology involves analyzing when the full field multiplication ($xy=z$) can be defined (either quantifier-free or existentially) using only addition, scalar multiplication, and absolute values.
   • For Real places, they show multiplication is quantifier-free definable.
   • For Complex places, they show the real subfield is definable, and multiplication is existentially definable under certain conditions.

3. Key Contributions and Results

A. Single Place Results (Theorem 1.1)

The authors classify the decidability of the theory of a global field $k$ with a single absolute value $|\cdot|_p$ based on the type of place $p$:

• Finite (Ultrametric) Places:
  • The theory $Th(k, +, 0, 1, (\cdot_\lambda), L_p, M_p)$ is decidable.
  • It admits Quantifier Elimination.
  • The theory of the global field is elementarily equivalent to the theory of its completion $\hat{k}_p$ (a local field).
• Real (Archimedean) Places:
  • The theory with only the order/absolute value relation $L_p$ ($Th(k, +, 0, 1, (\cdot_\lambda), L_p)$) is decidable and admits QE.
  • However, if the ternary relation $M_p$ (representing $|xy|_p = |z|_p$) is added, the theory becomes undecidable. This is because multiplication can be defined quantifier-free from $L_p$ and $M_p$ in the real case.
  • The existential theory with $M_p$ is Turing-equivalent to the existential theory of the field itself.
• Complex (Archimedean) Places:
  • The theory $Th(k, +, L_p)$ is undecidable. The complex absolute value allows the definition of the real subfield $k \cap \mathbb{R}$, which in turn allows defining the full field structure.
  • Existential Theory:
    • If $k \cap \mathbb{R}i = \{0\}$ (the field does not contain imaginary units), the existential theory is decidable.
    • If $k \cap \mathbb{R}i \neq \{0\}$, the existential theory is Turing-equivalent to the existential theory of the real subfield $k \cap \mathbb{R}$.

B. Families of Places (Theorem 1.2)

The authors generalize these results to sets of places $S_0$ (providing $L_p$) and $S_1$ (providing $M_p$).

• Decidability Criterion: The theory $Th(k, +, 0, 1, (\cdot_\lambda), (L_p)_{p \in S_0}, (M_p)_{p \in S_1})$ is decidable if and only if:
  1. $S_0$ contains no complex places.
  2. $S_1$ contains no infinite places (real or complex).
• Undecidability: If $S_0$ contains a complex place, or if $S_1$ contains any infinite place, the theory is undecidable.
• Axiomatization: In the decidable case, the authors provide an explicit axiomatization combining the theories of individual places with a "Weak Approximation" scheme. They also prove Model Completeness (and QE if $S_1 = \emptyset$).

4. Technical Details of Proofs

• Quantifier Elimination for Valued Vector Spaces: The core technical engine is Theorem 3.5, proving QE for the theory $T_{k,u}$ of vector spaces over $k$ with a discrete valuation. This relies on the structure of the value group (isomorphic to $\mathbb{Z}$) and the residue field.
• Reduction to Local Fields: For finite places, the global field's theory is shown to be the same as its completion's theory. Since the theory of local fields (like $\mathbb{Q}_p$) in the linear language is well-understood and decidable, the global result follows.
• Definability Arguments:
  • Real Case: Using the identity $x \ge 0 \iff |x-1| \le |x+1|$, the order is definable. Combined with $|xy|=|z|$, full multiplication is recovered.
  • Complex Case: They prove that the real subfield $k \cap \mathbb{R}$ is definable in $(k, +, L_\infty)$. If the field contains imaginary numbers, the full field structure becomes existentially definable, leading to undecidability.

5. Significance

• Boundary of Decidability: The paper precisely delineates the boundary between decidable and undecidable theories for global fields when moving from pure ring theory to linear theory with absolute values. It shows that the "complexity" of the absolute value (specifically the ability to define multiplication or the real subfield) is the deciding factor.
• Model Theory of Global Fields: It provides a rare example of a decidable theory for global fields that includes significant arithmetic structure (absolute values), contrasting with the undecidability of the full ring theory.
• Connection to Local Fields: It establishes a strong link between the model theory of global fields and their local completions in the linear setting, showing that for finite places, the global field behaves elementarily like its local counterpart.
• Existential Theory Insights: The results offer new perspectives on Hilbert's 10th problem. While the full existential theory of global fields remains open, the paper shows that restricting the language to linear operations plus absolute values yields decidability in many cases, except where the absolute value is strong enough to reconstruct the full ring multiplication.

In summary, Fehm and Touchard demonstrate that the linear theory of global fields is decidable provided one avoids complex places and avoids using absolute values to define multiplication (via the $M_p$ predicate for infinite places). The moment the language becomes powerful enough to define the full ring structure (which happens with complex places or real places with $M_p$), undecidability returns.