Voight: Quaternion algebras over global fields

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Full paper in Markdown.

John Voight, "Quaternion algebras over global fields," in Quaternion Algebras, Graduate Texts in Mathematics 288, Springer, 2021, pp. 217-240. https://doi.org/10.1007/978-3-030-56694-4_14

Overview

Voight classifies quaternion algebras over global fields by their local ramification. For $F=\mathbb Q$, Main Theorem 14.1.3 (p. 218) states that $B\mapsto\operatorname{Ram}B$ bijects isomorphism classes of quaternion algebras with finite even-cardinality sets of places, equivalently with positive squarefree discriminants. Hilbert reciprocity, Proposition 14.2.1 and equation (14.2.2) (p. 219), gives $\prod_v(a,b)_v=1$, hence the parity condition (Corollary 14.2.3, p. 219). Conversely, Proposition 14.2.7 (pp. 221–222) constructs an algebra with any prescribed allowable ramification set: using primes in arithmetic progressions (Theorem 14.2.9, p. 221), it chooses $q$ satisfying the quadratic-nonresidue and mod-8 conditions (14.2.11)–(14.2.12), then verifies that $(q^\diamond,D^\diamond\mid\mathbb Q)$ has exactly the desired local Hilbert symbols. Injectivity follows from the local-global equivalences of Proposition 14.3.1 (p. 223), proved through ternary quadratic forms and Hasse–Minkowski; thus global isomorphism can be checked at every completion—or all but one.

The quadratic-form component includes Legendre’s criterion for an isotropic diagonal ternary form (Theorem 14.3.4, pp. 223–224), the Hasse–Minkowski theorem over $\mathbb Q$ (Theorem 14.3.3, proof on p. 225), and local-global classification of quadratic forms (Corollary 14.3.7, p. 225). The proof proceeds by induction on dimension, reducing the ternary case to norm equations and using approximation plus a prime in an arithmetic progression to splice local representations in higher dimensions. As an integral application, Theorem 14.3.8 (Legendre–Gauss, p. 226) proves that $n\ge0$ is a sum of three integer squares exactly when $n\ne4^a(8b+7)$. Hasse–Minkowski first supplies a rational representation; integrality is then recovered using the Hamilton quaternion algebra and conjugacy of maximal orders. This integral conclusion is special and is not part of Hasse–Minkowski itself.

Sections 14.4–14.6 extend the framework to an arbitrary global field, after defining places, preferred absolute values and the product formula (14.4.6)–(14.4.7) (p. 228), rings of $S$-integers (Definition 14.4.17 and (14.4.18), p. 229), ramification (Definition 14.5.1, p. 230), and discriminant (Definition 14.5.4, p. 230). Main Theorem 14.6.1 (p. 231) gives the global classification by finite even-cardinality sets of noncomplex places. Its consequences include global Hilbert reciprocity (Corollary 14.6.2 and (14.6.3), p. 231), the local-global principle for quaternion algebras (Corollary 14.6.5, pp. 231–232), and the splitting/embedding criterion of Proposition 14.6.7 (p. 232): for separable quadratic $K/F$, an embedding $K\hookrightarrow B$ exists precisely when no ramified place of $B$ splits in $K$. The global Hasse–Minkowski theorem is recorded as Theorem 14.6.9 (p. 233). Unlike the self-contained rational treatment in §§14.2–14.3, the proofs of Main Theorem 14.6.1 and Theorem 14.6.9 are deferred to §26.8 and ultimately use analytic or class-field-theoretic input; Remark 14.6.10 and exact sequence (14.6.11) (p. 233) explain the classification through local Brauer invariants.

Finally, §14.7 determines reduced norm groups. If $\Omega$ is the set of ramified real places, Main Theorem 14.7.4 (p. 234) proves the Hasse–Schilling identity $\operatorname{nrd}(B^\times)=F^\times_{>_\Omega 0}$. Lemma 14.7.5 (p. 234) constructs locally irreducible quadratic polynomials of prescribed constant term; Lemma 14.7.6 and Corollary 14.7.8 (pp. 234–235) globalize them by density and weak approximation. The resulting quadratic extension is a field at every ramified place, so Proposition 14.6.7 embeds it in $B$, realizing the prescribed element as a reduced norm. The chapter’s scope is therefore structural and local-global: it classifies quaternion algebras, embeddings, quadratic forms, and norm groups, with explicit rational constructions, but does not develop counting or density estimates.

Relation to E940

This source bears on Problem 940.

Write E940’s set as

$$\mathcal P_r=\{m\in\mathbb Z_{\ge0}: v_p(m)=0\text{ or }v_p(m)\ge r\text{ for every prime }p\},\qquad \mathcal S_r=\bigcup_{k=0}^{r}\underbrace{(\mathcal P_r+\cdots+\mathcal P_r)}_{k\text{ summands}}.$$

E940 asks, for every $r\ge3$, whether infinitely many integers lie outside $\mathcal S_r$ and whether $\mathcal S_r$ has natural density zero. The chapter neither introduces $\mathcal P_r$ nor estimates $|\mathcal S_r\cap[1,X]|$; consequently none of its classification, reciprocity, or norm theorems proves a density statement for E940.

The nearest result is Theorem 14.3.8 (p. 226), an exact characterization of sums of three squares. It concerns quadratic variables and all integers, not sums of three $3$-powerful numbers.

Hasse–Minkowski concerns isotropy over a global field, whereas E940 imposes integral prime-exponent restrictions on each summand. The passage from rational to integral solutions in Theorem 14.3.8 uses a special maximal-order argument for three squares and does not extend here to higher powers or to $r$-powerful summands. The chapter’s relation to the unresolved density problem is therefore indirect and weak.