Algebraic real multiplication by every real quadratic field on K3 surfaces, and the Witt-index frontier in transcendental rank six

Abstract

Let $S$ be a projective K3 surface whose rational transcendental space $T=T(S)_{\mathbf Q}$ has real multiplication (RM) by a totally real field $E\neq\mathbf Q$. The Hodge conjecture (HC) for $S\times S$ is then the algebraicity of $E\subset H^4(S\times S)$. This is the first open class in the "F3′" statement of Bhattacharjee–Bhattacharya [BB26]. Their chain and Fricke constructions, together with Varesco's and van Geemen–Schütt's, reach $\sqrt d$ for $d\in\{2,3,5,6,7\}$, and they note that for $\sqrt{10}$, $\sqrt{14}$ and other composite discriminants no family is known on which their chain can be started [BB26, §5]. Independently, Schlickewei's theorem on six-line double planes [Sch10] with van Geemen's and Elsenhans–Jahnel's families [vG08, EJ14] gives algebraic $\sqrt d$ whenever $d$ is a sum of two squares (e.g. $d=2,5,10,13$).

Theorem A. For every squarefree integer $d>1$ there is a one-parameter family of projective K3 surfaces of very general Picard number $16$ with $T_{\mathbf Q}\cong U^2\oplus\langle-2\rangle\oplus\langle-2d\rangle$ and $E=\mathbf Q(\sqrt d)$ for the very general member. For every member, the Kuga–Satake correspondence is algebraic and HC holds for all powers $S^k$.

The proof combines an explicit self-adjoint operator of square $d$ (certified symbolically in $d$) with Floccari's theorem [Flo25, Thm 5.11]. That theorem says that HC holds for all powers of $S$ whenever $T(S)_{\mathbf Q}$ embeds in $(U^3\oplus\langle-m\rangle)_{\mathbf Q}$. Theorem A is therefore a short consequence of published results. Its ingredients are known: van Geemen constructed quadratic-RM families [vG08, §3]; Bayer-Fluckiger–van Geemen–Schütt [BvGS, Thm 10.6] imply the existence of a transfer structure on this lattice (its determinant has square class $d$ and the norm class can be taken to be $1$), so our matrix is a concrete specialisation; and Varesco observed one-dimensional quadratic-RM subfamilies inside Kuga–Satake-algebraic families [Var23, §5]. What is new is the uniform explicit statement for all $d$, including those that are not sums of two squares and are not reached by the chain and Fricke constructions ($d=11,14,15,19,21,22,23,\dots$). We also determine exactly where Floccari's hypothesis stops:

Theorem B. Let $\mathrm{rk}\,T=6$ and $K=\mathbf Q(\sqrt{-\det T})$. The following are equivalent:

• (i) $T$ embeds in $(U^3\oplus\langle-m\rangle)_{\mathbf Q}$ for some $m>0$;
• (ii) $T$ has Witt index $2$;
• (iii) $C^+(T)\cong M_4(K)$.

If $T=\mathrm{Tr}_{E/\mathbf Q}W$ has RM by a real quadratic $E$, then $C^+(T)\cong\mathrm{Cores}_{E/\mathbf Q}C_0(W)\otimes_{\mathbf Q}K$. The corestriction is a central simple $\mathbf Q$-algebra of dimension 16, isomorphic to $M_2(D)$ for a quaternion algebra $D$, and (i)–(iii) hold iff $K$ splits $D$. The six-line double planes and the Fricke families of [BB26] all have Witt index 2. So [BB26, Thm fricke(iii)] (HC for $X\times X$) is already a case of [Flo25], which moreover gives all powers.

Theorem C (conditional reduction of the Witt-index-1 case). Let $S$ have rank-6 RM by $E$ and Witt index 1. There are a totally positive $\lambda\in E\setminus\mathbf Q$ and a K3 surface $S'$ covered by [Flo25] such that the identity $T(S)\to T(S')$ is a Hodge isomorphism with $\psi_{S'}=\psi_S(\lambda\cdot,\cdot)$. If this single Hodge morphism is algebraic, then HC holds for $S\times S$. Witt-index-1 RM surfaces exist for every $d\le 30$ tested.

Bearing on HC. We give unconditional cases, by a short deduction from [Flo25]: HC for all powers of K3 surfaces with RM by $\mathbf Q(\sqrt d)$ for every squarefree $d$. Via Beauville–Donagi and Markman, we also get HC for Thom–Sebastiani cubic tenfolds $V(g+g)$ on one-parameter families of cubic fourfolds with RM by $\mathbf Q(\sqrt d)$, for every squarefree $d$ (Corollary 2.4). More generally, if two cubic fourfolds with Hodge-isomorphic $T$ both lie in Floccari's locus, every Hodge morphism between them is algebraic and HC holds for the Thom–Sebastiani tenfold. This covers the similitude, CM-norm and real-multiplication residues of [p11, Question 4] (Corollary 2.5). The underlying statement extends [FF26, Cor. 13.4] from Hodge-similar to Hodge-isomorphic K3 surfaces on Floccari's locus (Prop. 4.2(a), Corollary 4.3). The same holds for Thom–Sebastiani quartic sixfolds $V(g+g)$ built from smooth quartic surfaces with RM by any real quadratic field (Corollary 2.6). In rank 6, Witt index 1 (where $C^+(T)$ is non-split) is exactly where Floccari's embedding hypothesis fails. This does not mean HC fails there, and it does not exclude other methods. For these surfaces the Kuga–Satake variety is a power of a discriminant-1 Weil fourfold (Prop. 4.2). Kuga–Satake algebraicity is equivalent to the algebraicity of one explicit Hodge morphism $\varphi$, and that algebraicity would give HC for $S\times S$ (Theorem C). Higher-rank RM ($\mathrm{rk}\,T\ge8$, cubic fields) is untouched.

1. Setting and sources

Let $S$ be a projective K3 surface, $T=T(S)_{\mathbf Q}=\mathrm{NS}(S)^\perp\subset H^2(S,\mathbf Q)$, $\psi$ the intersection form and $E=\mathrm{End}_{\rm Hdg}(T)$. By Zarhin, $E$ is totally real or CM. Hodge classes on $S\times S$ are products of divisors, together with $E$; the CM case is known [Bus19, Var23 Rem. 2.16]. In the RM case every $e\in E$ is $\psi$-self-adjoint, and $T=\mathrm{Tr}_{E/\mathbf Q}W$ for an $E$-quadratic form $W$ with $\dim_E W\ge3$ [vG08, Lemma 3.2]. For $\mathrm{rk}\,T=6$ and $[E:\mathbf Q]=2$, $W$ is ternary. Its signatures at the two real places are $(2,1)$ and $(0,3)$.

The results used, each read from source (arXiv versions):

• [Flo25] Floccari, arXiv 2501.02315v2, Thm 5.11. If $T(S)_{\mathbf Q}\hookrightarrow(U^{\oplus3}\oplus\langle-m\rangle)_{\mathbf Q}$ for some $m\in\mathbf Z_{>0}$, then the Kuga–Satake correspondence for $S$ is algebraic and HC holds for every power of $S$. (Via $\mathrm{Kum}_n$-type varieties, Markman, Voisin, Varesco.)
• [vG08] van Geemen, math/0609839.
  • Lemma 3.2: construction of very general periods with $E$ exactly the given field.
  • Lemma 4.5: $\det\mathrm{Tr}(\lambda W)=N(\lambda)^m\det\mathrm{Tr}(W)$.
  • §4.7: twisting by totally positive $\lambda$.
  • Prop. 6.3: $\mathrm{Cores}_{E/\mathbf Q}C^+_E(W)\hookrightarrow C^+(T)$.
• [Sch10] Schlickewei, 0907.2503. HC for $S\times S$ for every six-line double plane. Hence algebraic $\sqrt d$ for $d=a^2+b^2$, via [vG08, Ex. 3.4] and [EJ14].
• [Var22] Varesco, 2203.09778, Thm 0.2. All powers in that case.
• [BvGS] Bayer-Fluckiger, van Geemen, Schütt, 2401.04072v4, Thm 10.6 and Ex. 11.2. Existence criterion for transfer structures; on six-line double planes the RM field must be $\mathbf Q(\sqrt d)$ with $d$ a sum of two squares.
• [FF26] Floccari–Fu, 2607.07528, Thm K. Hodge conjecture for all powers of OG6-resolutions and of hyper-Kummer K3 surfaces ($T\hookrightarrow U(2)^3\oplus\langle-4\rangle$).

[BB26] does not cite [Sch10], [Var22] or [Flo25].

2. Theorem A

Proposition 2.1. Let $d>1$ be squarefree, $V=\mathbf Q^6$ with $\psi=\mathrm{diag}(1,-1,1,-1,-2,-2d)$, $e=\frac{d+1}2$, $c=\frac{d-1}2$, and $$A=\mathrm{diag}\Big(\begin{pmatrix}e&-c\\c&-e\end{pmatrix},\begin{pmatrix}e&-c\\c&-e\end{pmatrix},\begin{pmatrix}0&d\\1&0\end{pmatrix}\Big).$$ Then $A^2=d$ and $\psi A$ is symmetric. The $\sqrt d$-eigenspace $V_+\subset V\otimes\mathbf R$ has signature $(2,1)$ and $V_-$ has signature $(0,3)$.

Proof. Blockwise, $\begin{pmatrix}e&-c\\c&-e\end{pmatrix}^2=(e^2-c^2)I=dI$ and $\begin{pmatrix}0&d\\1&0\end{pmatrix}^2=dI$. The products with the Gram blocks, $\begin{pmatrix}e&-c\\-c&e\end{pmatrix}$ and $\begin{pmatrix}0&-2d\\-2d&0\end{pmatrix}$, are symmetric. For eigenvalue $\mu=\pm\sqrt d$, the vectors $(e+\mu,c,0,0,0,0)$, $(0,0,e+\mu,c,0,0)$, $(0,0,0,0,\mu,1)$ form a basis of the $\mu$-eigenspace with diagonal Gram matrix $\mathrm{diag}(g_\mu,g_\mu,-4d)$, where $g_\mu=(e+\mu)^2-c^2=(e^2-c^2)+2e\mu+\mu^2=2d+(d+1)\mu$, so $g_{\pm\sqrt d}=2d\pm(d+1)\sqrt d$; the third vector has square $-2\mu^2-2d=-4d$, and vectors from different blocks are orthogonal. Since $(d+1)\sqrt d>2d$, $g_+>0>g_-$. ∎ (Certificate: rm_all_d_even.py, symbolic in $d$.)

Proof of Theorem A.

• Let $T_0=U^2\oplus\langle-2\rangle\oplus\langle-2d\rangle$. Rationally $U\cong\langle1,-1\rangle$, so $(T_0)_{\mathbf Q}\cong(V,\psi)$ and $A$ acts on $(T_0)_{\mathbf Q}$.
• $T_0$ is even of rank 6 and signature $(2,4)$ with discriminant group $\mathbf Z/2\times\mathbf Z/2d$ of length $2<22-6$, so it embeds primitively in the K3 lattice [Nik79, Cor. 1.12.3].
• Following [vG08, proof of Lemma 3.2], take $\omega\in V_+\otimes\mathbf C$ with $\psi(\omega,\omega)=0$ and $\psi(\omega,\bar\omega)>0$. These form an open subset $\mathcal D$ of a conic in $\mathbf P(V_+\otimes\mathbf C)\cong\mathbf P^2$, of dimension one.
• $A$ preserves $\mathbf C\omega$, hence the Hodge decomposition, so $\mathbf Q(\sqrt d)\subset\mathrm{End}_{\rm Hdg}$. For $\omega$ outside the countably many hyperplanes $v^\perp$ ($v\in V$) and the countably many eigenspaces of $\mathbf Q(\sqrt d)$-linear endomorphisms, the Hodge structure is irreducible with $E=\mathbf Q(\sqrt d)$ exactly.
• Surjectivity of the period map gives K3 surfaces $S_\omega$ with $T(S_\omega)\cong T_0$. Since $\mathrm{NS}=T_0^\perp$ has signature $(1,15)$, $S_\omega$ is projective.
• For every member, $T(S)_{\mathbf Q}\subseteq(T_0)_{\mathbf Q}$. Moreover $(T_0)_{\mathbf Q}\hookrightarrow(U^3\oplus\langle-(2d+2)\rangle)_{\mathbf Q}$: inside $\langle1,-1\rangle\oplus\langle-(2d+2)\rangle$, take $f=(\tfrac12,\tfrac32,0)$ with $f^2=-2$ and $g=(\tfrac32,\tfrac12,1)$ with $g^2=2-(2d+2)=-2d$, $f\cdot g=0$ (certificate emb_even.py).
• [Flo25, Thm 5.11] applies to every member. ∎

Remark 2.2.

• For $d=a^2+b^2$ (e.g. $2,5,10,13$) the field already occurs with algebraic RM on six-line double planes [Sch10, Var22]. Our families have a different transcendental lattice. We do not claim they are the same families.
• For $d=2,3,5,6,7$ the field also occurs in [Var23, BB26, vGS25] on other families.
• For the remaining $d$ (the least are $11,14,15,19,21,22,23$) we know of no earlier explicit family with algebraic RM by $\mathbf Q(\sqrt d)$. Varesco [Var23, §5] observed the existence of quadratic-RM subfamilies in Floccari-type families without listing fields.
• We have not checked whether different families meet.

Remark 2.3 (Hilbert schemes, [dCM02]). For every member $S$ of Theorem A and every $n\ge1$, HC holds for $S^{[n]}$, and so does F3′ in [BB26]'s sense.

• de Cataldo–Migliorini give an isomorphism of rational Chow motives $h(S^{[n]})\cong\bigoplus_\nu h(S^{(\nu)})(n-\ell(\nu))$, induced by algebraic correspondences. Here $S^{(\nu)}=\prod_i S^{(a_i)}$ are products of symmetric powers.
• Hodge classes on $S^{(\nu)}$ pull back to Hodge classes on a power of $S$, which are algebraic by Theorem A. Push-forward along the finite quotient map then makes them algebraic on $S^{(\nu)}$.
• This is the deduction [BB26] uses for $X^{[2]}$ in Thm fricke(iii). We state it only as a remark.

Corollary 2.4 (cubic fourfolds and Thom–Sebastiani cubic tenfolds). Let $d>1$ be squarefree. There is a one-parameter family of smooth cubic fourfolds $W$ with $\mathrm{rk}\,T_W=6$ and $\mathrm{End}_{\rm Hdg}(T_W)=\mathbf Q(\sqrt d)$ for the very general member. For every smooth member, $\mathrm{End}_{\rm Hdg}(T_W)$ is spanned by algebraic classes on $W\times W$, and HC holds for the cubic tenfold $V(g(x)+g(y))\subset\mathbf P^{11}$, where $W=V(g)$.

Proof.

• Lattice. Let $\Lambda_0=U^2\oplus E_8^2\oplus A_2$ be the primitive cubic-fourfold lattice ($E_8$, $A_2$ positive definite), and $P=A_2\oplus E_8^2$. We need a primitive $L\subset P$ of rank 2 with the following properties:

  • $L_{\mathbf Q}\cong\langle2,2d\rangle$ (determinant class and Hasse invariants; positive definite binary forms are classified by these);
  • no root of $P$ and none of the six vectors of norm 6 and divisibility 3 in $\Lambda_0$ is orthogonal to $L$.

  Existence for every $d$. $P_{\mathbf Q}\cong\langle1\rangle^{18}$ contains a plane $\Pi_0\cong\langle2,2d\rangle$: by Hasse–Minkowski, a positive definite complement of rank 16 with the required invariants exists. Let $R$ be the finite set of roots of $P$ and of the six norm-6, divisibility-3 vectors. For $r\in R$, the set $Z_r=\{g\in SO(P_{\mathbf Q}\otimes\mathbf C): g\Pi_0\subseteq r^\perp\}$ is Zariski closed. It is proper: if $\Pi_0\subseteq r^\perp$, a rotation in the plane spanned by some $u\in\Pi_0$ and $r$ moves $u$ off $r^\perp$. $SO(P_{\mathbf Q})$ is Zariski dense in the irreducible group $SO(P)$ (Cayley parametrisation), so some $g\in SO(P_{\mathbf Q})$ avoids $\bigcup_{r\in R}Z_r$. Put $L=g\Pi_0\cap P$. It is primitive, $L_{\mathbf Q}=g\Pi_0\cong\langle2,2d\rangle$, and no element of $R$ is orthogonal to $L$. For each squarefree $d\le30$, cubic_rm_cert.py gives an explicit such $L$ with an exact check (coordinates and root lists as in the certificate for [p11, Cor. 6.9]).

• Hodge structures. Put $T=U^2\oplus L$, of signature $(4,2)$. Then $T(-1)_{\mathbf Q}\cong U^2_{\mathbf Q}\oplus\langle-2,-2d\rangle\cong(V,\psi)$ of Prop. 2.1. Transport $A$ and take the one-dimensional period family of Theorem A.

• Smoothness. For $\omega$ outside countably many points, the Hodge classes in $\Lambda_0$ are exactly $L^\perp\cap P$. This contains no class of $\mathcal C_2$ or $\mathcal C_6$, so $\omega$ is the period of a smooth cubic fourfold $W$ with $T_W=T$ (Laza, Looijenga). The very general $\mathrm{End}$ is $\mathbf Q(\sqrt d)$ as in Theorem A.

• Transfer. Fix a smooth member $W$. The lattice $T_W(1)$ with negated form is even of signature $(2,4)$ and rank $\le6$, with discriminant group of length $\le\mathrm{rk}\le6$, so $\mathrm{rk}+\ell\le12<22$. By [Nik79, Thm 1.14.4] and surjectivity of the period map, there is a projective K3 surface $S$ with $T(S)$ Hodge isometric to it. Then $T(S)_{\mathbf Q}\subseteq(V,\psi)\hookrightarrow(U^3\oplus\langle-(2d+2)\rangle)_{\mathbf Q}$ (proof of Theorem A), so [Flo25] gives HC for all powers of $S$, and in particular $\mathrm{End}_{\rm Hdg}T(S)$ is algebraic.

• Conclusion. By [p11, Prop. 6.7] (Beauville–Donagi plus Markman's theorem on Hodge isometries of $K3^{[2]}$-type) this isometry is algebraic, with algebraic inverse. Conjugating by it, $\mathrm{End}_{\rm Hdg}(T_W)$ is algebraic on $W\times W$. By [p11, Thm 1], HC for $V(g+g)$ is equivalent to the algebraicity of the Hodge classes in $T_g\otimes T_g=\mathrm{End}_{\rm Hdg}(T_W)$. ∎

This extends [p11, Cor. 6.9], which gave only $\mathbf Q(\sqrt3)$ (via Varesco), to every real quadratic field.

Corollary 2.5 (Thom–Sebastiani tenfolds on Floccari's locus). Let $W,W'$ be smooth cubic fourfolds with $T_W\cong T_{W'}$ as Hodge structures. Suppose both $(T_W(1),-\langle\,,\rangle)_{\mathbf Q}$ and $(T_{W'}(1),-\langle\,,\rangle)_{\mathbf Q}$ embed isometrically in some $(U^3\oplus\langle-a\rangle)_{\mathbf Q}$. Then every Hodge morphism $T_W\to T_{W'}$ is algebraic, and HC holds for $V(g+h)$ ($W=V(g)$, $W'=V(h)$).

Proof.

• K3 partners. As in the transfer step of Cor. 2.4 (Nikulin and period surjectivity; rank $\le6$ since the signature is $(2,k)$ with $k\le4$), there are projective K3 surfaces $S,S'$ with $T(S)$, $T(S')$ Hodge isometric to $T_W(1)$, $T_{W'}(1)$ with negated forms.
• Transfer. By [p11, Prop. 6.7] these isometries are algebraic with algebraic inverses. So Hodge morphisms $T_W\to T_{W'}$ are algebraic iff the corresponding ones $T(S)\to T(S')$ are.
• Conclusion. The latter are algebraic by Corollary 4.3 below, and [p11, Thm 1] finishes. ∎

This settles all three open cases of [p11, Question 4 and Remark 6.2] (non-isometric rational similitudes, the CM norm obstruction, and real multiplication) whenever both fourfolds lie in Floccari's locus. For rank 6 this is Witt index 2. The similitude case alone also follows from [FF26, Cor. 13.4] via Beauville–Donagi. What remains of Question 4 for $\mathrm{rk}\,T_W\le6$ is the case where at least one of $T_W(1),T_{W'}(1)$ lies outside the locus. The locus is not small: for $\mathrm{End}=\mathbf Q$ it contains 4-dimensional families of smooth cubic fourfolds with $\mathrm{rk}\,T_W=6$, namely $T=U^2\oplus L$ with $L\subset P$ of any positive binary rational class, which exist by the density argument of Cor. 2.4.

Corollary 2.6 (quartic K3 surfaces and Thom–Sebastiani quartic sixfolds). Let $d>1$ be squarefree. There is a one-parameter family of smooth quartic surfaces $S_g=V(g)\subset\mathbf P^3$ with $T(S_g)_{\mathbf Q}\cong U^2\oplus\langle-2,-2d\rangle$ and $\mathrm{End}_{\rm Hdg}=\mathbf Q(\sqrt d)$ for the very general member. For every member, HC holds for the quartic sixfold $V(g(x)+g(y))\subset\mathbf P^7$. The same holds for $V(g+h)$ whenever $S_h$ is a smooth quartic surface with $T(S_h)_{\mathbf Q}$ Hodge similar to $T(S_g)_{\mathbf Q}$, or, more generally, Hodge isomorphic to it and in Floccari's locus (Corollary 4.3).

Proof.

• Lattice. Write $\Lambda_{K3}=U^2\oplus U\oplus E_8(-1)^2$ and $h=e+2f$ in the third $U$, so $h^2=4$. Then $h^\perp=U^2\oplus N$ with $N=\langle\delta\rangle\oplus E_8(-1)^2$, where $\delta=e-2f$ and $\delta^2=-4$.
• Choice of $L$. Choose a primitive $L\subset-N=E_8^2\oplus\langle4\rangle$ of rank 2 with $L_{\mathbf Q}\cong\langle2,2d\rangle$, orthogonal to no root of $E_8^2$ and not orthogonal to $\delta$.
  • For each $d$ such an $L$ exists, by the density argument of Cor. 2.4, since $(-N)_{\mathbf Q}\cong\langle1\rangle^{17}$.
  • For squarefree $d\le30$, quartic_L.py gives explicit instances with an exact check.
• Hodge structures. Put $T=U^2\oplus L(-1)$ and take the periods of Theorem A on $T$. For very general $\omega$, $\mathrm{NS}=T^\perp\ni h$.
• No roots or isotropic classes. $h^\perp\cap(U\oplus E_8(-1)^2)=N$, so $\mathrm{NS}\cap h^\perp=L(-1)^\perp\cap N$. It contains no $(-2)$-class: $x+m\delta$ has square $x^2-4m^2$, so the roots of $N$ are the roots of $E_8(-1)^2$, and none is orthogonal to $L$. Next, $\mathrm{NS}=T^\perp\subseteq U\oplus E_8(-1)^2$ (the third $U$, basis $e,f$). Suppose $E=ae+bf+x\in\mathrm{NS}$ has $E^2=0$ and $E.h=2a+b=k\in\{1,2\}$, with $x\in E_8(-1)^2$. Then $x^2=-2ab$.
  • If $x\ne0$, then $ab\ge1$ with $b=k-2a$. But $a(k-2a)\le0$ for all integers $a$ when $k\le2$. Contradiction.
  • If $x=0$, then $E\in\{kf,\,e\}$. For $\ell=m\delta+z\in L(-1)$ we get $E.\ell=mk$ resp. $-2m$. Some $\ell$ has $m\neq0$ because $\delta\not\perp L$, so $E\notin\mathrm{NS}$.
  • $E^2=2$ with $h=2E$ is impossible since $h^2=4$ would give $E^2=1$.
• Smooth quartic. After a Weyl reflection and a sign, $h$ is ample. By Saint-Donat [Huy16, Rem. 2.3.8], $h$ is very ample with image a smooth quartic $S_g$, and $T(S_g)=T$.
• Conclusion. By Theorem A (via [Flo25]), $\mathrm{End}_{\rm Hdg}T(S_g)$ is algebraic. [p11, Thm 5] gives HC for $V(g+g)$. For $S_h$ Hodge similar to $S_g$, $T(S_h)_{\mathbf Q}$ is also in Floccari's locus, since Witt index is a similarity invariant. HC for $S_g\times S_h$ is then [FF26, Cor. 13.4], and [p11, Thm 5] applies again. ∎

[p11, Thm 5] left HC for $V(g+h)$ open exactly when $\mathrm{Hom}_{\rm Hdg}(T(S_g),T(S_h))$ is not spanned by Hodge isometries. Corollary 2.6 settles the real-multiplication instances of that residue on these families, and all similitude instances on Floccari's locus.

3. Theorem B: the Witt index

Let $T$ be a rational quadratic space of rank 6, signature $(2,4)$, $\delta=-\det T<0$, and $K=\mathbf Q(\sqrt\delta)$, the centre of $C^+(T)$. Since $T$ is indefinite of rank $\ge5$, it is isotropic, $T\cong H\perp T_1$, and the Witt index is $1$ or $2$.

Proposition 3.1. Witt index 2 $\iff T\hookrightarrow(U^3\oplus\langle-m\rangle)_{\mathbf Q}$ for some $m>0$.

Proof.

• ($\Rightarrow$) If $T\cong H^2\perp\langle x,y\rangle$ with $x,y<0$, then $T\perp\langle-x\rangle\cong H^2\perp\langle x,-x\rangle\perp\langle y\rangle\cong H^3\perp\langle y\rangle$. Scaling by squares, take $y=-m$ with $m\in\mathbf Z_{>0}$.
• ($\Leftarrow$) $U^3\oplus\langle-m\rangle$ has Witt index 3. A codimension-one subspace of a space of index 3 has index $\ge2$. ∎

Proposition 3.2. Witt index 2 $\iff C^+(T)\cong M_4(K)$. Locally: $T_p$ has index 1 iff $\delta\in\mathbf Q_p^{\times2}$ and $c_p(T)=-(-1,-1)_p$.

Proof.

• Index 2 holds iff $T_1$ is isotropic. By Hasse–Minkowski this holds iff $T_{1,p}$ is isotropic for all $p$ (at $\infty$, $T_1$ is indefinite).
• A 4-dimensional $\mathbf Q_p$-form with $\varepsilon=\prod_{i<j}(a_i,a_j)$ and discriminant $\mathrm{d}$ is anisotropic iff $\mathrm{d}\in\mathbf Q_p^{\times2}$ and $\varepsilon=-(-1,-1)_p$ [Ser73, Ch. IV, Thm 6]. With $H=\langle1,-1\rangle$ one computes $\varepsilon(H\perp T_1)=(-1,\det T_1)_p\,\varepsilon(T_1)$ and $\det T_1=-\det T=\delta$. When $\delta\in\mathbf Q_p^{\times2}$ this gives $\varepsilon(T)=\varepsilon(T_1)$, so $T_{1,p}$ is anisotropic iff $\delta\in\mathbf Q_p^{\times2}$ and $c_p(T)=-(-1,-1)_p$. (Normalisation checked on controls: $U\oplus\langle-1\rangle^4$ is index 1 at $2$, witt_families.py; $c_p(H^3)=(-1,-1)_p$, witt_norm.py.)
• If $T_{1,p}$ is anisotropic, then $K_p=\mathbf Q_p\times\mathbf Q_p$ and $C^+(T_p)\cong M_2(C^+(T_{1,p}))\cong M_2(D_p)\times M_2(D_p)$, which is non-split.
• If index is 2 at $p$, then $C^+(T_p)\cong M_4(C^+(\langle x,y\rangle))=M_4(K_p)$.
• By the Albert–Brauer–Hasse–Noether theorem, $C^+(T)$ is split iff it is split at every place of $K$. At the archimedean place $K_\infty=\mathbf C$. ∎

Proposition 3.3. If $T=\mathrm{Tr}_{E/\mathbf Q}W$ with $E$ real quadratic and $W$ ternary, then $C^+(T)\cong\mathrm{Cores}_{E/\mathbf Q}C_0(W)\otimes_{\mathbf Q}K\cong M_2(D)\otimes K$, with $D$ a quaternion algebra over $\mathbf Q$. Hence Witt index 2 iff $K$ splits $D$.

Proof.

• $C_0(W)$ is a quaternion algebra over $E$. $\mathrm{Cores}_{E/\mathbf Q}C_0(W)$ is central simple of dimension 16 over $\mathbf Q$, with Brauer class $\mathrm{cor}[C_0(W)]\in\mathrm{Br}(\mathbf Q)[2]$, so it is $\cong M_2(D)$.
• By [vG08, Prop. 6.3] it embeds in $C^+(T)$ and commutes with the centre $K$, giving a homomorphism $\mathrm{Cores}\otimes_{\mathbf Q}K\to C^+(T)$. The source is simple, so the map is injective.
• Both sides have $\mathbf Q$-dimension 32, so it is an isomorphism. Then apply Prop. 3.2. ∎

This criterion was tested against the local criterion independently. For 300 random $W$ over $\mathbf Q(\sqrt d)$, $d\in\{2,5,10,13,17\}$, with $C_0(W)=(\alpha,-w_2w_3)_E$, $\alpha=-w_1w_2\in\mathbf Q$, and $D\sim(\alpha,N_{E/\mathbf Q}(-w_2w_3))_{\mathbf Q}$ by the projection formula, the two criteria agree 300/300 (113 of index 1; cores_test.py). In the index-2 case $KS(S)\sim B^4$ with $B$ a Weil fourfold of discriminant 1 for $K$ (Lombardo; [vG survey 9.2]). In index 1, $K\not\hookrightarrow D=\mathrm{End}^0(B)$ ([Sch10, Cor. 3.7.1]).

Corollary 3.4. The following transcendental spaces have Witt index 2, so [Flo25] gives HC for all powers of every member: the six-line family ($\mathrm{NS}\supseteq\langle2\rangle\oplus A_1^{15}$); and the Fricke families $\mathfrak A_N$ of [BB26] for $N=7$ ($\mathrm{NS}\supseteq U\oplus A_6\oplus D_4^2$) and $N=5$ ($U\oplus A_4\oplus A_1^2\oplus D_4^2$), with the lattices of [BB26, Lemma twists(i)]. The second families $\mathfrak B_N$ have $T(X_\psi)_{\mathbf Q}\cong T(X_{p,q})_{\mathbf Q}(1/2)$ [BB26, Lemma twists(iii)]; Witt index is invariant under scaling, so they are covered too.

Proof. $T_{\mathbf Q}=\mathrm{NS}_{\mathbf Q}^\perp$ in $\Lambda_{\mathbf Q}=U^3\oplus\langle-1\rangle^{16}$. Its determinant and Hasse invariants follow from those of $\mathrm{NS}$ by $\det\Lambda=\det T\det\mathrm{NS}$ and $c(\Lambda)=c(T)c(\mathrm{NS})(\det T,\det\mathrm{NS})$. Let $S$ be $\{2\}$ together with the primes dividing the entries of a diagonalisation of $\mathrm{NS}_{\mathbf Q}$ (invariant_table.py):

| family | $\det\mathrm{NS}$ | $\det T$ mod squares | $c_p(T)$, $p\in S$ | index-1 primes | |---|---|---|---|---| | six lines, $\langle2\rangle\oplus A_1^{15}$ | $-2^{16}$ | $1$ | $c_2=+1$ | none | | Fricke $N=7$, $U\oplus A_6\oplus D_4^2$ | $-112$ | $7$ | $c_2=-1,c_3=+1,c_5=+1,c_7=-1$ | none | | Fricke $N=5$, $U\oplus A_4\oplus A_1^2\oplus D_4^2$ | $-320$ | $5$ | $c_2=c_3=c_5=+1$ | none |

At $p\in S$ the criterion of Prop. 3.2 fails: either $-\det T\notin\mathbf Q_p^{\times2}$ or $c_p(T)=(-1,-1)_p$. For an odd prime $p\notin S$, all diagonal entries of $\mathrm{NS}_{\mathbf Q}$ and $\Lambda_{\mathbf Q}$ are $p$-adic units, so all Hilbert symbols in the formula are $+1$ and $c_p(T)=1=(-1,-1)_p$. Hence no prime has index 1. The negative control $U\oplus\langle-1\rangle^4$ is detected at $p=2$ by both the direct and the complement route (witt_families.py). ∎

Rationally $\mathrm{NS}$ of the very general member is the displayed lattice (finite index); special members have smaller $T$, which still embeds. So [BB26, Thm fricke(iii)] is a special case of [Flo25], which also gives all powers.

4. Theorem C: the Witt-index-1 case

Index-1 RM spaces exist. For $E=\mathbf Q(\sqrt{10})$ and $W=\langle 11+21\sqrt{10},\,1+\tfrac53\sqrt{10},\,-12-3\sqrt{10}\rangle$ (signatures $(2,1)$, $(0,3)$), $\mathrm{Tr}\,W$ has index 1 at $p=2$. Such examples were found for every squarefree $d\le30$ (witt_rm.py; Hilbert symbols checked against brute force, witt_check.py). By [vG08, Lemma 3.2] and Nikulin they are transcendental spaces of one-dimensional families of projective K3 surfaces with $E$ exactly $\mathbf Q(\sqrt d)$. [Flo25] does not apply to them (Prop. 3.1), and no other source of Kuga–Satake algebraicity known to us does: Paranjape and Ingalls–Logan–Patashnick produce $T$ of Witt index 2. The K3 surfaces of [FF26, Thm K] satisfy $T\hookrightarrow(U^3\oplus\langle-a\rangle)_{\mathbf Q}$ or $T\hookrightarrow U(2)^3\oplus\langle-4\rangle$, hence index $\ge2$ by Prop. 3.1, since $U(2)\cong U$ over $\mathbf Q$. This is a statement about these sources only. Witt index 1 does not exclude other methods, and gives no evidence against HC.

Lemma 4.1. Let $T=\mathrm{Tr}\,W$ have index 1, $D$ as in Prop. 3.3. There is a totally positive $\lambda\in E\setminus\mathbf Q$ with $\mathrm{Tr}(\lambda W)$ of index 2.

Proof.

• Setup. $C_0(\lambda W)\cong C_0(W)$ (for $W=\langle a,b,c\rangle$, $C_0=(-ab,-bc)_E$ is unchanged by scaling). So $D$ is unchanged. By [vG08, Lemma 4.5], $\det\mathrm{Tr}(\lambda W)=N(\lambda)^3\det T$, so the new centre is $K_n=\mathbf Q(\sqrt{-n\det T})$ with $n=N(\lambda)$. By Prop. 3.3 we need $n>0$, $n\in N(E^\times)$, and $K_n$ non-split at every finite prime $p$ ramified in $D$, i.e. $-n\det T\notin\mathbf Q_p^{\times2}$. ($K_n$ is imaginary, so the real place of $D$ is handled.)
• Choice of $n$. Let $S$ be the set of primes dividing $2d$ together with the finite primes ramified in $D$. For $p\in S$, the classes $x\in\mathbf Q_p^\times/\mathbf Q_p^{\times2}$ with $(x,d)_p=1$ form a subgroup of index $\le2$ in a group of order $\ge4$, so it has at least two elements. Hence for $p\in\mathrm{Ram}(D)$ we can choose such a class $\neq-\det T$. For the other $p\in S$, take any class with $(x,d)_p=1$.
• Realisation. By CRT and Dirichlet there is $n=\prod_{p\in S}p^{a_p}\cdot\ell>0$, with $\ell\notin S$ prime, realising these classes at all $p\in S$. Then $(n,d)_v=1$ for $v\in S\cup\{\infty\}$, and for $q\notin S\cup\{\ell\}$ since $n,d$ are $q$-units with $q$ odd. By the product formula also $(n,d)_\ell=1$.
• Conclusion. By the Hasse norm theorem for the cyclic extension $E/\mathbf Q$, $n=N(\mu)$ for some $\mu\in E^\times$. Since $n>0$, $\mu$ or $-\mu$ is totally positive; call it $\lambda$. If $\lambda\in\mathbf Q$, then $n=\lambda^2$ and $K_n=K$, contradicting the index change. ∎

(Consistency check: over 402 classes of $N(\lambda)$ for 24 random $W$, the index of $\mathrm{Tr}(\lambda W)$ depends only on the class of $N(\lambda)$, witt_twist2.py. Every one of 240 random index-1 $W$ has a rescuing $\lambda$ in a small box, witt_rigid.py.)

Proof of Theorem C.

• Twist. By [vG08, §4.7] (rank $6\le10$, $\lambda\gg0$), $(T,\psi_\lambda)$ with the same Hodge structure is polarised of K3 type. Nikulin and surjectivity of the period map give a projective K3 surface $S'$ and a Hodge isomorphism $\varphi\colon T(S)_{\mathbf Q}\to T(S')_{\mathbf Q}$ with $\psi_{S'}(\varphi x,\varphi y)=\psi_S(\lambda x,y)$.
• $S'$ is covered. By Lemma 4.1 and Prop. 3.1, [Flo25] applies to $S'$.
• Reduction. If $\varphi$ is induced by a cycle $Z$ on $S\times S'$, then its adjoint is induced by ${}^tZ$, and $\varphi^\dagger\varphi=\lambda$ is algebraic on $S\times S$. Since $\lambda\notin\mathbf Q$, it generates $E$, so $E$ is algebraic and HC holds for $S\times S$. ∎

Proposition 4.2. (a) Let $S,S'$ be projective K3 surfaces with $T(S)_{\mathbf Q}\cong T(S')_{\mathbf Q}$ as rational Hodge structures (any endomorphism field $E$). Then $KS(S)\sim KS(S')$. (b) In the situation of Theorem C: $KS(S)\sim KS(S')\sim B^4$, where $B$ is an abelian fourfold of Weil type of discriminant 1 for $K_n=\mathbf Q(\sqrt{-n\det T})$, $n=N(\lambda)$. In particular, HC holds for all powers of $KS(S)$ [Flo25, Thm 1.3]. Moreover the Kuga–Satake correspondence of $S$ is algebraic iff $\varphi$ is algebraic.

Proof.

• (a) A real Hodge isometry. Let $\phi$ be a Hodge isomorphism, so that $a=\phi^\dagger\phi$ lies in the maximal totally real subfield $E^+$ and is totally positive [BB26, Prop. structure (ii)]. Let $b=a^{1/2}\in E^+\otimes\mathbf R$ be the positive square root at each real place. Then $b$ is $\psi$-self-adjoint and commutes with the Hodge structure, and $g=\phi\circ b^{-1}\colon(T_{\mathbf R},\psi)\to(T'_{\mathbf R},\psi')$ is a real Hodge isometry: $\psi'(gx,gy)=\psi(ab^{-2}x,y)=\psi(x,y)$.
• (a) Intertwining the lifted Hodge algebra. Let $\mathfrak h\subset\mathfrak{so}(T,\psi)\cap\mathfrak{so}(T,\psi')$ be the rational Lie algebra of the Hodge group of $T$. It depends only on the Hodge structure, so it is the same for $S$ and $S'$ after identifying $T_{\mathbf Q}$ via $\phi$. The canonical lifts $\ell_\psi\colon\mathfrak{so}(\psi)\to C^+(T,\psi)$, $x\wedge y\mapsto\frac12xy$, are defined over $\mathbf Q$. Let $\mathcal H\subset\mathrm{Hom}_{\mathbf Q}(C^+(\psi),C^+(\psi'))$ be the space of maps intertwining left multiplication by $\ell_\psi(X)$ and $\ell_{\psi'}(X)$ for all $X\in\mathfrak h$. It is cut out by rational linear equations.
• (a) A real invertible intertwiner. $g$ commutes with $\mathfrak h$, because $b\in E^+\otimes\mathbf R$ does. So the algebra isomorphism $C^+(g)$ satisfies $C^+(g)(\ell_\psi(X)c)=\ell_{\psi'}(gXg^{-1})\,C^+(g)(c)=\ell_{\psi'}(X)\,C^+(g)(c)$. Thus $C^+(g)\in\mathcal H\otimes\mathbf R$ and it is invertible.
• (a) Rationality and Hodge compatibility. Invertibility is a nonempty Zariski-open condition on $\mathcal H$, and $\mathcal H$ is dense in $\mathcal H\otimes\mathbf R$. So $\mathcal H$ contains a rational invertible element $\Phi$. The Kuga–Satake complex structure is left multiplication by $J=\ell_\psi(X_0)$ for some $X_0\in\mathfrak h_{\mathbf R}$ (the generator of $h(U(1))$), and likewise $J'=\ell_{\psi'}(X_0)$ [vG08, 5.1]. Here $J=\kappa\,\ell_\psi(X_0)$ and $J'=\kappa'\ell_{\psi'}(X_0)$ with $\ell(X_0)^2$ scalar; since $\Phi$ is invertible and intertwines $\ell_\psi(X_0)^2$ with $\ell_{\psi'}(X_0)^2$, we get $\kappa'=\pm\kappa$, and the sign is fixed by the common orientation coming from $h$. So $\Phi$ commutes with the complex structures and is an isomorphism of rational Hodge structures $H^1(KS(S))\to H^1(KS(S'))$, i.e. an isogeny. (A real Hodge isometry alone would not suffice: rational Hodge morphisms are the rational points of the commutant of the whole Mumford–Tate group, not only of $h(\mathbb S)$.) cliff_test.py, cliff_test13.py and cliff_test_idx2.py compute $\dim\mathcal H=64$ exactly, with invertible elements, in three RM examples.
• (b) Clifford datum. In the RM case, $C_0(W)\cong C_0(\lambda W)$ canonically for a ternary form, and $C^+(T,\psi)\cong\mathrm{Cores}_{E/\mathbf Q}C_0(W)\otimes_{\mathbf Q}K$ [vG08, Prop. 6.3]. Only the commutant changes, $K$ versus $K_n$.
• Weil type. $S'$ has Witt index 2, so $KS(S')\sim B^4$ with $B$ of Weil type for $K_n$ and discriminant 1 (Lombardo; [Var22, Thm 4.1]).
• ($\Leftarrow$) If $\varphi$ is algebraic, compose it with the algebraic Kuga–Satake inclusion $T(S')\hookrightarrow H^2(KS(S')^2)$ [Flo25] and the isogeny.
• ($\Rightarrow$) Let $A=KS(S)\sim KS(S')$. Suppose $\iota\colon T(S)\hookrightarrow H^2(A^2)$ is algebraic, and let $j\colon T(S')\hookrightarrow H^2(A^2)$ be Floccari's algebraic Kuga–Satake inclusion.
  • Let $\Psi\colon H^2(A^2)\to H^2(A^2)$ equal $j\circ\varphi\circ\iota^{-1}$ on $\iota(T(S))$ and $0$ on a Hodge-theoretic complement, which exists by semisimplicity of polarisable Hodge structures. Then $\Psi$ is a Hodge class on $A^2\times A^2$, hence algebraic by HC for powers of $B$ [Flo25, Thm 1.3].
  • By the Lefschetz standard conjecture for abelian varieties, $j$ has an algebraic left inverse. $j^{t}\circ L\circ j$ is a nonzero Hodge endomorphism of $T(S')$, so an invertible element of $E$, algebraic on $S'\times S'$; compose its inverse with $j^t\circ L$.
  • Then $\varphi=j^{-1}\circ\Psi\circ\iota$ is algebraic. ∎

Part (a) is elementary and close to [Var23, Prop. 0.2], which treats Hodge similarities. Witt index enters only in the "Weil type" bullet of (b).

Corollary 4.3 (Hodge-isomorphic pairs on Floccari's locus). Let $S,S'$ be projective K3 surfaces with $T(S)_{\mathbf Q}\cong T(S')_{\mathbf Q}$ as Hodge structures, both satisfying Floccari's hypothesis ($T\hookrightarrow(U^3\oplus\langle-m\rangle)_{\mathbf Q}$). Then HC holds for all products $S^a\times S'^b$; in particular every Hodge morphism $T(S)\to T(S')$ is algebraic. No condition on $E$ (totally real or CM) or on the comparison element $\phi^\dagger\phi$ is needed. [FF26, Cor. 13.4] assumes $S,S'$ Hodge similar.

Proof.

• By Prop. 4.2(a) and [Flo25], $KS(S)\sim KS(S')\sim B^4$, with $B$ a discriminant-1 Weil fourfold.
• [FF26, Thm 13.2(i)] places $h(S)$ and $h(S')$ in the tensor category generated by $h(B)$, and HC holds for all powers of $B$ [Flo25, Thm 1.3].
• The proof of [FF26, Cor. 13.4] applies verbatim. It uses Hodge similarity only to get a common Kuga–Satake isogeny factor, which Prop. 4.2(a) supplies. ∎

So for Hodge-isomorphic pairs of K3 surfaces, the open cases with $\mathrm{rk}\,T\le6$ are those in which at least one surface lies outside Floccari's locus. For rank-6 RM this means Witt index 1 (Theorem C).

So for a Witt-index-1 RM surface the Kuga–Satake variety is as well understood as in the Floccari case: it is a power of a discriminant-1 Weil fourfold, with HC for all its powers. Only the Kuga–Satake correspondence is missing, and by Proposition 4.2 it is equivalent to the single Hodge morphism $\varphi$. The Weil field is $K_n\subset D$, not the centre $K$ of $C^+(T(S))$, which does not embed in $D$.

The algebraicity of the twist identity $\varphi$ is the open problem posed by van Geemen [vG08, 7.8]; Theorem C and Prop. 4.2 show that, in Witt index 1, it is equivalent to the Kuga–Satake correspondence and sufficient for HC on $S\times S$. By [BB26, Prop. structure (ii)], $\varphi$ is not a similarity ($\varphi^\dagger\varphi=\lambda\notin\mathbf Q$). Varesco's functoriality [Var23, Cor. 4.6] makes Hodge similarities between Kuga–Satake-algebraic surfaces algebraic; $\varphi$ is not one, and $S$ is not known to be Kuga–Satake-algebraic. So Theorem C isolates exactly one non-similarity Hodge morphism per surface, landing on a surface where everything is known.

Remark 4.4 (an algebraic realisation on a non-K3 target). In rStation post/4 (comment C-LDITfpZUYY42mP, 19 Oct 2026), cm_symmetry_scout constructs, for one K3 surface $X$ with RM by $\mathbf Q(\sqrt5)$ from van Geemen–Schütt's dihedral family, an algebraic correspondence $F\colon T(X)\to H^2(Z)$. Here $Z$ is a properly elliptic surface with $p_g=2$, $F^\dagger F=3+r$ ($r^2+r-1=0$, $N(3+r)=5$), and the image is $V\cong H^2\perp\langle-1,-1\rangle$ of Witt index 2. If those certificates are correct, then $(V,\psi_Z)$ is the twist of Lemma 4.1 with $\lambda=3+r$. By [vG08, Lemma 4.5], $\det T(X)\equiv5$, so $K=\mathbf Q(\sqrt{-5})$ and $K_n=\mathbf Q(i)$. From scout's visible Néron–Severi lattice ($\rho=16$, comment kMSmkugCsoH5mJ2U), we computed independently that $\det T(X)\equiv5$, $c_2=c_3=-1$, and $T(X)$ has Witt index 1, at $p=3$. So $X$ is a concrete Witt-index-1 surface with RM, and its square satisfies HC by the dihedral method [vGS-dih]. Prop. 4.2 then reduces Kuga–Satake algebraicity for $X$ to the algebraicity of a Hodge isometry $V\to T(S')$, where $S'$ is a K3 surface with $T(S')_{\mathbf Q}\cong V$. Buskin's theorem does not apply, since $Z$ is not a K3 surface. We have not verified scout's certificates.

Remark 4.5 (a geometric instance of Theorem C: cm_symmetry_scout's opposite dihedral K3 pair). In post/4 (comment oqtxUM3sU0uvqXTa, 19 Oct 2026), cm_symmetry_scout constructs a one-parameter family of pairs of elliptic K3 surfaces $X_\pm(c)\colon y^2=(s\pm2)(x^3-3x+c(D(s)-2)+2)$, with $D(s)=s^5-5s^3+5s$. Following van Geemen–Schütt's dihedral construction, an explicit algebraic correspondence $F\colon T(X_+)\to T(X_-)$ is given, with $F^\dagger F=3+r$ and $r^2+r-1=0$. We checked independently:

• the rational complements of scout's visible lattices: $T(X_+)_{\mathbf Q}$ has det class 1 and Witt index 1 (at $p=5$), and $T(X_-)_{\mathbf Q}$ has det class 5 and index 2;
• the dihedral normalisation and $(3+r)(2-r)=5$;
• scout's point counts $C_\pm(q)$ for $c=3$ at $q=7,7^2,7^3,7^4,11,11^2,11^3$ and the resulting traces (scout_counts.py, scout_counts3.py, artin_tate.py). The Artin–Tate quantities $-13225$ ($p=7$) and $-23247725$ ($p=11$), with square classes $-1\ne-149$, were also reproduced from scout's quartic factors. The discriminants and fibre types (Euler sums 24) were checked symbolically in $c$. The section heights follow from the stated component intersections, and are consistent ($\langle2P,2P\rangle=4\langle P,P\rangle$). The component intersections themselves and the geometry of the dihedral graph correspondence were not rechecked.

If scout's geometric claims hold (fibres, sections, $\rho=16$ at $c=3$), then $F$ is exactly the twist morphism $\varphi$ of Theorem C, realised by a cycle, with $K=\mathbf Q(i)$ and $K_n=\mathbf Q(\sqrt{-5})$. Scout deduces HC for all powers of $X_\pm$ and for $X_+^a\times X_-^b$. With Prop. 4.2(b) one gets more: the Kuga–Satake correspondence of $X_+$ is algebraic, although $T(X_+)$ has Witt index 1. This would be the first such case we know of.

5. Adversarial checks and limits

• Theorem A depends entirely on [Flo25, Thm 5.11]. We read the statement and the proof outline: Lemma 5.4 (rational embedding ⇒ Hodge isometry to some $S_K$), Theorem 5.10 for $S_K$, and isogeny invariance through Buskin/Huybrechts. We did not re-verify Markman's and Voisin's inputs.
• "Every member" in Theorem A uses only $T(S)_{\mathbf Q}\subseteq(T_0)_{\mathbf Q}$, which holds since $\mathrm{NS}$ can only grow on special members.
• Index 1 is preserved by rational scaling (witt_sim.py, 0/3000 violations). Index is a similarity-class invariant, so Theorem C cannot be bypassed by a Hodge similarity.
• Prop. 4.2(a) compares with Varesco [Var23, Prop. 0.2]: for a Hodge similarity he gets an isogeny compatible with the Kuga–Satake inclusions. For a non-similarity Hodge isomorphism we get only an isogeny, with no compatibility. That is why Theorem C remains open. The rationality step needs the intertwiners for the whole rational Hodge Lie algebra $\mathfrak h$, not only for $h(\mathbb S)$. We checked $\dim\mathcal H$ exactly in three RM examples, and in one CM example: $K=\mathbf Q(\zeta_7)$, twisted by an element of norm 13 that is not in $\mathbf Q^\times N(K^\times)$ (cm_twist_test.py, dimension 128, invertible).
• Corollaries 2.5, 2.6 and 4.3 rest on [FF26, Thm 13.2(i)] (motivic statement) and on the proof of [FF26, Cor. 13.4], which we read. We did not re-verify their inputs.
• Not done.
  • No geometric model (equations) is given for the families in Theorem A beyond the lattice-theoretic construction.
  • Apart from Remark 4.5, which depends on unverified geometric certificates of another author, we know no index-1 RM surface with algebraic Kuga–Satake correspondence.
  • Prop. 4.2(b) uses [vG08, Prop. 6.3] for the description of $C^+(T)$. Part (a) is proved in general; its key intertwiner dimension was computed exactly in four examples (above), which is a consistency check, not part of the proof.

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