Voight: Quaternion algebras over global fields

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corollary_14_2_3: The parity consequence of Hilbert reciprocity over the rationals: for every quaternion algebra B over Q, the set Ram B of places where B is ramified is finite and has even cardinality.

corollary_14_6_2: Hilbert reciprocity over a global field F with char F not 2, deduced from Voight's Main Theorem 14.6.1: for all nonzero a and b in F, the product of the Hilbert symbols (a,b)_v over all places v of F equals 1.

main_theorem_14_1_3: Voight's classification over the rationals: B -> Ram B is a bijection from quaternion algebras over Q up to isomorphism to finite sets of places of Q of even cardinality, and Sigma -> (product of the primes in Sigma) is a bijection from those sets to the positive squarefree integers, the composite being B -> disc B.

main_theorem_14_6_1: Voight's classification of quaternion algebras over a global field F: the map B -> Ram B is a bijection from quaternion algebras over F up to isomorphism to the finite sets of noncomplex places of F of even cardinality; the proof is deferred to the book's Section 26.8.

main_theorem_14_7_4: The Hasse-Schilling norm theorem as Voight proves it: for a quaternion algebra B over a global field F, with Omega the set of real places ramified in B, the group nrd(B^x) of reduced norms equals the group of nonzero elements of F positive at every place of Omega.

proposition_14_2_1: Hilbert reciprocity over the rationals as Voight proves it from quadratic reciprocity: for all nonzero rationals a and b, the product of the local Hilbert symbols (a,b)_v over all places v of Q, the primes and infinity, equals 1.

proposition_14_6_7: Voight's local-global principle for splitting and embeddings: for a finite separable extension K of a global field F, K splits a quaternion algebra B if and only if every completion K_w does; when K has degree 2 this is also equivalent to K embedding in B, to local embeddings at every place, and to K_v being a field at every ramified place v of B.

theorem_14_3_3: The Hasse-Minkowski theorem over the rationals as Voight proves it: a quadratic form Q over Q is isotropic if and only if its completion Q_v is isotropic for every place v of Q.

theorem_14_3_8: The Legendre-Gauss three-square theorem with Voight's quaternion proof: an integer n >= 0 is a sum of three integer squares if and only if n is not of the form 4^a(8b+7) with a, b integers.

theorem_14_6_9: The Hasse-Minkowski theorem over a global field as Voight records it: a quadratic form Q over a global field F is isotropic over F if and only if Q_v is isotropic over F_v for every place v of F; the proof is deferred to the book's Section 26.8.

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The copy read for this card is the book's Chapter 14, 24 pages (PDF p. n is printed p. 216+n). That chapter prints "© The Author(s) 2021" in the footer of its first page and, on its last page, "This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/)", the Creative Commons Attribution-NonCommercial 4.0 license.

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 Corollaries 14.3.6 (p. 224) and 14.3.7 (p. 225), the local-global principles for ternary forms and for equivalence of quadratic forms; Proposition 14.3.1 (p. 223, proved on pp. 225–226) records that 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 for $\operatorname{char}F\ne2$ (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.

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages; no proof is checked step by step, and the proofs of Main Theorem 14.6.1 and Theorem 14.6.9 lie outside the chapter.

Results.

• Main Theorem 14.1.3 (p. 218): over $\mathbb Q$, $B\mapsto\operatorname{Ram}B$ is a bijection onto finite sets of places of even cardinality, and onto squarefree $D>0$ through $\operatorname{disc}B$.
• Proposition 14.2.1 (p. 219): Hilbert reciprocity over $\mathbb Q$, $\prod_v(a,b)_v=1$ for all $a,b\in\mathbb Q^\times$.
• Corollary 14.2.3 (p. 219): $\operatorname{Ram}B$ is finite of even cardinality for every quaternion algebra $B$ over $\mathbb Q$.
• Theorem 14.3.3 (p. 223): Hasse--Minkowski over $\mathbb Q$.
• Theorem 14.3.8 (p. 226): Legendre--Gauss, $n\ge0$ is a sum of three squares if and only if $n\ne4^a(8b+7)$.
• Main Theorem 14.6.1 (p. 231): over a global field, $B\mapsto\operatorname{Ram}B$ is a bijection onto finite sets of noncomplex places of even cardinality.
• Corollary 14.6.2 (p. 231): Hilbert reciprocity over a global field with $\operatorname{char}F\ne2$.
• Proposition 14.6.7 (p. 232): local-global principle for splitting fields and quadratic embeddings.
• Theorem 14.6.9 (p. 233): Hasse--Minkowski over a global field.
• Main Theorem 14.7.4 (p. 234): Hasse--Schilling, $\operatorname{nrd}(B^\times)=F^\times_{>_\Omega0}$.

Bears on.

• #940: the chapter contains no result about $r$-powerful numbers; its reciprocity and local-global theorems bear on the problem only indirectly, as the section below records.

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. Even if a proposed E940 argument reduced some auxiliary condition to the rational solvability of a quadratic form, Theorem 14.3.3 (p. 223; proof p. 225) or Theorem 14.6.9 (p. 233) could replace that rational solvability question by local ones. Proposition 14.6.7 (p. 232) could likewise test a quadratic-field embedding or norm construction through ramified places, and Main Theorem 14.7.4 (p. 234) could characterize reduced norms by signs at ramified real places. These tools might certify individual auxiliary representations, but they supply neither uniform integral control nor bounds for the number of represented integers.

In particular, 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 is therefore background for possible local obstructions and norm-form reformulations; its relation to the unresolved density problem is indirect and weak.

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