Serre's Three letters to Walter Feit on group representations and quaternions' (2008) proves that the faithful irreducible two-dimensional complex representation $\bar\rho$ of the quaternion group $Q_8$ is realizable over the ring of integers $O_K$ of $K=\mathbb{Q}(\sqrt{-N})$ if and only if $N=x^2+2y^2$ when 2 is inert in $K$ ($N\equiv 3 \bmod 8$), and, when 2 is ramified in $K$, if and only if $N=x^2+2y^2$ or $N=x^2+y^2$ (a genuine disjunction: $N=6=2^2+2\cdot 1^2$ is integral by the criterion but is not a sum of two squares, while $N=5=1^2+2^2$ is a sum of two squares but is not of the form $x^2+2y^2$), but the proof — via Steinitz classes, genus theory, and Brauer-group/Hilbert-symbol computations — is non-constructive and produces no explicit lattice. The problem asks for explicit constructions: for every positive instance of the criterion, write down generators of $Q_8$ inside $\mathrm{GL}_2(O_K)$ (equivalently a free $O_K$-lattice stable under the representation), including the dyadically ramified case, and do the same for the companion groups $2A_4$ and $2^{-}S_3$ whose integrality criteria are now known. Later literature (2023–2026) confirms the construction problem is open: specialists in exactly this subfield state (January and August 2026) that they can count the integral forms but have no way of distinguish one from the other without explicitly writing down a representation', and that for Schur-index-2 representations like Q8Q_8 `there is some work to be done'.

Background

A complex representation of a finite group GG that is realizable over a number field kk need not be realizable over the ring of integers OkO_k: there may be no OkO_k-lattice stable under the representation. For a representation ρ ⁣:G→GLn(k)\rho\colon G\to\mathrm{GL}_n(k), an integral form of ρ\rho is a GLn(Ok)\mathrm{GL}_n(O_k)-conjugacy class of integral representations (homomorphisms G→GLn(Ok)G\to\mathrm{GL}_n(O_k)) that become GLn(k)\mathrm{GL}_n(k)-conjugate to ρ\rho over kk; the basic question, posed in Serre's 1997 letters to Walter Feit (published as J.-P. Serre, Three letters to Walter Feit on group representations and quaternions', J. Algebra 319 (2008) 549–557), is whether the set of integral forms can be empty, i.e. whether $\rho$ can fail to admit any integral realization. The prototype is the faithful irreducible two-dimensional complex representation $\bar\rho$ of the quaternion group $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ (order 8, $i^2=j^2=k^2=ijk=-1$). Its character is rational-valued (traces $2,-2,0$) and has Schur index 2 at the real place, hence Schur index 2 over $\mathbb{Q}$ (Brauer–Speiser), so $\bar\rho$ is not realizable over $\mathbb{Q}$ in dimension 2; over the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-N})$, $N$ a positive squarefree integer with $N\not\equiv 7\bmod 8$ (so that $K$ embeds in the rational quaternion algebra $D=(-1,-1)_{\mathbb{Q}}$ and splits it, by Gauss's theorem on sums of three squares), the Schur index drops to 1 and $\bar\rho$ becomes realizable as a two-dimensional $K$-representation. Serre proved (Propositions 1 and 3 of the letters; re-derived as Lemma 2 and Theorem 1 of Schmid 2023) that $\bar\rho$ is realizable over $O_K$ — equivalently, some basis exhibits generators of $Q_8$ as matrices in $\mathrm{GL}_2(O_K)$, i.e. there is a free $O_K$-lattice stable under $\bar\rho$ — if and only if $N=x^2+2y^2$ when 2 is inert in $K$ (equivalently $N\equiv 3\bmod 8$), and, when 2 is ramified in $K$, $N=x^2+2y^2$ or $N=x^2+y^2$: in the ramified case the obstruction in the genus group $C_K/C_K^2$ (ideal classes modulo squares) is only defined up to the class of the prime ideal above 2, whence the second representability condition; the disjunction is genuine — $N=6=2^2+2\cdot 1^2$ is integral by the criterion yet is not a sum of two squares, while $N=5=1^2+2^2$ is a sum of two squares yet is not of the form $x^2+2y^2$. When $N\equiv 7\bmod 8$, 2 splits in $K$ and $\bar\rho$ is not realizable over $K$ at all. Schmid's Theorem 1 states the criterion equivalently in embedding form: $\bar\rho$ is realizable over $O_K$ precisely when $\mathbb{Q}(\sqrt{2},\sqrt{N})$ embeds into a $Q_8$-field (a Galois extension of $\mathbb{Q}$ with Galois group $Q_8$; by Witt's theorem, $\mathbb{Q}(\sqrt{a},\sqrt{b})$ embeds into a $Q_8$-field iff $(-1,-1)=(-a,-b)$ in $\mathrm{Br}_2(\mathbb{Q})$) or $N$ is a sum of two integer squares. The proof machinery is arithmetic and non-constructive: Steinitz's theorem (a finitely generated torsion-free module over the Dedekind domain $O_K$ is determined up to isomorphism by its rank and a Steinitz ideal class, and it is free exactly when that class is trivial), genus theory of $C_K/C_K^2$ with its genus characters, the quaternion (Hilbert) symbol, the 2-torsion $\mathrm{Br}_2(\mathbb{Q})$ of the Brauer group (the group of similarity classes of finite-dimensional central simple algebras over $\mathbb{Q}$), and the maximal orders of $D$ containing $O_K$; the obstruction is a class $c_K\in C_K/C_K^2$ whose triviality, computed via the Hilbert symbol $\nu_K(c_K)=(D)\cdot(d_D,-N)$ and Brauer's formula, is equivalent to integrality. Feit had found in 1974 (unpublished; rediscovered by Cliff–Ritter–Weiss 1992) the example $K=\mathbb{Q}(\sqrt{-35})$, where $35=5\cdot 7$ with $5\equiv 5\bmod 8$ and $7\equiv 7\bmod 8$ (so $35$ is represented neither by $x^2+2y^2$ nor by $x^2+y^2$), so the criterion correctly predicts non-integrality. What the letters do not supply is any explicit lattice in the positive cases: the existence proofs never produce matrices. A sufficient normalization for an individual instance is to write $-1=a^2+b^2$ with $a,b\in O_K$ (then $u=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ and $v=\begin{pmatrix}a&b\\b&-a\end{pmatrix}$ generate a copy of $Q_8$ in $\mathrm{GL}_2(O_K)$), which is elementary for small $N$ but has no known closed form valid uniformly in $N$; it is sufficient but not necessary ($N=34=5^2+3^2$ is integral by the criterion, yet $a^2+b^2=-1$ has no solution in $\mathbb{Z}[\sqrt{-34}]$ because the negative Pell equation $x^2-34y^2=-1$ is unsolvable), so a general construction must normalize lattices by other means. The hard case is the ramified-2 (dyadic) one, where 2 is not invertible in $O_K$ and half-integral elements enter. Later work has extended the criteria (Schmid 2023: the basic spin character of the double cover $2A_4$ of the tetrahedral group (the binary tetrahedral group $\mathrm{SL}_2(\mathbb{F}_3)$) is realizable over $O_K$ iff $N=x^2+2y^2$; that of the double cover $2^{-}S_3$ of the symmetric group $S_3$ iff $N=x^2+3y^2$ where either $x,y$ are integers or both lie in $\frac{1}{2}+\mathbb{Z}$ (a Davenport–Cassels-type variant)), and the Arenas-Carmona school has computed the exact cardinality of the set of integral forms for $Q_8$ via Bruhat–Tits trees (the combinatorial tree on which $\mathrm{GL}_2$ over a local field acts), obtaining $2h_k^{(2)}$ forms when $N\equiv 3\bmod 8$, $4h_k^{(2)}$ when 2 is ramified and the class of the dyadic prime ideal is a square in the class group, and either $h_k^{(2)}$ or $3h_k^{(2)}$ when it is not, where $h_k^{(2)}$ is the cardinality of the maximal exponent-2 subgroup of the ideal class group of $k$ — but counting does not produce constructions, and in the last, ambiguous subcase the authors state they have no way of distinguish one from the other without explicitly writing down a representation'; analogous counts for the Hurwitz group SL2(F3)\mathrm{SL}_2(\mathbb{F}_3) and the dicyclic group C3⋊C4C_3\rtimes C_4 of order 12 are indicated but not carried out. The third letter additionally asks for more complicated examples'; in Schmid 2023, Section 1 (immediately after his Theorem 2 on characters realizable over $K$ but not over $O_K$) the request concerns more complicated examples of that non-integrality phenomenon, and globally irreducible representations in the sense of Gross (irreducible representations that remain irreducible under reduction modulo every prime) — notably the basic spin characters (the faithful irreducible characters of the Schur double covers $2^{\pm}A_n$, $2^{\pm}S_n$ of the alternating and symmetric groups; Tiep determined those that are globally irreducible) over their quadratic splitting fields — appear only as Schmid's own suggested candidates (Candidates could be obtained via globally irreducible representations as introduced by Gross'); Malinin (2018) obtains conditions on KK for a finite subgroup of GL2(K)\mathrm{GL}_2(K) to be conjugate into GL2(OK)\mathrm{GL}_2(O_K) and studies globally irreducible representations, without solving the Q8Q_8 construction problem. Here the genus/Brauer techniques face a structural obstacle visible already in Schmid's re-derivation: the Q8Q_8 character stays irreducible modulo every odd prime but becomes reducible modulo 2, and for larger groups reducibility modulo primes dividing the group order can break the transfer of the technique.

Problem Statement

Given K=Q(−N)K=\mathbb{Q}(\sqrt{-N}) with NN a positive squarefree integer, and ρˉ\bar\rho the faithful irreducible two-dimensional complex representation of the quaternion group Q8={±1,±i,±j,±k}Q_8=\{\pm 1,\pm i,\pm j,\pm k\} (i2=j2=k2=ijk=−1i^2=j^2=k^2=ijk=-1): (1) For every NN satisfying Serre's positive criterion — N=x2+2y2N=x^2+2y^2 when 2 is inert in KK (N≡3 mod 8N\equiv 3\bmod 8), and, when 2 is ramified in KK, N=x2+2y2N=x^2+2y^2 or N=x2+y2N=x^2+y^2 (the case N≡7 mod 8N\equiv 7\bmod 8, where 2 splits in KK and ρˉ\bar\rho is not realizable over KK at all, does not arise) — give an explicit construction of a free OKO_K-lattice stable under ρˉ\bar\rho, equivalently explicit matrices in GL2(OK)\mathrm{GL}_2(O_K) generating a subgroup isomorphic to Q8Q_8 realizing ρˉ\bar\rho; the construction must in particular settle the ramified-2 (dyadic) case, including the subcase that the 2026 counting results cannot distinguish without writing down a representation. (2) Do the same for the companion groups 2A42A_4 (the binary tetrahedral group SL2(F3)\mathrm{SL}_2(\mathbb{F}_3)) and 2−S32^{-}S_3 (the dicyclic group of order 12, C3⋊C4C_3\rtimes C_4) in their two-dimensional faithful representations over KK, in the positive cases of Schmid's criteria (N=x2+2y2N=x^2+2y^2 resp. N=x2+3y2N=x^2+3y^2 with the half-integral variant). The problem comprises the Q8Q_8 core (1) and component (2) for both companion groups 2A42A_4 and 2−S32^{-}S_3.

References

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