Quaternion lattices and quaternion fields

Let Q8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_8$$\end{document} be the quaternion group of order 8 and χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }$$\end{document} its faithful irreducible character. Then χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }$$\end{document} can be realized over certain imaginary quadratic number fields K=Q(-N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K={\mathbb Q}\bigl (\sqrt{-N}\bigr )$$\end{document} but not over their rings of integers (Feit, Serre); here N is a positive square-free integer. We show that this happens precisely when Q(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb Q}\bigl (\sqrt{N}\bigr )$$\end{document} but not Q(2,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb Q}\bigl (\sqrt{2}, \sqrt{N}\bigr )$$\end{document} can be embedded into a Q8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_8$$\end{document}-field over the rationals (Galois with group Q8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_8$$\end{document}) and N is not a sum of two integer squares. In particular, we get that χ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }$$\end{document} cannot be integrally realized if N is (properly) divisible by some prime q≡7(mod8)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q\equiv 7\,({\textrm{mod}\,}8)$$\end{document}.

We use the results obtained by Serre [13], and a celebrated theorem of Witt [16]: A real biquadratic number field Q √ a, √ b can be embedded into a Q 8 -field over the rationals if and only if (−1, −1) = (−a, −b) in Br 2 (Q).
One knows that in Q 8 -fields over the rationals at least two (finite) primes are ramified, and those where only two primes are ramified are described explicitly in [9]. In fact, either p = 2 and some prime q ≡ 1, 3 (mod 8) ramify, or two distinct primes p ≡ q ≡ 1 (mod 4). (For p = 2, q = 3, one of the two Q 8 -fields is Q (2 + √ 2)(3 + √ 3) already given by Dedekind.) By genus theory (developed by Gauss [5]), the order of C K /C 2 K equals 2 s−1 where s is the number of primes which are ramified in K. Theorem 2. Suppose the faithful irreducible character of Q 8 can be realized over K = Q √ −N but not over R = O K . Then at least two rational primes must ramify in K, and just two are ramified precisely when N = 2q or N = pq for primes p ≡ 5 (mod 8) and q ≡ 7 (mod 8).
A similar example is also given when N = pq for primes p ≡ 3 (mod 8) and q ≡ 7 (mod 8), but then 2, p, and q are ramified in K = Q √ −pq (and C K /C 2 K has order 4). In his third letter, Serre [13] asked for "more complicated examples". Candidates could be obtained via globally irreducible representations as introduced by Gross [6]. For this reason, we shall discuss the arguments given by Serre in some detail.

Definite rational quaternion algebras. Let
be a definite quaternion algebra over Q (a > 0, b > 0), and let (D) = (−a, −b) be its class in Br 2 (Q). This class is determined by the local Hilbert symbols (−a, −b) p over the p-adic fields Q p for all rational primes p (knowing that (−a, −b) ∞ = −1 over R = Q ∞ ). Recall that D is said to be ramified at p (including p = ∞) provided the completion Q p ⊗ D is a skew field, the unique non-split quaternion algebra over Q p , just in which case (−a, −b) p = −1. By Hilbert's reciprocity law [11,Thm. III.3], the number of such primes is finite and odd. Let Following Eichler [3] (and Serre [13]), we define the signed discriminant (in German: "Grundzahl") d D = d (−a,−b) = −(product of all primes p where D is ramified at). Here the minus sign indicates that D is ramified at ∞. Eichler's definition has the advantage that the isomorphism type of D is determined by d D . For basic properties of (definite) quaternion algebras, we refer to Lam [7]. In particular, there is a natural involutory anti-automorphism α →ᾱ on D, which induces on every (imaginary) quadratic subfield the nontrivial Galois automorphism over Q, and we may define the (reduced) trace and norm. Also, the fact that D does not split (is a skew field) may be expressed by the property that the quadratic (nondegenerate) diagonal form [a, b, ab] resulting from the pure quaternions in D is not isotropic over Q (does not represent zero) [ [a, b, ab] represents N , that is, if N = ax 2 + by 2 + abz 2 for rationals x, y, z. This happens precisely when −N is not a square in Q * p for all primes p dividing d D or, equivalently, when (D) = (−N, −n) in Br 2 (Q) for some positive integer n, which may be chosen such that N and n belong to different nontrivial cosets of Q * modulo squares.
Proof. We can embed K into D if and only if there exist rationals x, y, z such that N = ax 2 + by 2 + cz 2 . In fact, then we assign to √ −N the pure quaternion xi + yj + zk and observe that Q1 ⊕ Q(xi + yj + zk) is a (commutative) subfield of D (each nonzero element being invertible as its norm is positive). Of course, one can embed K in D in many distinct ways (all resulting subfields being conjugate by the Skolem-Noether theorem), its number being related to h K (as already observed by Gauss).
The Davenport-Cassels lemma gives a sufficient condition for the statement that N can be represented by [a, b, ab] over Z. In particular this holds when (D) = (−1, −1) where N = x 2 + y 2 + z 2 with integers x, y, z. By a classical result of Gauss [5], this is possible if and only if N ≡ 7 (mod 8) (see [   (D) = (−N, −n), where n may be chosen as asserted (in view of Dirichlet's prime number theorem and the fact that the primes plus −1 give rise to a basis for the F 2 -space Q * /Q * 2 ).

Genus theory.
Let again D be a definite quaternion algebra over the rationals, with class (−a, −b) in Br 2 (Q). Let K = Q( √ −N ) be embedded in D, thus splitting D, and let R = O K be the ring of integers of K, its unique maximal Z-order. Suppose L is a full R-lattice in D (of R-rank 2). Since R is a Dedekind domain and L torsion-free, up to isomorphism L is determined by its rank and its Steinitz class (L) ∈ C K [2,Thm. 4.13]. In particular, L is a free R-module if and only if (L) = 1. If (L) = (a) 2 ∈ C 2 K for some fractional ideal a of K, then (a −1 L) = (a) −2 (L) = 1 and so a −1 L is a free R-module. Thus we are led to consider the quotient group C K /C 2 K , and to apply genus theory (see Zagier [17,Sect. 12] for a nice expositionà la Gauss). The principal genus theorem tells us that an integral ideal a of K has class (a) ∈ C 2 K if and only if (Na, −N ) = 1 in Br 2 (Q) (see also [4,Prop. 2.12]). Here Na = |R/a| is the absolute norm of a. We obtain an injective homomorphism N ). The image of ν K consists of all those classes of quaternion algebras which are ramified only at those (finite) primes ramified in K. Serre [13] has proved in his last letter to Feit (Propositions 1, 3) that the following holds.
Here R p ∼ = Z p ⊗ R is a discrete valuation ring with maximal ideal πR p , say, and quotient field K p ∼ = Q p ⊗ K. In particular, L p ∼ = M p as R p G-lattices; if this is true for all primes p, then L and M are said to be in the "same genus" (for lattices) [2,Sect. 31]. This notion is weaker, however. In order to get (global) lattices in the same genusà la Gauss, we need to know that L p = α p M p for each prime p and some α p ∈ K * p , because then L = aM where Vol. 120 (2023)

Quaternion lattices and quaternion fields 257
Such an α p ∈ K * p exists when the reduction L p /πL p is an irreducible [R p /πR p ]G-module, that is, when χ remains irreducible as a Brauer character mod p (see [12,Ex. 15.3]).
Replacing the ring R by a maximal order R in D, the representation of G on V is called globally irreducible provided the reductions L/pL are irreducible for all (two-sided) maximal ideals p of R . There are quite a lot such representations; the ones known so far are listed in [15]. In particular, Tiep [14] determined all globally irreducible representations associated to basic spin characters of 2 ± S n and 2A n (where we use the Atlas notation: in 2 − S n , transpositions are lifted to elements of order 4). The concept of (basic) spin characters has been introduced by Schur [10]. The basic spin character of 2 + S n and 2 − S n is rational-valued when n is odd, and that of 2A n when n is even.
It should be mentioned that there are also globally irreducible representations (G, V ) where the corresponding End G (V ) is isomorphic to Q (treated by Thompson; see [12,Ex. 15.4]) or to an imaginary quadratic number field.  N ). The character χ remains irreducible mod p for all odd primes p but not for p = 2. Thus the Steinitz class of any full RQ 8 -lattice in D is determined by (H) up to multiplication with squares and powers of (q), where q is the prime ideal of R above 2. If 2 is inert in K, then q = 2R is principal, and each RG-lattice affording χ belongs to the genus of H. Otherwise 2 is ramified in K and Nq = 2, so that ν K ((q)C 2 K ) = (2, −N ). If (2, −N ) = 1, we get the same statement as before. Otherwise there is an RQ 8 -lattice in D which is free over R precisely when Arch. Math.

Proof of Theorem 2.
By hypothesis, the faithful irreducible character χ of Q 8 can be realized over K = Q √ −N but not over R = O K . By genus theory (à la Gauss), then at least two primes must ramify in K. From the proof of Theorem 1, we know that N can be neither written as N = x 2 + y 2 nor as N = x 2 + 2y 2 with integers x, y. Thus in the first case, N is divisible by a prime q ≡ 3 (mod 4), and in the second case by a prime q ≡ 1, 3 (mod 8). This follows from classical results due to Fermat and Euler (describing the odd prime divisors of such sums with gcd(x, y) = 1). It follows that χ cannot be integrally realized if N is divisible by a prime q ≡ 7 (mod 8).
Suppose that N is even. Then 2 is ramified in K and in Q( √ N ). Let q be an odd prime such that Q √ 2, √ q cannot be embedded into a Q 8 -field over the rationals. Then q ≡ 1, 3 (mod 8) by the results in [9]. By Theorem 1 and our minimality assumption, N = 2q. If q ≡ 5 (mod 8), then N = 2q is a sum of two integer squares and so χ can be realized over R, against our hypothesis. Thus q ≡ 7 (mod 8), as required. So let N be odd in what follows. If N is a prime, then necessarily N ≡ 1 (mod 4) because only then 2 ramifies in K. But then N is a sum of two integer squares, which is excluded. Therefore from Theorem 1 and the results in [9], we can infer that there are two odd primes p = q dividing N where not both are congruent to 1 mod 4. Moreover, the assumption on K ensures that N = pq. Observe that N ≡ 1, 3, 5 (mod 8).
Assume first that N ≡ 1 (mod 8). Then either p ≡ q ≡ 3 (mod 8) or p ≡ q ≡ 5 (mod 8). In the first case, p and q can be written as x 2 +2y 2 with integers x, y (Fermat-Euler). But then (−2, N) = (−2, p) · (−2, q) = 1 in Br 2 (Q) and so N can be written in this manner too, which is excluded. If p ≡ q ≡ 5 (mod 8), then p, q and hence N = pq can be written as sums of two integer squares, which again is excluded.
Assume next that N ≡ 5 (mod 8). Since not both primes are congruent to 1 (mod 4), by appropriate notation, we get that p ≡ 3 (mod 8) and q ≡ 7 (mod 8). Thus χ cannot be realized over R. However, in this case 2, p, and q ramify in K.
Let finally N ≡ 3 (mod 8). In this case, 2 is unramified in K, thus χ is not realizable over R if and only if N = pq cannot be written as x 2 + 2y 2 with integers x, y. Hence we cannot have p ≡ 1 (mod 8), q ≡ 3(mod 8), or vice versa (Fermat-Euler). Choosing notation appropriately, we therefore have p ≡ 5 (mod 8) and q ≡ 7 (mod 8), as required.
6. An example. Let us call a finite group G quaternionic provided it admits a faithful irreducible character χ such that the corresponding simple component of the real group algebra RG is isomorphic to the Hamiltonian quaternion skew field H. We shall see that then G = Q 8 , 2A 4 , or 2 − S 3 , and that χ is the basic spin character of these groups. words, the Frobenius-Schur indicator ind(χ) = −1, that is, g∈G χ(g 2 ) = −|G| (e.g. see [12,Prop. 39]). Since χ(1) = 2 is a divisor of |G|, there are t ≥ 1 involutions in G. Let r be the number of elements g ∈ G for which g 2 is an involution with χ(g 2 ) = −2 (so g 2 is scalar multiplication with −1), and let s be the number of elements g ∈ G for which g 2 has order 3 (implying that χ(g 2 ) = −1). Using that χ is faithful and real-valued, one gets that g χ(g 2 ) ≥ 0, the sum being taken over the remaining elements g = 1 of G. Consequently Thus there are no such remaining elements. It follows that G is a {2, 3}-group and that χ is rational-valued.
Since the Schur index m(χ) = 2 (over Q) and m ∞ (χ) = 2, there is an irreducible QG-module V affording 2χ and D = End G (V ) is a definite quaternion algebra over Q. Indeed (G, χ) gives rise to a globally irreducible representation (see Gross [6,Sect. 6] and Nebe [8, Thm. 6.1]). But this was already known to Eichler [3]  It is immediate that the basic spin characters of 2A 4 and 2 − S 3 remain irreducible as Brauer characters modulo every prime. Hence letting K = Q( √ −N ) split D, application of Lemma 2 yields the following.
The basic spin character of 2A 4 can be realized over O K if and only if N = x 2 + 2y 2 with integers x, y.
The basic spin character of 2 − S 3 can be realized over O K if and only if N = x 2 + 3y 2 where either x, y are integers or both are in 1 2 + Z (Davenport-Cassels).
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