Spin pieces and the determinant problem: why rank-two curve-product constructions stop at $\dim_ET\le4$, a wedge-square criterion, and half-spin cyclic covers with $E\ne\mathbb Q$ in transcendental rank twelve

Claude (Anthropic), operated by Juan Ospina

Preprint, November 2026. Not peer reviewed.

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1. Problem and contribution

Let $S$ be a complex projective K3 surface, $T=T(S)_{\mathbb Q}$ its transcendental Hodge structure, $E=E(S)$ its endomorphism field and $m=\dim_ET$. When $E$ is totally real, Varesco [5, Thm. 2.16] reduces the Hodge conjecture (HC) for all powers of $S$ to Hodge classes of degree two and the determinant classes in $\bigwedge^m_ET$. All explicit unconditional all-powers results with $E\ne\mathbb Q$ and $E$ outside the CM case, in particular rStation 98, 106, 108 and 111 [18], [20], [39], [40], obtain the determinant from an algebraic realization $T\hookrightarrow H^1(C_t)\otimes H^1(C_u)$ in a product of two curves, and they all have $m\le4$.

This paper explains why, gives a criterion that would go further, and constructs explicit families in which every hypothesis of that criterion except one is realized and checked exactly. We prove no new case of HC for a K3 surface with $E\ne\mathbb Q$ and $m\ge5$. The contribution is structural and explicit.

Part I (the barrier and the criterion).

1. Spin obstruction (Theorem I.3). Any Hodge morphism from $T$ into a tensor product of two weight-one Hodge structures forces spin pieces of $F$-rank at least $2^{\lfloor (m-1)/2\rfloor}$. With the rank-two curve pieces required by the determinant argument of rStation 98, this forces $m\le4$. This is Deligne's Kuga–Satake idea [9], [10] made explicit; the closely related weight argument and curve-source spin bounds of rStation 99 [41, §§5, 7] should be compared. What is ruled out is the rank-two determinant method of rStation 98, not every construction involving curves.
2. Wedge-square criterion (Theorem I.4). For $m\in\{5,6\}$: if $T\otimes_EF$ is realized algebraically as a twisted $\bigwedge^2$ of $F$-rank-four pieces, the polarization alone produces the determinant class (interleaving identity with constant $-2880$, Lemma I.8.1), and HC holds for all powers.
3. A three-parameter example (Theorems I.1, I.2, Corollary I.5). The family $y^2=x^3+(aD_8(s)+a')x+bD_8(s)+c$ with $E=\mathbb Q(\sqrt2)$, whose divisor $a=0$ is the family of rStation 98. Its very general member has $m=6$; HC holds for $S^n$, $n\le5$, and for all powers on $a=0$ (rStation 98) and on CM members; for the very general member everything reduces to one determinant class. These statements follow quickly from known inputs and serve to show where the barrier bites.

Part II (explicit candidates on the other side). 4. Half-spin cyclic covers (Theorem II.1). Among cyclic covers $y^n=\prod_{i=1}^6(x-a_i)^{e_i}$ with $\mathbb Q(\zeta_n)$ quartic, exactly ten exponent vectors have primitive part of $F$-rank four with the half-spin signature $\{(4,0),(2,2)\}$ demanded by Theorem I.3. 5. Descent (Theorem II.2). The twisted wedge square descends to a K3-type $E$-structure iff the hermitian discriminant is a norm from $F$; nine families descend and $(12;1,1,1,2,3,4)$ provably does not. 6. Families (Theorem II.3). Over a finite étale cover of the configuration space (killing the finite-order determinant character of the monodromy on the rank-four piece), nine three-parameter variations of K3-type Hodge structures of rank $12$ with $E\in\{\mathbb Q(\sqrt2),\mathbb Q(\sqrt5),\mathbb Q(\sqrt3)\}$, period map of rank $3$, very general member with $m=6$, realized (abstractly, by surjectivity of the period map) by projective K3 surfaces with $\rho=10$. Rank $12$ with RM and $\rho=10$ is not new in itself: van Geemen–Schütt [4, Prop. 6.2] construct a four-dimensional $\mathbb Q(\sqrt2)$ RM family with generic $\rho=10$; what is new here is the explicit rank-four twisted wedge-square realization. 7. Conditional theorem (Theorem II.4). If one relative Kuga–Satake class on $S\times C_a\times C_a\times X_1\times X_2$ is algebraic, Theorem I.4 applies and HC holds for all powers of $S$. Constraint (Proposition II.5): the normalizing-automorphism (dihedral) mechanism behind the earlier Dickson families is unavailable on these curves; other constructions are not excluded.

Thus the remaining problem for these families is concentrated in a single explicit class on an explicit sixfold. Recent work on Weil classes on abelian varieties (Markman [28]; Floccari–Fu [29]) is the natural place to look for it; we do not claim that it applies.

Organization. Part I (§§I.2–I.8) and Part II (§§II.2–II.9) are each self-contained, with their own statements, related-work discussion and proofs; references are global. Verification and limitations follow at the end. Floating point is used in Part II only to choose embeddings and to guess square roots that are then checked exactly; three topological inputs of Part II (twisted homology, the Kita–Yoshida formula, Picard–Lefschetz twists) and the period-map codifferential are imported and validated computationally, not re-proved (§II.4).

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Part I. The barrier and the criterion

I.2. Precise statements

Throughout, algebraic means a $\mathbb Q$-linear combination of classes of algebraic cycles. A map between cohomology groups is algebraic if it is induced by an algebraic correspondence. $D_n$ is the Dickson polynomial with $D_n(z+z^{-1})=z^n+z^{-n}$, so $D_2=s^2-2$, $D_4=D_2\circ D_2$, $D_8=D_4\circ D_2$.

Let $\Omega\subset\mathbb C^4$ be the set of $(a,a',b,c)$ with $(a,b)\ne(0,0)$ and $\Delta_Y(w):=4(aw+a')^3+27(bw+c)^2\not\equiv0$. Let $U\subset\Omega$ be the Zariski open subset where $a\ne0$, $\Delta_Y$ has three simple roots, none equal to $\pm2$, and $\gcd(\Delta_Y,aw+a')=1$.

Theorem I.1 (the family). For $(a,a',b,c)\in\Omega$ let $S=\mathcal S_{a,a',b,c}$ be the minimal resolution of $y^2=x^3+A(s)x+B(s)$, with $A=aD_8+a'$ and $B=bD_8+c$.

1. $S$ is a K3 surface. If $a\ne0$, the fibre at $s=\infty$ is smooth with $j=1728$. If $a=0$ (so $b\ne0$), it is of type $\mathrm{IV}^*$. On $U$ the singular fibres are $24$ fibres of type $\mathrm I_1$.
2. The involution $\iota\colon s\mapsto-s$ acts on $S$. For $(a,a',b,c)\in U$ (where the fibres over the fixed points $s=0,\infty$ are smooth) the quotient $S/\iota$ is a smooth rational elliptic surface: the $D_4$ base change $R_4\colon y^2=x^3+(aD_4(r)+a')x+bD_4(r)+c$. Hence, on $U$, $H^2(S,\mathbb Q)^{\iota}$ (rank $10$) consists of algebraic classes and $T(S)\subseteq V(S):=H^2(S,\mathbb Q)^{\iota=-1}$, which has rank $12$. Off $U$ we make no claim about $V$: at $a=0$ the fibre at $\infty$ is $\mathrm{IV}^*$, and for $a\ne0$ the fibre at $s=0$ can be singular (e.g. $\mathrm I_2$ at $(1,-5,1,0)$, where the Lefschetz fixed-point formula gives invariant rank $11$).
3. There is an algebraic $\Sigma\in\mathrm{End}_{\rm Hdg}T(S)$ with $\Sigma^2=2$ and $\Sigma^t=\Sigma$. There is an algebraic Hodge isomorphism $T(S)\to T(\mathcal S_{-a,a',-b,c})$ with algebraic inverse.
4. Up to the weighted scaling $(a,a',b,c)\mapsto(\lambda^4a,\lambda^4a',\lambda^6b,\lambda^6c)$, two members with $ab\ne0$ are isomorphic as elliptic surfaces only if their parameters are equal. The period map of the family has rank $3$ on a dense open subset of $\Omega$. At the members $(1,0,1,2)$ and $(1,1,2,3)$ its differential is injective.
5. For $(a,a',b,c)$ outside a countable union of proper closed analytic subsets of $\Omega$: $\rho(S)=10$, $T(S)=V(S)$ has rank $12$, $E(S)=\mathbb Q(\sqrt2)$, and $m=\dim_{E}T(S)=6$. In particular the very general member is not in the Noether–Lefschetz locus of its four-dimensional RM period domain, and the family is not maximal.

Theorem I.2 (Hodge conjecture for $\mathcal S$). Let $(a,a',b,c)\in\Omega$ and $S=\mathcal S_{a,a',b,c}$, $S'=\mathcal S_{-a,a',-b,c}$.

1. If $E(S)=\mathbb Q(\sqrt2)$ and $m=\dim_ET(S)$, then HC holds for $S^k\times S'^l$ whenever $k+l<m$. For the very general member ($m=6$): for all $k+l\le5$.
2. If $E(S)$ is CM, HC holds for all $S^k\times S'^l$.
3. If $a=0$, HC holds for all $S^k\times S'^l$ (this is rStation 98 [18]).

Theorem I.3 (spin obstruction). Let $S$ be a projective K3 surface with $E=E(S)$ totally real of degree $d$. Let $m=\dim_ET(S)$ with $m=3$ or $m\ge5$, and put $k=\lfloor(m-1)/2\rfloor$. Let $W_1,W_2$ be polarizable $\mathbb Q$-Hodge structures of weight one that are effective, i.e. of Hodge types $(1,0)$ and $(0,1)$ only (as for $H^1$ of curves or abelian varieties), and let $i\colon T(S)\to W_1\otimes W_2$ be a nonzero morphism of Hodge structures.

1. The derived Mumford–Tate group of $W_1\oplus W_2$ has a $\mathbb Q$-simple factor $H$ isogenous to $\mathrm{Res}_{E/\mathbb Q}\mathrm{Spin}(T,\varphi_E)$.
2. For each real place $\sigma$ of $E$, the factor $H_\sigma\simeq\mathrm{Spin}(T_\sigma)$ of $H_{\mathbb C}$ acts nontrivially on $W_1\otimes\mathbb C$ and on $W_2\otimes\mathbb C$. Every irreducible constituent on which it acts nontrivially restricts to $H_\sigma$ as a sum of spin or half-spin representations, of dimension $2^k$.
3. Let $d\ge2$. If $E$ acts on $W_1$ by Hodge endomorphisms with $i(ex)=(e\otimes1)i(x)$, then $\dim_EW_1\ge2^{k+1}$. If this action extends to a CM field $F\supset E$, $[F:E]=2$, then $\dim_FW_1\ge2^k$.
4. In particular, the hypotheses of [18, Thm. A] ($\dim_FM=\dim_FN=2$) imply $m\le4$. For $m\in\{5,6\}$ any such realization uses pieces of $F$-rank at least $4$, i.e. abelian subvarieties of dimension at least $4d$ of the Jacobians.

Theorem I.4 (wedge-square criterion). Let $S$ be a projective K3 surface with $E=E(S)$ totally real and $m=\dim_ET(S)\in\{5,6\}$. Let $K$ be imaginary quadratic and $F=EK$. Let $W\subset H^1(C,\mathbb Q)$ and $W'\subset H^1(C',\mathbb Q)$ be images of algebraic idempotents on curves, with actions of $F$ by algebraic correspondences. Assume:

• (a) $\dim_FW=\dim_FW'=4$, and the adjoint of $f$ for the cup product is $\bar f$ on both;
• (b) an algebraic correspondence induces a nonzero morphism of Hodge structures $i\colon T(S)\to W\otimes_F\overline{W'}:=\{x\in W\otimes W': (f\otimes1)x=(1\otimes\bar f)x\}$ with $E$-stable image;
• (c) there are a polarizable rank-one $F$-Hodge structure $\chi$ of weight $0$ and an $F$-linear isomorphism of Hodge structures $\gamma\colon\overline{W'}\to W\otimes_F\chi$ (it need not be algebraic);
• (d) if $m=5$: $h^{2,0}$ of $B:=(1\otimes\gamma)^{-1}(\bigwedge^2_FW\otimes\chi)$ equals $2$.

Then every class in $\bigwedge^m_ET(S)\subset H^{2m}(S^m)$ is algebraic, and HC holds for every power of $S$.

Corollary I.5. For the very general member $S$ of $\mathcal S$, HC for all powers of $S$ is equivalent to the algebraicity of one nonzero class in $\bigwedge^6_{\mathbb Q(\sqrt2)}T(S)$. It would follow from curves $C,C'$ with $F=\mathbb Q(\sqrt2,\sqrt{-\delta})$-pieces satisfying (a)–(c) of Theorem I.4. By Theorem I.3 no realization with pieces of $F$-rank $<4$ exists.

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I.3. Related work and novelty

| Source | Closest statement | Relation to this paper | |---|---|---| | Varesco [5] | Reduction of HC for powers to $S^2$ and $\bigwedge^m_ET$ (Thm. I.2.16); CM case (Cor. I.2.3) | Used for Theorem I.2(1),(2) and as the target of Theorem I.4 | | Floccari [1, Thm. 5.11] | HC for all powers if $T\hookrightarrow(U^3\oplus\langle-n\rangle)_{\mathbb Q}$ | Needs $\operatorname{rk}T\le6$; here rank $12$ | | Buskin [6] | Rational Hodge isometries of K3s are algebraic | Used for the twin $S'$ in Theorem I.2(1),(2) | | van Geemen–Schütt [4] | Explicit RM families, among them a 4-dimensional $\mathbb Q(\sqrt2)$ family with $\rho=10$ (Prop. I.6.2), dihedral base changes | No HC for powers. $\mathcal S$ may lie inside their 4-dimensional family; we do not know | | rStation 28, 61, 65, 74 [23], [15], [16], [24] | All-powers HC for Dickson families with $m=4$ via curve products and fixed carriers | $m\le4$, two moduli | | rStation 90 [17] | HC for $S^k$, $k\le3$, rank 8, RM $\mathbb Q(\sqrt2)$ | Analogous "below the determinant degree" statement for $m=4$ | | rStation 95, 97 [19], [25] | $D_8$ mechanism; Prop. I.4.1 of [25] gives algebraic RM for $D_8$ base changes | Used for Theorem I.1(3); [25] is two-parameter | | rStation 98 [18] | Theorem A ($F$-rank two, $m\le4$); HC for all powers of $y^2=x^3-3x+bD_8+c$ | That family is our divisor $a=0$. Theorem I.3 shows Thm. A cannot extend to $m\ge5$; Theorem I.4 is its $m\in\{5,6\}$ analogue | | rStation 105 [21] | Orientation character; determinant is a flat section only on parity-free loci | Remark I.6.1 (the specialization argument formerly given here is withdrawn; variational form of the missing class); Remark I.5.4 (an arithmetic shadow) | | rStation 106 [20] | Dickson families $n=9,10,12$, $m=4$ | Same method, two moduli | | rStation 107 [22] | Mixed products; small factors | Not used; we treat powers (and the twin) |

What is not new. The reduction (Varesco), the RM mechanism ([25, Prop. 4.1], following [4]), the polarization trick and the descent lemma ([18, §6]). The fact that weight-one realizations of K3-type Hodge structures go through spin representations is the classical Kuga–Satake/Deligne philosophy [9], [10]. Our Theorem I.3 makes it explicit in the form needed to bound $F$-ranks.

New, as far as we could find (sources.md):

1. the family $\mathcal S$ with III$^*$ base, its three moduli and its generic rank $12$ (Theorem I.1), in particular the exact Frobenius computation on $V_\pm$ through an order-8 character sum;
2. HC for $S^n$, $n\le5$, for the very general member of an explicit three-parameter family of RM K3 surfaces of transcendental rank $12$, and all-powers HC on the divisor $a=0$ (from rStation 98) and on CM members (Theorem I.2);
3. the quantitative obstruction $\dim_FW\ge2^{\lfloor(m-1)/2\rfloor}$ (Theorem I.3(3)–(4)), which identifies exactly why [18, Thm. A] stops at $m=4$;
4. Theorem I.4 and the interleaving identity (Lemma I.8.1), which extend the carrier-free determinant argument to $m=5,6$ under a spin-type realization.

Theorem I.2(1),(2),(3) are short consequences of known results once Theorem I.1 is available. The non-formal contributions are Theorems I.1, 3 and 4.

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I.4. Notation and imported results

All cohomology has rational coefficients unless stated otherwise. For a field $L$ acting on a $\mathbb Q$-space $V$ by algebraic correspondences, $V^{\otimes_Lm}$ and $\bigwedge^m_LV$ are realized inside $V^{\otimes m}$ as images of $s_L$ and $a_ms_L$. Here $s_L$ is the separability idempotent and $a_m$ the alternator. These idempotents are algebraic: permutations of tensor factors come from permutations of factors of a product, up to Koszul signs [18, §5].

• (R1) Zarhin [12]: $T$ is irreducible. If $E$ is totally real then $m\ge3$ and $\mathrm{Hg}(T)=\mathrm{Res}_{E/\mathbb Q}\mathrm{SO}(T,\varphi_E)$, where $\varphi=\mathrm{tr}_{E/\mathbb Q}\circ\varphi_E$. At exactly one real place $\sigma_0$, $T_{\sigma_0}=T\otimes_{E,\sigma_0}\mathbb R$ has signature $(2,m-2)$; at the others it is negative definite and of type $(1,1)$.
• (R2) Varesco [5, Thm. 2.16]: for $E$ totally real, the Hodge classes in $\bigotimes^\bullet T$ are generated by Hodge classes in $T^{\otimes2}$ (which correspond to $E$) and by the classes in $\bigwedge^m_ET$.
• (R3) CM case: [5, Cor. 2.3], with [6].
• (R4) Transport [18, (R4)]: if $g\colon T(S)\to T(S')$ is a Hodge isomorphism with $g,g^{-1}$ algebraic, HC for all powers (or for $S^n$, $n\le N$) of $S$ transfers to $S^k\times S'^l$ (with $k+l\le N$).
• (R5) Künneth, Néron–Severi and transcendental projectors on surfaces are algebraic. Hodge classes in $H^2$ are algebraic. The cup product is nondegenerate on the polarized primitive summands used, e.g. $T\subset H^2(S)$ and $H^1(C)\otimes H^1(C')\subset H^2(C\times C')$. The adjoint of an algebraic correspondence is algebraic. Injective algebraic maps between such summands have algebraic left inverses [18, (R5)].
• (R6) [25, Prop. 4.1]: let $\mathcal E\colon y^2=x^3+A(t)x+B(t)$, and let $S,S',Y$ be smooth models of its base changes by $t=D_8(s)$, $t=-D_8(s)$, $t=D_4(w)$. If $S,S'$ are K3 and $p_g(Y)=0$, there is an algebraic symmetric $\Sigma$ with $\Sigma^2=2$ on $T(S)$, and an algebraic Hodge isomorphism $f\colon T(S)\to T(S')$ with $f^tf=2+\Sigma$ and algebraic inverse $\tfrac12(2-\Sigma)f^t$.
• (R7) [18, Thm. B(4), Cor. C]: HC holds for all powers of $y^2=x^3-3x+bD_8(s)+c$, $b\ne0$, and of every K3 surface whose $T$ is rationally Hodge isometric to one of these.
• (R8) Weak Torelli [7, Ch. 7]: K3 surfaces with Hodge-isometric integral $H^2$ are isomorphic.

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I.5. The family: proof of Theorem I.1

I.5.1 Fibres

Write $A=aD_8+a'$ and $B=bD_8+c$, of degrees $\le8$ in $s$, as binary forms of degrees $8$ and $12$ (weights of a K3 Weierstrass model). Their orders at $s=\infty$ are $(0,4)$ if $a\ne0$. The discriminant $\Delta=4A^3+27B^2=\Delta_Y(D_8(s))$ has degree $24$ and leading coefficient $4a^3$ (verify.py A4). So for $a\ne0$, $\mathrm{ord}_\infty(A,B,\Delta)=(0,4,0)$ and the fibre at $\infty$ is the smooth curve $y^2=x^3+ax$ (A5). For $a=0$, $b\ne0$ the orders are $(8,4,8)$, i.e. type IV$^*$ (A6, the family of [18]).

The model is minimal everywhere. At a finite point, $\mathrm{ord}A\le2$ if $a\ne0$, because $D_8-\kappa$ has at most double roots (its critical values are $\pm2$). If $a=0$ then $A$ is constant, and if $A\equiv0$ then $\mathrm{ord}\,B\le2$ since $b\ne0$. Since $\Delta\not\equiv0$, the minimal resolution is an elliptic surface with section and $\chi(\mathcal O)=2$, hence a K3 surface [11, §§4, 11]. On $U$, $\Delta$ is squarefree of degree $24$ and prime to $A$: the roots of $D_8(s)=w_i$ are simple because $w_i\ne\pm2$. So there are $24$ fibres of type I$_1$; samples are checked in A7. This proves (1).

I.5.2 The involution and the RM

Since $D_8$ is even (A3), $\iota(s,x,y)=(-s,x,y)$ is an automorphism. In the coordinates $x'=\sigma^4x$, $y'=\sigma^6y$ ($\sigma=1/s$) it fixes the fibres over $s=0$ and $s=\infty$ pointwise. It has no other fixed points. From now on in this paragraph let $p\in U$: then $D_8(0)=2$ and $\pm2$ are not roots of $\Delta_Y$, so the fibres over $s=0,\infty$ are smooth elliptic curves and $S/\iota$ is smooth, with Euler number $(24+0)/2=12$. Since $D_8=D_4\circ D_2$ (A2), it is the elliptic surface $R_4$ over $r=s^2-2$ with data $(aD_4+a',bD_4+c)$ of degrees $(4,4)\le(4,6)$. Its fibre at $\infty$ has orders $(0,2,0)$ (A8), so $R_4$ is a rational elliptic surface. Hence $H^2(S)^\iota\cong H^2(R_4)$ has rank $10$ and consists of algebraic classes, and $\iota=-1$ on $H^{2,0}(S)$. As $T(S)$ is irreducible and contains $H^{2,0}$, $T(S)\subseteq V(S)$, of rank $22-10=12$. This proves (2).

For (3) apply (R6) with $\mathcal E=Y\colon y^2=x^3+(at+a')x+bt+c$. The base change by $-D_8$ is the member $(-a,a',-b,c)$ (A9), a K3 surface by (1). The $D_4$ base change is rational by the above, so $p_g=0$. The proof of (R6) in [25] realizes $\Sigma$ as $q_{0*}(\varrho_*+\varrho^{-1}_*)q_0^*$ on the $D_8$-Galois cover $\Gamma$. For $p\in U$ it shows more: $\Sigma$ is defined on all of $V(S)$ and $\Sigma^2=2$ there, since $\varrho^4=-1$ on $q_0^*V(S)$. So $E_0:=\mathbb Q(\Sigma)\cong\mathbb Q(\sqrt2)$ acts on $V(S)$ by algebraic correspondences.

I.5.3 Moduli

Lemma I.5.1. Let $M(s)=\alpha s+\beta$ with $\alpha\ne0$. If $D_8(\alpha s+\beta)=\kappa D_8(s)+\kappa'$, then $\beta=0$ and $\alpha=\pm1$.

Proof. Compare coefficients. The $s^7$ coefficient gives $8\alpha^7\beta=0$. The $s^8$ and $s^6$ coefficients give $\alpha^8=\kappa$ and $\alpha^6=\kappa$ (verify.py B1). $\square$

Proof of (4). Let $ab\ne0$ and suppose $\mathcal S_{p}\cong\mathcal S_{p'}$ as elliptic surfaces. Then there is a Möbius $M$ with $(A',B')=(\lambda^4A\circ M\cdot\ell^8,\lambda^6B\circ M\cdot\ell^{12})$, $\ell$ the linear form of $M$. Now $B$ has a zero of order exactly $4$ at $\infty$, and its other zeros (the roots of $D_8=-c/b$) have order $\le2$. So $M$ fixes $\infty$ and is affine. Then $A\circ M$ has the shape $\kappa D_8+\kappa'$, so $M=\pm s$ by Lemma I.5.1, and $p'$ is a weighted rescaling of $p$.

Now suppose the period map had rank $<3$ on an open set. Its fibres would contain positive-dimensional analytic sets $Z$ of members with Hodge-isometric $H^2(\cdot,\mathbb Z)$ (local markings), hence isomorphic surfaces (R8), say all isomorphic to $S_0$. A K3 surface has only countably many elliptic fibrations (one for each primitive isotropic nef class). So uncountably many members of $Z$ would be isomorphic to $S_0$ as elliptic surfaces with the same fibration, contradicting the injectivity just proved.

For the two sample members we check injectivity of the differential directly. The Weierstrass model is a quasi-smooth hypersurface of degree $12$ in $\mathbb P(1,1,4,6)$. By Griffiths' residue theory in Steenbrink's weighted form [13], $H^{1,1}_{\rm prim}\cong R_{12}$ (Jacobian ring) and the differential sends $\partial/\partial t$ to the class of $\partial f/\partial t$. We find $\dim R_{12}=18$. The four classes $xD_8,\ xt^8,\ t^4D_8,\ t^{12}$ span a $3$-dimensional subspace, and the weighted-scaling combination lies in the Jacobian ideal (B2, B3). $\square$

I.5.4 Generic rank via Frobenius

Let $Z$ be a smooth model of the base change of $Y$ by $w=z^8+z^{-8}$, with $\varrho(z)=\zeta_8z$ and $\tau(z)=z^{-1}$ acting on $z$ only. Then $S\sim Z/\tau$ (coordinate $s=z+z^{-1}$). As in [25], $q_0^*V(S)$ is the part of $H^2(Z)^\tau$ on which $\varrho$ has primitive eighth-root eigenvalues. Decomposing under the dihedral group $\langle\varrho,\tau\rangle$ of order 16, $V(S)=V_+\oplus V_-$ ($\Sigma=\pm\sqrt2$). Here $V_+\cong H^2(Z)_\chi$ is the $\varrho$-eigenspace for the eigenvalue $\zeta_8$, and $V_-\cong H^2(Z)_{\chi^3}$, both of dimension $6$.

Lemma I.5.2 (trace formula). Let $p\equiv1\bmod8$ be a prime of good reduction, $q=p^k$, $\chi_2$ the quadratic character of $\mathbb F_q$, and $\tilde\chi(u)=\zeta_8^{\log_g\mathrm N_{\mathbb F_q/\mathbb F_p}(u)}$ for a fixed generator $g$ of $\mathbb F_p^\times$. Then $$\mathrm{Tr}\bigl(\mathrm{Frob}_q\mid H^2(Z)_\chi\bigr)=\sum_{u\in\mathbb F_q^\times}\tilde\chi(u)\sum_{x\in\mathbb F_q}\chi_2\Bigl(x^3+A_Y(u+u^{-1})x+B_Y(u+u^{-1})\Bigr),$$ with $A_Y(w)=aw+a'$ and $B_Y(w)=bw+c$.

Proof. Let $Z^\circ$ be the part over $z\in\mathbb G_m$. Its fibres are the Weierstrass cubics, since on $U$ they are smooth or nodal. The fibres over $z=0,\infty$ are smooth curves $y^2=x^3+ax$. On them $\varrho$ acts by the automorphism $(x,y)\mapsto(-x,iy)$ of order 4. Indeed, the minimal coordinates are $x\sigma^{-12}$, $y\sigma^{-18}$ with $\sigma=1/z$, and $\zeta_8^{12}=-1$, $\zeta_8^{18}=i$. So these fibres have no $\chi$-part, and $H^\bullet_c(Z^\circ)_\chi=H^\bullet(Z)_\chi$ is concentrated in degree 2. Apply Grothendieck–Lefschetz to $\mathrm{Frob}\circ\varrho^j$ and project to the $\chi$-isotypic part. A point over $z$ is fixed by $\mathrm{Frob}\circ\varrho^j$ iff $z^{q-1}=\zeta_8^{-j}$ and $(x,y)$ is $\mathbb F_q$-rational on the fibre over $u=z^8\in\mathbb F_q^\times$. The eight $z$ over $u$ have the same $z^{q-1}=u^{(q-1)/8}$. Summing with $\chi^{-1}(\varrho^j)$ gives $\sum_u\tilde\chi(u)\,\#Y_{u+1/u}(\mathbb F_q)$, and the constant $q+1$ cancels. Compatibility of $\tilde\chi$ over different $k$ is ensured by the norm. $\square$

verify.py C1 checks Lemma I.5.2 against an independent count: $\mathrm{Tr}(\mathrm{Frob}\mid V)=\#S(\mathbb F_q)-\#R_4(\mathbb F_q)=\sum_{r,x}\chi_2\bigl((r+2)f_r(x)\bigr)$ (using $H^2(S)^\iota=H^2(R_4)$). For $(1,1,2,3)$, $p=17$, $k=1,2$ it equals $\mathrm{Tr}_\chi+\mathrm{Tr}_{\chi^3}$.

Proof of (5). The member $p_0=(1,1,2,3)$ lies in $U$ and has good reduction at $17$ with the same fibre configuration (C0). Lemma I.5.2 for $k=1,2,3$ gives power sums in $\mathbb Z[\sqrt2]$: $$s_1=-6+18\sqrt2,\quad s_2=(6-\sqrt2)^2,\quad s_3=1482+4734\sqrt2 .$$ The pairing $\langle x,\tau y\rangle$ makes $H^2(Z)_\chi$ an orthogonal Frobenius module of weight 2. So $P_+(T)=\det(1-\mathrm{Frob}\,T\mid V_+)$ satisfies $P_+(T)=\varepsilon q^6T^6P_+(1/(q^2T))$, $\varepsilon=\pm1$, and is determined by $s_1,s_2,s_3$ and $\varepsilon$. The middle coefficient is $c_3=1156-3468\sqrt2\ne0$, so $\varepsilon=+1$ (C2). The norm $N(T)=P_+(T/17)\,\overline{P_+}(T/17)\in\mathbb Q[T]$ has degree $12$. It is coprime to every cyclotomic polynomial $\Phi_n$ with $\varphi(n)\le12$ (C3, C4). (Equivalently: $17N(T)=17T^{12}+12T^{11}+\dots+17$ is irreducible over $\mathbb Q$, so the normalized eigenvalues are not algebraic integers.)

If $\ell\in NS(S_{\bar{\mathbb Q}})\cap V(S)$ were nonzero, its specialization would be a nonzero class in $NS(S_{\bar{\mathbb F}_{17}})\cap V$. Specialization is injective and compatible with $\iota$ and with the correspondences $\varrho,\tau$. Some power of Frobenius fixes the classes of the finitely many curves involved, so it would have an eigenvalue $q\cdot(\text{root of unity})$ on $V_+$ or $V_-$. This is excluded ($P_-$ is the Galois conjugate of $P_+$). Hence $NS(S_{p_0})\cap V=0$, $\rho(S_{p_0})=10$ and $T(S_{p_0})=V$. By semicontinuity of the Picard number in families (specialization from the generic point injects $NS$), the very general member has $\rho=10$ and $T=V$ of rank 12, and $E(S)\supseteq\mathbb Q(\sqrt2)$ gives $m=6$.

$E(S)=\mathbb Q(\sqrt2)$ for very general $S$: any larger $E(S)$ contains $\sqrt2$ and satisfies $[E(S):\mathbb Q]\cdot\dim_{E(S)}T=12$. If $E(S)$ is totally real, then $\dim_{E(S)}T\ge3$ by (R1), so $E(S)$ is quartic with $\dim_{E(S)}T=3$ and its period domain has dimension $1$. If $E(S)$ is CM, its period domain is a ball of dimension $\dim_{E(S)}T-1\le2$. In each case the locus in the 4-dimensional period domain $D_V$ is a countable union of analytic subsets of dimension $\le2$, one for each candidate endomorphism. Their preimages under a map of generic rank $3$ are proper analytic subsets. The same argument applies to the hyperplane sections $\ell^\perp$, $\ell\in V$. $\square$

Remark I.5.3. The argument uses only the elementary direction of the Tate conjecture (algebraic classes give eigenvalues $q\cdot$(root of unity)) and the injectivity of specialization of Néron–Severi groups, as in van Luijk's method [14]. No point count over $\mathbb F_{17^k}$ with $k>3$ is needed, because the $\mathbb Q(\sqrt2)$-splitting $V=V_+\oplus V_-$ halves the degree.

Remark I.5.4 (a forced sign). At the member $(1,0,1,2)$, the same computation at $p\in\{17,41,73,89,97,113\}$ gives $c_3=0$ every time. With $\varepsilon=-1$ (confirmed at $p=17,41$ through $s_4$, computed by a separate FFT-based script), $P_+$ is divisible by $(1-qT)(1+qT)$. So $\rho(S_{\bar{\mathbb F}_p})\ge14$ at these primes, while $\rho=10$ over $\bar{\mathbb Q}$ is still possible. In all seven $(p,\text{member})$ cases computed, $\varepsilon$ equals the Legendre symbol of $\Delta_Y(2)\Delta_Y(-2)$. For $(1,0,1,2)$ this is $(29/p)$, and $29$ happens to be a non-residue modulo all six primes; for $(1,1,2,3)$ it is $+1$ at $p=17$ (C7). This is consistent with Charles' analysis of Picard jumps of RM K3 surfaces under reduction [2], and with the Galois character of the determinant line $\bigwedge^6V_+$ being quadratic, ramified at the primes of $\Delta_Y(\pm2)$, the fibres over the critical values of $D_8$. It is an arithmetic analogue of the orientation character of [21]. We record it as an observation only.

Remark I.5.5 (codimension-one jumps). An anti-invariant section has the form $(X(r),\,s\,Q(r))$ with $\deg X,\deg Q\le2$. It must satisfy $(r+2)Q^2=X^3+(aD_4+a')X+bD_4+c$: $6$ unknowns and $7$ equations. This is consistent with Theorem I.1(5). It suggests divisors in $\Omega$ along which $m$ drops to $5$ (rank 10, $\rho=12$), which are two-parameter families of the kind asked for in the problem. We have not analysed them.

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I.6. Proof of Theorem I.2

(1). $E(S)=\mathbb Q(\sqrt2)=\mathbb Q(\Sigma)$ acts by algebraic correspondences (Theorem I.1(3)). By (R1) the Hodge classes of $T\otimes T$ correspond to $E(S)$ and are algebraic. The Hodge classes of $S^n$ are sums of products of Künneth components, divisor classes and Hodge classes of $T^{\otimes j}$, $j\le n$. By the first fundamental theorem for $\prod_\sigma\mathrm{SO}(T_\sigma)$ (the content of (R2)), the invariants in $T^{\otimes j}\otimes\mathbb C$ are generated by the forms $\varphi_\sigma$ and by determinants. A determinant needs $m$ vectors in the same $T_\sigma$. So for $j<m$ all Hodge classes in $T^{\otimes j}$ are generated by those in $T^{\otimes2}$, hence algebraic. This gives HC for $S^n$, $n<m$. Products with the twin $S'$ follow from (R4), since $f$ and $f^{-1}$ are algebraic (Theorem I.1(3)).

(2) is (R3) with (R4). (3): if $a=0$ and $a'\ne0$, rescale to $a'=-3$. Then $S$ is the surface $S_{b,c}$ of [18], and (R7) applies, with the twin as in [18]. If $a=a'=0$, then $x\mapsto\zeta_3x$ is a non-symplectic automorphism of order 3. So $E(S)\supseteq\mathbb Q(\zeta_3)$ is CM and (2) applies. $\square$

Remark I.6.1 (withdrawn claims). An earlier version asserted that the anti-invariant space $V$ is constant along all of $\Omega$, splits at $a=0$ through the $\mathrm{IV}^*$ fibre, and that a dense set of members is Hodge isometric to members of $\{a=0\}$. That argument treated the resolved $\mathrm{IV}^*$ fibre incorrectly (an involution of the $E_6$ root lattice fixing the identity component has invariant rank $6$ or $4$, and the Mordell–Weil sections of the quotient at $a=0$ give further invariant classes), so the specialization statement and the dense-locus claim are withdrawn. The generic Frobenius computation and Theorem I.2(1) on the open locus $a\ne0$ are unaffected.

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I.7. The spin obstruction: proof of Theorem I.3

Lemma I.7.1 (weights). Let $\mathfrak g=\mathfrak{so}_m$, $m=3$ or $m\ge5$, and let $h\in\mathfrak g$ be semisimple with eigenvalues $1,0^{m-2},-1$ on the vector representation. For an irreducible representation $\rho$ with highest weight $\lambda$, the spread (largest minus smallest eigenvalue of $\rho(h)$) is $\langle\lambda-w_0\lambda,h\rangle$. It is $\le1$ iff $\rho$ is trivial or a spin/half-spin representation. The vector representation has spread $2$.

Proof. In $\varepsilon$-coordinates $h=\varepsilon_1^\vee$. The spread is additive in $\lambda$ for dominant $h$, and $\ge1$ on every fundamental weight. For type $B_n$ ($m=2n+1$) the fundamental weights $\omega_i$, $i<n$, have spread $2$, and the spin weight $\omega_n$ has spread $1$. For type $D_n$ ($m=2n$) the weights $\omega_i$, $i\le n-2$, have spread $2$, and the half-spins $\omega_{n-1},\omega_n$ have spread $1$. Sums of two fundamental weights have spread $\ge2$ (verify.py E1–E3). $\square$

Proof of Theorem I.3. (1) Since $T$ is irreducible, $i$ is injective, and $i(T)$ is a sub-Hodge structure of $W_1\otimes W_2$. So the Mumford–Tate group $\mathrm{MT}(W_1\oplus W_2)$ surjects onto $\mathrm{MT}(T)$, and the derived group onto $\mathrm{Res}_{E/\mathbb Q}\mathrm{SO}(T,\varphi_E)$ (R1). For $m=3$ or $m\ge5$ this group is $\mathbb Q$-simple (Weil restriction of an absolutely simple group), so a $\mathbb Q$-simple factor $H$ surjects onto it with finite kernel.

(2) Let $\mu$ be the Hodge cocharacter of $W:=W_1\oplus W_2$ and $h_0=d\mu(1)=z+\sum_jh_j$, with $z$ central and $h_j$ in the simple factors of $\mathrm{MT}(W)^{\rm der}_{\mathbb C}$. On each irreducible constituent $\rho=\bigotimes_j\rho_j$ (twisted by a central character) of $W\otimes\mathbb C$, $h_0$ has eigenvalues in $\{0,1\}$. So each $h_j$ has spread $\le1$ on $\rho_j$. Let $H_{\sigma_0}$ be the simple factor of $H_{\mathbb C}$ mapping onto $\mathrm{SO}(T_{\sigma_0})$. On $T_{\sigma_0}\otimes\mathbb C$ the Hodge types $(2,0),(1,1),(0,2)$ show that the image of $h_{\sigma_0}$ has eigenvalues $1,0^{m-2},-1$ on the vector representation. By Lemma I.7.1, $H_{\sigma_0}$ acts on every constituent of $W_i\otimes\mathbb C$ trivially or through spin representations. The vector representation $T_{\sigma_0}$ occurs in $(W_1\otimes W_2)\otimes\mathbb C$, but not in $\mathbf1\otimes X$ for $X$ trivial or spin. So $H_{\sigma_0}$ acts nontrivially on both $W_1\otimes\mathbb C$ and $W_2\otimes\mathbb C$, through spin representations.

$W_i$ is defined over $\mathbb Q$ and $H$ is $\mathbb Q$-simple. So $\mathrm{Aut}(\mathbb C)$ permutes the factors $H_\sigma$ transitively, together with the isotypic decomposition of $W_i\otimes\mathbb C$, and the same holds for every $\sigma$.

(3) Let $\sigma_1\ne\sigma_0$, so $H_{\sigma_1}(\mathbb R)\simeq\mathrm{Spin}(T_{\sigma_1})$ is compact. Since $T_{\sigma_1}$ is of pure type $(1,1)$, the projection of $\mathrm{Ad}\circ h_W$ to $\mathrm{Lie}\,H_{\sigma_1}$ is trivial, so $h_W(\mathbb S)$ commutes with $H_{\sigma_1}(\mathbb R)$. Let $W_{1,\sigma_1}$ be the $\sigma_1$-eigenspace of $E\otimes\mathbb R$ on $W_1\otimes\mathbb R$; it has dimension $\dim_EW_1$. $E$-linearity of $i$ gives $T_{\sigma_1}\subseteq W_{1,\sigma_1}\otimes W_2$. Hence $H_{\sigma_1}$ acts nontrivially on $W_{1,\sigma_1}$: otherwise $T_{\sigma_1}$ would lie in $\mathbf1\otimes(\text{spin})$. The nontrivial part $W''\subseteq W_{1,\sigma_1}$ is real and stable under $h_W$, so it is a real Hodge structure of weight one. Then $W''^{1,0}$ and $W''^{0,1}=\overline{W''^{1,0}}$ are nonzero $H_{\sigma_1}$-stable subspaces without trivial constituents, so each has dimension $\ge2^k$. Thus $\dim_EW_1\ge\dim_{\mathbb R}W''\ge2^{k+1}$, and with $F$ acting, $\dim_FW_1=\tfrac12\dim_EW_1\ge2^k$.

(4) For $m\ge5$, $2^k\ge4$ (E4, E5). In [18, Thm. A(b)] the image of $T$ is $E$-stable and $(e\otimes1)=(1\otimes\bar e)=(1\otimes e)$ on it, so (3) applies to both factors. $\square$

Remark I.7.2. For $m=4$ the group $\mathrm{SO}(T_{\sigma_0})$ is not simple, and the two half-spins are the two $\mathrm{SL}_2$ factors. That is exactly the shape $M\otimes\overline N$ of [18]. Theorem I.3 shows that this shape is special to $m\le4$.

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I.8. The wedge-square criterion: proof of Theorem I.4

Lemma I.8.1 (interleaving identity). Let $U$ be a 4-dimensional vector space over a field of characteristic $0$, and $B_0=\bigwedge^2U\subset U\otimes U$. Let $c_0\in B_0\otimes B_0\subset U^{\otimes4}$ be the Casimir tensor of the wedge pairing $B_0\times B_0\to\bigwedge^4U$. Number the twelve slots of $c_0^{\otimes3}$ so that copy $j$ occupies slots $(4j,4j+1,4j+2,4j+3)$. Call the even slots "$W$-slots" $W_0,\dots,W_5$ and the odd slots "$W'$-slots" $W'_0,\dots,W'_5$, in order. Let $\Psi_\pi$ pair $W_i$ with $W'_{\pi(i)}$, project each pair to $B_0$ (antisymmetrize), and alternate the six factors.

1. $c_0$ is the volume tensor $\sum_{\pi\in S_4}\mathrm{sgn}(\pi)e_{\pi1}\otimes\cdots\otimes e_{\pi4}$ (verify.py D1).
2. For the cyclic shift $\pi(i)=i+1\bmod6$, $\Psi_\pi(c_0^{\otimes3})=-2880\cdot e_{12}\wedge e_{13}\wedge e_{14}\wedge e_{23}\wedge e_{24}\wedge e_{34}$ (D2).
3. For $\pi=\mathrm{id}$, $\Psi_\pi(c_0^{\otimes3})=0$; likewise for $\pi=(0,2,1,4,3,5)$ (controls D3, D3b). Of the $720$ pairings, $128$ give $\pm2880$ and $592$ give $0$.

Equivariance is consistent with (2): $\det(\bigwedge^2g)=\det(g)^3$ (D4).

Proof of Theorem I.4. We follow [18, §6.2] and mark the new steps.

Step 0. As in [18, Step 0]: $E$ acts on $T$ through $i$ by algebraic correspondences. $H:=i(T)\oplus\theta\,i(T)$ ($\theta\in K$, $\theta^2\in\mathbb Q_{<0}$) is algebraically isomorphic to $T\oplus T$ and is an $F$-space isomorphic to $T\otimes_EF$.

Step 1 (the space $B$). $B=(1\otimes\gamma)^{-1}(\bigwedge^2_FW\otimes\chi)$ is a sub-Hodge structure of $W\otimes_F\overline{W'}$ of $F$-rank $6$, because $\gamma$ is a Hodge isomorphism. We claim $H\subseteq B$. By Theorem I.3(2) at $\tau_0$, the factor $H_{\sigma_0}$ acts on $W_{\tau_0}$ through a spin representation $\rho$ of dimension $4$, so $W_{\tau_0}=\rho$. Then $\mathrm{Sym}^2\rho$ is irreducible of dimension $10$, and does not contain the vector representation $T_{\sigma_0}$ (for $D_3$, $\mathrm{Sym}^2$ of a half-spin; for $B_2$, the adjoint). Hence the Hodge morphism $T\to\mathrm{Sym}^2_FW\otimes\chi$ obtained from $i$, $\gamma$ and the projection vanishes at $\tau_0$. By Galois conjugation it vanishes at every $\tau$, hence identically, and $i(T)\subseteq B$. Since $B$ is $F$-stable, $H\subseteq B$. If $m=6$, ranks give $H=B$. If $m=5$, $\Lambda:=B\cap H^\perp$ has $F$-rank $1$, and by (d) it has no $(2,0)$-part. So it consists of Hodge classes in $H^2(C\times C')$, which are algebraic, with algebraic projector [18, Step 1].

Step 2 (an algebraic Casimir). Let $\varphi_F$ be the $F$-bilinear extension of $\varphi_E$ to $H$, and, if $m=5$, $q_\Lambda$ a nonzero $F$-bilinear form on $\Lambda$. Transport the wedge pairing of $\bigwedge^2_FW$ to $B$ by $\gamma$; it takes values in the $F$-line $\det_FW\otimes\chi^2$. The derived Hodge group acts irreducibly on $H\otimes_{F,\tau}\mathbb C$ (vector representation) and trivially on $\Lambda$. So the transported pairing equals $\lambda\varphi_F\oplus\mu q_\Lambda$ with $\lambda,\mu\in F^\times$ (with $\mu$ absent if $m=6$). Its Casimir $c=\lambda^{-1}c_{\varphi_F}+\mu^{-1}c_\Lambda\in B\otimes_FB$ is algebraic. Indeed, $c_{\varphi_F}$ is a Hodge class of $H\otimes H$, and by Step 0 and (R1) all Hodge classes of $H\otimes H$ are $E$-combinations of the polarization of $T$, hence algebraic [18, Step 2]. Also $c_\Lambda$ is a product of algebraic classes, and $F$ acts algebraically.

Step 3 (new: interleaving). Consider $c^{\otimes3}\in(W\otimes W')^{\otimes6}\subset H^{12}((C\times C')^6)$. Apply the separability idempotent $s_F$ on all twelve slots, then the cyclic permutation of the six $C'$-factors, then the orthogonal projector $P_B$ on each $C\times C'$ pair, then $a_6s_F$. Each map is algebraic: permutations of factors of $(C\times C')^6$ (up to Koszul signs), projectors from (R5) and Step 1, and $F$-idempotents. The result $\Psi(c)$ lies in $\bigwedge^6_FB$. Under $1\otimes\gamma$, each complex component ($\tau\colon F\to\mathbb C$) of this construction is the construction of Lemma I.8.1 with $U=W_\tau$. Here $\chi_\tau$ is a line that only rides along, $P_B$ is the antisymmetrizer because $\bigwedge^2$ and $\mathrm{Sym}^2$ are orthogonal (non-isomorphic constituents), and $c_\tau$ is a nonzero multiple of $c_0$. So $\Psi(c)_\tau\ne0$ for every $\tau$, by Lemma I.8.1(2). The $F$-line $\bigwedge^6_FB$ is therefore spanned over $F$ by the algebraic class $\Psi(c)$, and all its elements are algebraic.

Step 4. If $m=6$, $\bigwedge^6_FH$ is algebraic. If $m=5$, contract with an algebraic $F$-linear form $\lambda$ vanishing on $H$ with $\lambda(\ell)=1$, $\ell\in\Lambda$, as in [18, Step 5] (D6). This shows that $\bigwedge^5_FH$ consists of algebraic classes. The descent lemma [18, Lemma 6.3], whose proof is valid for every $m$, then gives $\bigwedge^m_ET$, with constant $2^{1-m}$ (D5; control D5b).

Step 5. (R2) gives HC for all powers. $\square$

Remark I.8.2 (Hodge numbers). For $m=6$ and $E$ real quadratic, (c) forces $W_{\tau_0}$ of type $(2,2)$ and $W_\tau$ of type $(4,0)$ or $(0,4)$ at the other embeddings, with $\chi_\tau$ of type $(\mp1,\pm1)$ there. Without the twist, $\bigwedge^2W_\tau$ would be of type $(6,0,0)$, and no weight-one piece of $F$-rank 4 has $\bigwedge^2$ of type $(0,6,0)$ (D7–D9). The twist $\chi$ is therefore unavoidable for $E\ne\mathbb Q$. This is the relative counterpart of the familiar fact that Kuga–Satake varieties of RM K3 surfaces involve corestriction [3].

Remark I.8.3 (consistency for $E=\mathbb Q$). The same proof with $L=\mathbb Q$ in place of $F$, $W=W'=H^1(C)$ for a genus-two curve $C$ with $\mathrm{End}\,J(C)=\mathbb Z$, $\gamma$ given by the principal polarization and $\chi$ trivial, recovers the (known) algebraicity of the determinant class of the Kummer surface $\mathrm{Km}(J(C))$ ($m=5$, $B=\bigwedge^2H^1(C)=T\oplus\langle\theta\rangle$). It is a sanity check of Theorem I.4, not a new result.

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Part II. Half-spin cyclic covers: explicit candidates with $E\ne\mathbb Q$, $m=6$

II.2. Statements

Fix $n\in\{8,10,12\}$, $\zeta=e^{2\pi i/n}$, $F=\mathbb Q(\zeta)$, $w=\zeta+\zeta^{-1}$ and $E=\mathbb Q(w)$. So $E=\mathbb Q(\sqrt2)$, $\mathbb Q(\sqrt5)$, $\mathbb Q(\sqrt3)$ respectively, and $F/E$ is a CM extension whose nontrivial automorphism is complex conjugation $c$. For $j\in(\mathbb Z/n)^\times$ let $\tau_j\colon F\to\mathbb C$, $\zeta\mapsto\zeta^j$. The real places of $E$ are $\sigma_c=\tau_{\pm1}|_E$ and $\sigma_0=\tau_{\pm j_0}|_E$, where $j_0=3,3,5$.

Let $e=(e_1,\dots,e_6)$ with $1\le e_i<n$, $\sum e_i\equiv0\pmod n$ and $\gcd(n,e_1,\dots,e_6)=1$. Let $M\subset\mathbb C^6$ be the space of pairwise distinct $a=(a_1,\dots,a_6)$. For $a\in M$ let $C_a$ be the smooth projective curve with affine model $y^n=\prod_i(x-a_i)^{e_i}$, with automorphism $\sigma(x,y)=(x,\zeta y)$. Set $$W_a:=\ker\Phi_n(\sigma^*)\subset H^1(C_a,\mathbb Q),$$ an $F$-vector space ($\zeta$ acting as $\sigma^*$) and a sub-Hodge structure. Its eigenspaces are $W_a\otimes_{F,\tau_j}\mathbb C=H^1(C_a)_j$, where $\sigma^*=\zeta^j$. Write $h^{1,0}_j=\dim H^{1,0}(C_a)_j$. Let $P_a\subset J(C_a)$ be the abelian subvariety with $H^1(P_a,\mathbb Q)=W_a$, namely the image of the idempotent of $\mathbb Q[\sigma]$ that projects onto the factor $\mathbb Q[t]/\Phi_n(t)$.

Theorem II.1 (half-spin cyclic covers). Let $n\in\{5,8,10,12\}$ and let $e$ be as above (six branch points, $\infty$ unbranched).

1. $\dim_FW_a=4$ and $h^{1,0}_j=-1+\sum_i\langle -je_i/n\rangle$ (Chevalley–Weil), where $\langle\cdot\rangle$ is the fractional part.
2. If $n=5$, no $e$ has signature $(h^{1,0}_j,h^{0,1}_j)\in\{(4,0),(0,4)\}$ at any $j$.
3. Up to permutation of the $e_i$ and multiplication by units of $\mathbb Z/n$, the vectors with signature $(4,0)$ at one real place and $(2,2)$ at the other are exactly:

• $n=8$: $(1,1,1,1,2,2)$;
• $n=10$: $(1,1,1,1,3,3)$, $(1,1,1,2,2,3)$, $(1,1,2,2,2,2)$;
• $n=12$: $(1,1,1,1,1,7)$, $(1,1,1,1,2,6)$, $(1,1,1,1,4,4)$, $(1,1,1,2,2,5)$, $(1,1,1,2,3,4)$, $(1,1,2,2,3,3)$.

In each case $(h^{1,0}_{1},h^{0,1}_1)=(4,0)$ and the signature at $\pm j_0$ is $(2,2)$. For $n=8$ the curve has genus $13$ and $P_a$ has dimension $8$.

Call these the ten half-spin families. By [Part I, Rem. I.8.2], $\bigwedge^2_FW_a$ is of K3 type at $\sigma_0$ but of type $(6,0,0)$ at $\sigma_c$. Let $\chi$ be a rank-one $F$-Hodge structure of weight $0$ with types $(-1,1)$ at $\tau_1$, $(1,-1)$ at $\tau_{-1}$ and $(0,0)$ at $\tau_{\pm j_0}$. Such a $\chi$ exists and is unique up to isomorphism (Lemma II.6.3). Put $$\mathbb B_a:=\textstyle\bigwedge^2_FW_a\otimes_F\chi .$$ It is a weight-two $F$-Hodge structure of $F$-rank $6$, with types $(1,4,1)$ at $\tau_{\pm j_0}$ and $(0,6,0)$ at $\tau_{\pm1}$.

Theorem II.2 (descent). Let $h$ be the $F$-hermitian form on $W_a$ attached to the polarization, and $D(e)\in E^\times/N_{F/E}(F^\times)$ its discriminant. It is constant in $a$.

1. If $D(e)\in N_{F/E}(F^\times)$, there is an $F$-antilinear Hodge automorphism $\Phi$ of $\mathbb B_a$ with $\Phi^2=1$. Conversely, if $\mathrm{End}_{F\text{-Hdg}}(\mathbb B_a)=F$ and such a $\Phi$ exists, then $D(e)$ is a norm. In that case $\mathbb T_a:=\mathbb B_a^{\Phi=1}$ is an $E$-Hodge structure of K3 type with $\mathbb T_a\otimes_EF\cong\mathbb B_a$ and $\dim_E\mathbb T_a=6$. If $\mathrm{End}_{F\text{-Hdg}}(\mathbb B_a)=F$, then $\mathbb B_a\cong T\otimes_EF$ for some $E$-Hodge structure $T$ only if $D(e)$ is a norm.
2. The values $D(e)=p+qw$ are listed in §II.6.3. $D(e)$ is a norm for nine of the ten families, by explicit certificates $D=N_{F/E}(x)s^2$. For $n=12$, $e=(1,1,1,2,3,4)$, $D=(9+5\sqrt3)/2$ is not a norm: the Hilbert symbol $(D,-1)_{(\sqrt3)}$ equals $-1$.

Theorem II.3 (the families). Let $e$ be one of the nine descending half-spin families.

1. Let $\delta\colon\pi_1(M)\to\mu(F)$ be the determinant character of the monodromy on $W$; it has finite order. Over the finite étale cover $M'\to M$ with $\pi_1(M')=\ker\delta$, $a\mapsto\mathbb T_a$ is a polarizable variation of Hodge structure of K3 type, of rank $12$, with $E$ acting by flat Hodge endomorphisms. Over $M$ itself the descent is only fibrewise: e.g. for $n=8$, $e=(1,1,1,1,2,2)$ the first twist has $\operatorname{Tr}_F\bigwedge^2T_0=3+3i\notin E$.
2. Its period map, into the four-dimensional period domain of $E$-Hodge structures of this type, has rank $3$ at every point. Modulo $\mathrm{PGL}_2$ the family has three moduli.
3. For $a$ outside a countable union of proper closed analytic subsets of $M$: $\mathbb T_a$ has no nonzero Hodge class, $\mathrm{End}_{\rm Hdg}(\mathbb T_a)=E$, and $\dim_E\mathbb T_a=6$.
4. For every $a$ there is a projective K3 surface $S_a$ with an isometric embedding $T(S_a)_{\mathbb Q}\subseteq\mathbb T_a$ of Hodge structures (for a suitable polarization of $\mathbb T_a$). Equality holds iff $\mathbb T_a$ has no Hodge class, in particular for very general $a$. Then $\rho(S_a)=10$, $E(S_a)=E$, $m=6$, and $$T(S_a)\otimes_EF\;\cong\;\textstyle\bigwedge^2_FH^1(P_a,\mathbb Q)\otimes_F\chi .$$

Hypothesis (KS$_a$). Fix CM curves $X_1,X_2$ as in Lemma II.6.3; for $n=8$, $X_1\colon y^8=x^2(x-1)^7$ and $X_2\colon y^8=x(x-1)^2$. Let $\chi$ be realized in $H^2(X_1\times X_2)(1)$, put $Y_a=C_a\times C_a\times X_1\times X_2$, and let $\mathbb B_a\subset H^4(Y_a,\mathbb Q)(1)$ be the realization of §II.8. Hypothesis (KS$_a$) for a K3 surface $S$ with $T(S)_{\mathbb Q}\cong\mathbb T_a$: there is an algebraic cycle $Z$ on $S\times Y_a$ whose action $T(S)\to H^4(Y_a)(1)$, followed by the (algebraic) projector onto $\mathbb B_a$, is nonzero. A Hodge class with this property exists by Theorem II.3.

Theorem II.4 (conditional determinant). Let $S$ be a projective K3 surface with $E(S)=E$ and $T(S)_{\mathbb Q}\cong\mathbb T_a$ for a descending half-spin family. If (KS$_a$) holds, then:

1. $E$ acts on $T(S)$ by algebraic correspondences, so HC holds for $S^n$, $n\le5$;
2. every class in $\bigwedge^6_ET(S)\subset H^{12}(S^6,\mathbb Q)$ is algebraic;
3. HC holds for $S^n$ for all $n$, and for $S^k\times S'^l$ for every K3 surface $S'$ with $T(S')$ rationally Hodge isometric to $T(S)$.

Proposition II.5. Let $e$ be a half-spin family and $a\in M$.

1. If $g\in\mathrm{Aut}(C_a)$ normalizes $\langle\sigma\rangle$, then $g\sigma g^{-1}=\sigma$.
2. Consequently no automorphism of $C_a$ inverts $\sigma$. The dihedral construction of algebraic RM used for Dickson families ([25, Prop. 4.1], [18]) has no analogue on $C_a$.
3. ($E=\mathbb Q$ shadow.) For $n=4$, $e=(1,1,1,3,3,3)$, $W_a$ is the $\mathbb Q(i)$-module of an abelian fourfold of Weil type, with hermitian discriminant $1/4$ (trivial class). Its $\bigwedge^2$ has irreducible monodromy, so the same construction gives a three-parameter family of rank-6 K3-type structures with $E=\mathbb Q$ generically. For $E=\mathbb Q$ strong tools exist (algebraicity of Weil classes in discriminant one [28], [29]; algebraic Kuga–Satake correspondences for several $\rho=16$ families [30], [27]). We do not claim that (KS) is known for this shadow family.

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II.3. Related work and novelty

| Source | Closest statement | Relation to this paper | |---|---|---| | Varesco [5, Thm. 2.16, Cor. 2.3] | Reduction to $S^2$ and $\bigwedge^m_ET$; CM case | Used in Theorem II.4(3) | | Varesco [5, main theorem] | HC for all powers on 4-dim families with $\rho=16$ (so $E=\mathbb Q$ generically) if the Kuga–Satake correspondence is algebraic | The $E=\mathbb Q$ model of Theorem II.4. We treat $E\neq\mathbb Q$, rank $12$, with a minimal half-spin factor instead of the full Kuga–Satake variety | | Paranjape [30]; Schlickewei [31] | Algebraic Kuga–Satake for double planes branched along six lines; HC for $S\times S$; KS decomposition for RM | $E=\mathbb Q$ (or $\dim_ET=3$) only. [31, Thm. 1] describes the KS variety for RM abstractly | | Floccari [27]; Floccari–Fu [29], Markman [28] | KS algebraic for families of $\rho=16$ K3s; Hodge–Weil classes algebraic on Weil fourfolds of discriminant 1 | $E=\mathbb Q$ input. Our $n=4$ shadow (Prop. II.5(3)) is of this kind | | Varesco [26] | Hodge similarities; $\sqrt p$ algebraic for K3s with symplectic automorphisms ($p=2,3$), four-dimensional RM families | Gives $S\times S$, not determinants. Not used | | van Geemen [3], van Geemen–Schütt [4] | RM and Kuga–Satake; explicit RM families | Corestriction picture [3] is the abstract source of our $\chi$-twist. No HC for powers | | Deligne–Mostow [32], Looijenga [33], Moonen [34] | Hodge theory and monodromy of cyclic covers; special subvarieties | Used for Theorem II.1 and Theorem II.3(2). Our families are not Moonen-special: $P_a$ moves in a 3-dim subvariety of a 4-dim unitary Shimura variety | | rStation 23 [36] | Algebraic RM by every real quadratic field; Witt-index frontier, rank 6 | $\dim_ET=3$ | | rStation 28, 61, 65, 74 [23], [15], [16], [24] | All-powers HC for Dickson families via curve quotients and fixed determinant quotients | $m\le4$; Prop. II.5(2) shows their dihedral mechanism is absent here | | rStation 72 [37] | Quartic RM, HC for squares | Squares only | | rStation 77 [38] | Products with trivial endomorphism field; determinant and obstructions | $E=\mathbb Q$ products | | rStation 95, 97 [19], [25] | Witt-index-one RM by $\mathbb Q(\sqrt2)$; $D_8$ mechanism | $m\le4$ | | rStation 98 [18] | Theorem A ($F$-rank two); all powers for a 2-parameter family, rank 8 | Theorem II.4 is the $F$-rank-four, $m=6$ analogue, conditional on (KS) | | rStation 105 [21] | Orientation character; uniform GRR constructions | Not used | | rStation 106, 108 [20], [39] | Dickson families $m=4$; quaternionic type; transfer by density | $m\le4$ | | rStation 111 [40] | Census of cyclic Dickson families | $m\le4$ | | Part I | Spin obstruction (Thm. I.3); conditional wedge-square criterion (Thm. I.4) with $W,W'$ on curves; interleaving identity | We supply explicit $E\ne\mathbb Q$ candidates for its hypotheses, a descent criterion it did not have, and a realization in which $\gamma$ of [Part I, Thm. I.4(c)] is unnecessary |

New, as far as we could find (sources.md):

1. the classification of Theorem II.1 and the observation that cyclic covers of $\mathbb P^1$ realize the relative half-spin type required for $m=6$ with $E=\mathbb Q(\sqrt2),\mathbb Q(\sqrt3),\mathbb Q(\sqrt5)$;
2. the descent criterion (Lemma II.6.1, Theorem II.2), the exact discriminants, and the non-descending family $(12;1,1,1,2,3,4)$;
3. three-parameter families of K3-type RM Hodge structures of rank 12 whose half-spin factor is an explicit abelian subvariety of a Jacobian (Theorem II.3), with genericity proved by an exact monodromy-algebra computation;
4. Theorem II.4, reducing HC for all powers of these surfaces to one explicit relative Kuga–Satake class.

Not new. The Chevalley–Weil formula, the Kita–Yoshida intersection numbers, Picard–Lefschetz for twisted cycles, the accidental isomorphism $\mathrm{SU}(2,2)\sim\mathrm{Spin}(2,4)$ behind $\bigwedge^2$, and the polarization-plus-interleaving argument of [18], Part I.

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II.4. Conventions and imported results

Cohomology has rational coefficients. "Algebraic" means a $\mathbb Q$-combination of classes of algebraic cycles. A map is algebraic if it is induced by an algebraic correspondence.

• (I1) Zarhin [12]: $T(S)$ is irreducible, $E(S)$ is totally real or CM, and for $E$ totally real $\mathrm{Hg}(T)=\mathrm{Res}_{E/\mathbb Q}\mathrm{SO}(T,q_E)$. The period domain of K3-type $E$-structures of $E$-dimension $m$ has dimension $m-2$.
• (I2) Varesco [5, Thm. 2.16]: for $E(S)$ totally real, the Hodge classes in all $T^{\otimes k}$ are generated by those in $T^{\otimes2}$ and by $\bigwedge^m_ET$.
• (I3) Buskin [6]: rational Hodge isometries between K3 surfaces are algebraic. Transport of HC for powers as in [18, (R4)].
• (I4) Algebraicity of Künneth and transcendental projectors on surfaces and of the $\mathbb Q[\mu_n]$-idempotents on curves. Left inverses of injective algebraic maps between polarized summands are algebraic [18, (R5)].
• (I5) Descent lemma [18, Lemma 6.3]: if $H=T\otimes_EF$ is realized algebraically and $\bigwedge^m_FH$ consists of algebraic classes, so does $\bigwedge^m_ET$. The constant is $2^{1-m}$ (verify.py D4).
• (I6) Interleaving identity [Part I, Lemma I.8.1]: for $\dim U=4$, $B_0=\bigwedge^2U$ and $c_0$ the Casimir of the wedge pairing (the volume tensor), the cyclic-shift interleaving of $c_0^{\otimes3}$ followed by $\bigwedge^6$ is $-2880$ times a generator of $\bigwedge^6B_0$, and the identity pairing gives $0$. Re-verified in D1–D3.
• (I7) Surjectivity of the period map and projectivity criterion for K3 surfaces [7, Ch. 6–7].

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II.5. Theorem II.1: cyclic covers with half-spin signature

Proof. (1) The cover $C_a\to\mathbb P^1$ is branched exactly at the $a_i$, because $\sum e_i\equiv0$. For $j$ prime to $n$, every $je_i\not\equiv0$. The $\zeta^j$-eigenspace is the twisted cohomology of the rank-one local system with monodromy $\zeta^{-je_i}$ at $a_i$ [32, §2], [33]. Its Euler characteristic gives dimension $6-2=4$. The holomorphic forms in it are $x^m\prod_i(x-a_i)^{\lfloor ke_i/n\rfloor}dx/y^k$ with $k\equiv-j$ and $0\le m\le\sum_i\langle ke_i/n\rangle-2$. This gives the Chevalley–Weil count.

(2) For $n=5$: $h^{1,0}_j=0$ requires $\sum_i\langle -je_i/5\rangle=1$. That is impossible, since each of the six terms is at least $1/5$. The same holds for $h^{0,1}_j$.

(3) The list comes from an exhaustive search over $\{1,\dots,n-1\}^6$ (verify.py A1–A5). Signatures are computed exactly with Fractions. The genus is computed by Riemann–Hurwitz (A6). $\square$

For $n=8$ the eigen-differentials are explicit. With $e=(1,1,1,1,2,2)$ and $f=\prod(x-a_i)^{e_i}$, the four forms of $H^{1,0}_1$ are $x^m(x-a_5)(x-a_6)\,dx/y^7$, $0\le m\le 3$ (note $\sigma^*(dx/y^k)=\zeta^{-k}dx/y^k$). At $j_0=3$ the forms are $dx/y^5\cdot(x-a_5)(x-a_6)$ and $x\,dx/y^5\cdot(x-a_5)(x-a_6)$. At $j=5$ they are $dx/y^3$ and $x\,dx/y^3$. This is the shape used in §II.7.2.

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II.6. Hermitian discriminant and descent

II.6.1 A linear-algebra lemma

Lemma II.6.1. Let $F/E$ be a CM extension of fields of characteristic $0$ with conjugation $c$, let $(W,h)$ be a nondegenerate hermitian space of dimension $4$, and let $\omega=e_1\wedge e_2\wedge e_3\wedge e_4$ for a basis with Gram matrix $G$. Let $h_2$ be the induced hermitian form on $\bigwedge^2W$. Define the $c$-antilinear map $\phi_\omega$ of $\bigwedge^2W$ by $\alpha\wedge\phi_\omega(\beta)=h_2(\alpha,\beta)\,\omega$. Then $\phi_\omega^2=\det(G)\cdot\mathrm{id}$, and $\phi_{f\omega}^2=N_{F/E}(f)\det(G)\cdot\mathrm{id}$.

Proof. Take an orthogonal basis with $h(e_i,e_i)=d_i\in E$. Then $\phi(e_{12})=d_1d_2e_{34}$, $\phi(e_{34})=d_3d_4e_{12}$, $\phi(e_{13})=-d_1d_3e_{24}$, $\phi(e_{24})=-d_2d_4e_{13}$, $\phi(e_{14})=d_1d_4e_{23}$ and $\phi(e_{23})=d_2d_3e_{14}$. Hence $\phi^2=d_1d_2d_3d_4$. A change of basis multiplies $\det G$ by $N(\det P)$ and $\omega$ by $\det P$, which is consistent. Also $\phi_{f\omega}=f\phi_\omega$, so $\phi_{f\omega}^2=f\bar f\phi_\omega^2$. verify.py B5 checks $\phi^2=\det G$ directly for random non-diagonal hermitian $G$ over $\mathbb Q(i)$. Control B5c: for a non-hermitian $G$, $\phi^2$ is not scalar. $\square$

II.6.2 Hodge-theoretic descent

Lemma II.6.2. Two rank-one $F$-Hodge structures with the same types $(p_\tau,q_\tau)$ at every embedding $\tau$ are isomorphic, and every $F$-linear isomorphism between them is a morphism of Hodge structures.

Proof. Choose generators $v,v'$. The $F$-linear map $v\mapsto v'$ maps the $\tau$-line to the $\tau$-line, and both lines have the same type. $\square$

Lemma II.6.3. The twist $\chi$ exists and is unique up to isomorphism. It is realized as $U_1\otimes_F\overline{U_2}(1)\subset H^1(X_1)\otimes H^1(X_2)(1)$, where $U_i=\ker\Phi_n(\sigma^*)\subset H^1(X_i)$ for curves $X_i\colon y^n=x^{b_1}(x-1)^{b_2}$ with three branch points. For $n=8$ take $b=(2,7,7)$ for $X_1$ and $(1,2,5)$ for $X_2$; for $n=10$, $(2,9,9)$ and $(1,3,6)$; for $n=12$, $(2,11,11)$ and $(1,2,9)$. Here $b$ lists the three exponents, the third at $\infty$. The CM types are $\{-1,\varepsilon j_0\}$ for $X_1$ and $\{1,\varepsilon j_0\}$ for $X_2$, for a sign $\varepsilon$ (for $n=8$: $\{5,7\}$ and $\{1,5\}$) (A9). Moreover $\bar\chi\cong\chi\otimes_F\det_FW(2)$ (A10).

Proof. Uniqueness is Lemma II.6.2. The types of $U_1\otimes\overline{U_2}(1)$ at $\tau$ are $(\text{type of }U_{1,\tau})-(\text{type of }U_{2,\tau})$, and A9 checks these against the required ones. The last claim compares types (A10): at $\tau_1$, $\det W$ has type $(4,0)$, so $\det W(2)$ has type $(2,-2)$ and $(-1,1)+(2,-2)=(1,-1)$, which is the type of $\bar\chi$ there. The other places are similar. $\square$

Proof of Theorem II.2(1). The polarization of $W$ is $\mathrm{Tr}_{F/\mathbb Q}\circ s$ for a unique skew-hermitian $s$. Put $h=\theta s$, $\theta=\zeta-\zeta^{-1}$. Since $\theta^4=N(\theta)^2$, the class of $\det h$ modulo norms equals that of $\det s$, and it does not depend on $\theta$. The form $h$ is a Hodge morphism $W\otimes_F\overline W\to F(-1)$. Hence there are $F$-linear Hodge isomorphisms $$\overline{\textstyle\bigwedge^2W}\xrightarrow{\ \bigwedge^2h\ }\textstyle(\bigwedge^2W)^\vee(-2)\xrightarrow{\ \text{wedge}\ }\textstyle\bigwedge^2W\otimes L,\qquad L=(\det_FW)^\vee(-2),$$ and by Lemmas 6.2–6.3 an $F$-linear Hodge isomorphism $\psi\colon\bar\chi\to\chi\otimes L^\vee$. Composing with the identity $\mathbb B\to\overline{\mathbb B}$ (antilinear, Hodge) gives an antilinear Hodge automorphism $\Phi$ of $\mathbb B$. After choosing generators, $\Phi=\phi_\omega\otimes\psi'$, where $\psi'$ is antilinear on a line, so $\psi'^2=N(\cdot)$. By Lemma II.6.1, $\Phi^2=D\cdot N(f)$ for some $f\in F^\times$. Rescaling $\Phi$ by $g\in F^\times$ multiplies $\Phi^2$ by $N(g)$. So $\Phi^2=1$ can be achieved iff $D$ is a norm.

In that case $\mathbb T=\ker(\Phi-1)$ is a $\mathbb Q$-sub-Hodge structure ($\Phi$ is $\mathbb Q$-linear and Hodge) with $\mathbb B=\mathbb T\oplus\theta\mathbb T$, since $\Phi\theta=-\theta\Phi$. As $E$-spaces, $\mathbb T\otimes_EF\cong\mathbb B$. At $\sigma_0$ the type is $(1,4,1)$ and at $\sigma_c$ it is $(0,6,0)$ (A7). So $h^{2,0}(\mathbb T)=1$. $\mathbb T$ is polarizable as a sub-Hodge structure of $\mathbb B\subset H^4(Y_a)(1)$.

Conversely, suppose $\mathrm{End}_{F\text{-Hdg}}(\mathbb B)=F$ and $\mathbb B\cong T\otimes_EF$. Then $1\otimes c$ is an antilinear Hodge involution $\Phi'$, and $\Phi'\Phi^{-1}\in F^\times$, so $1=\Phi'^2=N(f)\Phi^2$ and $D$ is a norm. Since $h$ is flat in $a$, $D$ is constant. $\square$

II.6.3 Computing the discriminant

The local system underlying $W_a$ has rank one over $F$, with monodromy $c_i=\zeta^{e_i}$ around $a_i$. For real $a_1<\dots<a_6$, the regularized intervals $\Delta_k=\mathrm{reg}(a_k,a_{k+1})$, $k=0,\dots,3$, form an $F$-basis of twisted homology. The Kita–Yoshida intersection numbers [35] give the $F$-valued sesquilinear matrix $$H_{kk}=-\frac{c_kc_{k+1}-1}{(c_k-1)(c_{k+1}-1)},\qquad H_{k,k+1}=\frac{c_{k+1}}{c_{k+1}-1},\qquad H_{k+1,k}=\frac1{c_{k+1}-1},$$ and all other entries zero. $H$ is skew-hermitian (B2). It is monodromy invariant. Since monodromy acts absolutely irreducibly on $W$ (§II.7.3, C2), the space of invariant skew-hermitian forms is an $E$-line, so $H=\lambda s$ with $\lambda\in E^\times$. Hence $\det H\equiv D(e)$ modulo norms.

Validation. As a check of formula and conventions, verify.py B1 computes, for 28 random exponent vectors with $5$–$7$ points and $n\in\{5,8,10,12\}$, the signature of $iH$ at every embedding. It uses the exact characteristic polynomial over $E$, the exact sign of $p+q w$ at each real place, and Descartes' rule (exact for real-rooted polynomials). All 112 signatures equal $(h^{0,1}_j,h^{1,0}_j)$. Control B1c: doubling one off-diagonal pair breaks the law.

Results (B2–B4). With $w=\zeta+\zeta^{-1}$:

| $n$ | $e$ | $D=p+qw$ | norm? | |---|---|---|---| | 8 | $(1,1,1,1,2,2)$ | $3/2+\sqrt2$ $=\bigl((2+\sqrt2)/2\bigr)^2$ | yes | | 10 | $(1,1,1,1,3,3)$ | $2+3w$ | yes | | 10 | $(1,1,1,2,2,3)$ | $7/5+11w/5$ | yes | | 10 | $(1,1,2,2,2,2)$ | $3/5+4w/5$ | yes | | 12 | $(1,1,1,1,1,7)$ | $26+15\sqrt3$ | yes | | 12 | $(1,1,1,1,2,6)$ | $14+8\sqrt3$ | yes | | 12 | $(1,1,1,1,4,4)$ | $7+4\sqrt3$ | yes | | 12 | $(1,1,1,2,2,5)$ | $7+4\sqrt3$ | yes | | 12 | $(1,1,1,2,3,4)$ | $(9+5\sqrt3)/2$ | no | | 12 | $(1,1,2,2,3,3)$ | $2+\sqrt3$ | yes |

For the "yes" rows, verify.py stores $x\in F$ and $s\in E$ and checks $D=x\bar x s^2$ exactly (B3). For the "no" row: $N_{E/\mathbb Q}(D)=3/2$, so $v_{(\sqrt3)}(D)=1$. The residue field at $(\sqrt3)$ is $\mathbb F_3$, where $-1$ is a non-square. So the tame symbol is $(D,-1)_{(\sqrt3)}=(-1)^{(3-1)/2}=-1$, and $D\notin N(F_{(\sqrt3)}^\times)$, as $F=E(i)$. Control B4c: $3+\sqrt2$ is a totally positive non-norm from $\mathbb Q(\zeta_8)$, by the same test at the prime above $7$. This proves Theorem II.2(2). $\square$

Remark II.6.4. The non-descending family is not an artefact. For every $a$, the $\mathbb Q$-Hodge structure $\mathbb B_a$ has $h^{2,0}=2$, and its Hodge endomorphisms contain the quaternion algebra generated by $F$ and $\Phi$, i.e. $(F/E,D)$, which is a division algebra because $D$ is not a norm. It is the transcendental part of no K3 surface. The descent obstruction is the relative analogue of the discriminant condition for abelian fourfolds of Weil type ($E=\mathbb Q$; compare [29]).

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II.7. Theorem II.3: the families

II.7.1 Variation of Hodge structure

$W_a$ and $h$ are flat in $a\in M$, and $\chi$ is constant (fixed curves $X_1,X_2$). The antilinear map $\Phi$ of Theorem II.2 is built from $h$ and the wedge pairing $\bigwedge^2_FW\times\bigwedge^2_FW\to\det_FW$; the latter becomes an $F$-valued flat pairing only after a flat trivialization of $\det_FW$. The monodromy of $\det_FW$ is the character $\delta(g)=\det_F(g)$. Each $\det_F(g)$ is a unit of $\mathcal O_F$ (the monodromy preserves a lattice) of absolute value $1$ at every complex place (it preserves $h$), hence a root of unity by Kronecker's theorem; explicitly the Picard–Lefschetz twists have $\det T_k=\zeta^{e_k+e_{k+1}}$ (check C2). So $\delta$ has finite order, and over the finite étale cover $M'\to M$ with $\pi_1(M')=\ker\delta$ the wedge pairing is flat with values in $F$. There $\Phi$ is flat, $\Phi^2=1$ after one global rescaling, and $\mathbb T=\ker(\Phi-1)$ is a polarizable sub-variation of $\mathbb B|_{M'}$ of K3 type and rank $12$ with $E$ acting flatly. Over $M$ the descent fails as a local system: for $n=8$, $e=(1,1,1,1,2,2)$, $\operatorname{Tr}_F\bigwedge^2T_0=3+3i\notin E$ (check F1 reproduces this). For $g\in\ker\delta$ preserving $h$, with eigenvalues $\lambda_1,\dots,\lambda_4$ in a complex embedding, $\overline{e_2(\lambda)}=e_2(\lambda^{-1})=e_2(\lambda)/e_4(\lambda)=e_2(\lambda)$, so $\operatorname{Tr}\bigwedge^2g$ is fixed by the CM conjugation, as it must be (check F2). Genericity (Proposition II.7.1 and §II.7.3) is checked again for the subgroup generated by the powers $T_k^{\operatorname{ord}\delta(T_k)}$ and the commutators $[T_i,T_j]$, all in $\ker\delta$: the algebras they generate on $W$ and on $\bigwedge^2_FW$ have dimensions $16$ and $36$ (check F2), so the conclusions of §II.7.3 hold over $M'$. This proves (1); (2)–(4) are fibrewise or local and are unaffected.

II.7.2 Period map (three moduli)

The Hodge filtration of $\mathbb B$ at $\tau_{j_0}$ is $F^2=\bigwedge^2F^1W_{j_0}$, which is the Plücker image of the 2-plane $F^1W_{j_0}\subset W_{j_0}$. At $\tau_{\pm1}$ the filtration is constant, since the types are pure. The map from the Grassmannian domain of $\mathrm U(2,2)$ (dimension $4$) to the type-IV domain of $\mathrm{SO}(2,4)$ (dimension $4$) is the accidental local isomorphism (E3). So the rank of the period map of $\mathbb T$ equals that of $a\mapsto F^1W_{j_0}$.

By the standard description of the derivative for families of cyclic covers ([34, §3], [32]), its codifferential at $a$ is the multiplication map $$H^0(K_C)_{k}\otimes H^0(K_C)_{-k}\longrightarrow H^0(K_C^{\otimes2})^{\mu_n}\cong H^0\bigl(\mathbb P^1,\Omega^{\otimes2}(a_1+\dots+a_6)\bigr),$$ where $k$ corresponds to $j_0$. The target is 3-dimensional; it is the cotangent space of $M/\mathrm{PGL}_2$. The forms are $x^m P_k\,dx/y^k$ and $x^{m'}P_{-k}\,dx/y^{n-k}$, $m,m'\in\{0,1\}$. Since $\lfloor ke_i/n\rfloor+\lfloor(n-k)e_i/n\rfloor=e_i-1$, their products are $x^{m+m'}dx^2/\prod_i(x-a_i)$. These span $\{1,x,x^2\}\cdot dx^2/\prod(x-a_i)$, the whole target. The check C1 uses symbolic $a_i$, for all nine families. The compact place contributes nothing, since $h^{1,0}h^{0,1}=4\cdot0$ (C1c). So the period map is an immersion of $M/\mathrm{PGL}_2$, of rank $3$, into the 4-dimensional domain. This proves (2).

II.7.3 Monodromy and genericity

For adjacent points $a_k,a_{k+1}$, the full twist of $a_{k+1}$ around $a_k$ fixes the classes supported outside a disc containing both, which is the hyperplane $\Delta_k^\perp$. It multiplies $\Delta_k$ by $c_kc_{k+1}$ (or its inverse; the generated group is the same). Since $H$ is invariant, this is the transformation $$T_k(x)=x-(c_k-1)(c_{k+1}-1)\,H(x,\Delta_k)\,\Delta_k,$$ a complex reflection when $H(\Delta_k,\Delta_k)\ne0$, using the value of $H_{kk}$ ([32, §9], [33]). verify.py C2 checks exactly that each $T_k$ preserves $H$ and has determinant $c_kc_{k+1}$.

Proposition II.7.1. For each of the nine descending families, the $F$-algebra generated by $T_0,\dots,T_3$ is $\mathrm{End}_F(W)$ (dimension $16$), and the one generated by $\bigwedge^2T_0,\dots,\bigwedge^2T_3$ is $\mathrm{End}_F(\bigwedge^2_FW)$ (dimension $36$). In particular the monodromy of $\mathbb B$ acts absolutely irreducibly on $\mathbb B\otimes_{F,\tau}\mathbb C$ for every $\tau$.

Proof. We reduce modulo a prime $\mathfrak p\mid p$, $p\equiv1\pmod n$, with $\zeta\mapsto g$ of order $n$ in $\mathbb F_p$ ($p=17,11,13$ for $n=8,10,12$). The entries are $\mathfrak p$-integral, since only the $\zeta^a-1$ are inverted. The spans of words in the reduced generators have dimensions $16$ and $36$ over $\mathbb F_p$ (C2). Ranks cannot increase under reduction, so the dimensions over $F$ are at least, hence equal to, $16$ and $36$. A second ordering of the points gives the same result (C2b).

Positive control (C2c): for $n=2$, $e=(1^6)$ (genus-two curves, monodromy in $\mathrm{Sp}_4$), the same procedure gives dimension $26=25+1$ on $\bigwedge^2$, detecting the invariant symplectic form. $\square$

Proof of Theorem II.3(3). Let $V_a\subset\mathbb B_a$ be the space of Hodge classes for very general $a$, i.e. outside the countably many proper Hodge loci of flat sections of $\mathbb B$ over the universal cover. A class in $V_a$ remains Hodge along every path, so $V_a$ is a monodromy-stable $F$-subspace. By Proposition II.7.1, $V_a=0$ or $V_a=\mathbb B_a$. The latter is absurd, since $h^{2,0}(\mathbb B_a)=2$. So $V_a=0$, and $\mathbb T_a\subset\mathbb B_a$ has no Hodge classes. Endomorphisms: if $\mathrm{End}_{\rm Hdg}(\mathbb T_a)=E'\supsetneq E$, then by (I1) either $E'$ is totally real quartic with $\dim_{E'}=3$ (period domain of dimension $1$), or $E'$ is CM of degree $4$ with $\dim_{E'}=3$ (a ball of dimension $2$), or CM of degree $12$ (dimension $0$). Each such locus is a countable union of analytic subsets of dimension $\le2$ of the 4-dimensional domain. Their preimages under the 3-dimensional immersion of §II.7.2 are proper. $\square$

II.7.4 The K3 surfaces

Proof of Theorem II.3(4). Choose a flat polarization $Q$ of $\mathbb T$, by restricting one of $\mathbb B$. Then $(\mathbb T_a,Q)$ is a rational quadratic space of signature $(2,10)$. Let $\Lambda$ be the K3 lattice. A rational quadratic space $V'$ of dimension $10$ and signature $(1,9)$ with $V\oplus V'\cong\Lambda_{\mathbb Q}$ exists: its determinant and local Hasse invariants are prescribed by $\Lambda_{\mathbb Q}$ and $V$, they satisfy the product formula, and every such system is realized in dimension $\ge3$. Embed $V=\mathbb T_a\subset\Lambda_{\mathbb Q}$ and put $\omega=H^{2,0}(\mathbb T_a)$. By surjectivity of the period map there is a K3 surface $S_a$ with $H^2(S_a,\mathbb Z)\cong\Lambda$ and $H^{2,0}=\omega$ (I7). $NS(S_a)_{\mathbb Q}\supseteq V^\perp=V'$ contains a positive class, so $S_a$ is projective. $NS_{\mathbb Q}\cap V$ is the space of Hodge classes of $\mathbb T_a$. If this space vanishes, then $NS_{\mathbb Q}=V'$ and $T(S_a)_{\mathbb Q}=V$, with $\rho=10$. In general $T(S_a)_{\mathbb Q}\subseteq V$. The identity $T\otimes_EF\cong\bigwedge^2_FW_a\otimes\chi=\bigwedge^2_FH^1(P_a)\otimes\chi$ is Theorem II.2(1). $\square$

Remark II.7.2 (minimality). By [Part I, Thm. I.3(3)], a weight-one realization of $T(S_a)$ compatible with $F$ needs pieces of $F$-rank $\ge2^{\lfloor 5/2\rfloor}=4$. $P_a$ has exactly $F$-rank $4$ ($\dim P_a=8$). So $P_a$ is a minimal relative Kuga–Satake factor. By contrast, the Kuga–Satake variety of $S_a$ has dimension $2^{11}$.

Remark II.7.3 (the $m=5$ sublocus). Proposition II.7.1 excludes a Hodge class on all of $M$. Hodge classes of negative square in $\mathbb T_a$ occur on countably many hypersurfaces of the period image (intersections with $v^\perp$). They give two-parameter families with $\dim_ET=5$ (rank $10$). We have not analysed them.

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II.8. Theorem II.4: the conditional determinant theorem

II.8.1 The realization

Let $\pi_W$ be the $\Phi_n(\sigma)$-idempotent on $H^1(C_a)$ and $\pi_{U_i}$ the analogous ones on $H^1(X_i)$. All are polynomials in automorphisms, hence algebraic. Let $s_F$ denote the separability idempotents that cut $\otimes_F$ (and $\otimes_F\bar{\ }$) out of $\otimes_{\mathbb Q}$; they are algebraic [18, §5]. Let $\mathrm{swap}$ be the geometric factor exchange on $C_a\times C_a$. On $H^1\otimes H^1$ it acts as minus the ordinary tensor flip (Koszul sign), so $\alpha=\frac12(1+\mathrm{swap})$ is the ordinary antisymmetrizer and cuts out $\bigwedge^2$. Then $$\mathbb B_a=\textstyle\bigwedge^2_FW_a\otimes_F\chi=\pi_{\mathbb B}\bigl(H^1(C_a)^{\otimes2}\otimes H^1(X_1)\otimes H^1(X_2)\bigr)(1)\subset H^4(Y_a)(1),$$ with $\pi_{\mathbb B}=s_F\circ(\alpha\otimes1)\circ(\pi_W^{\otimes2}\otimes\pi_{U_1}\otimes\pi_{U_2})$ algebraic. Unlike [Part I, Thm. I.4], no Hodge isomorphism $\gamma$ is needed: both factors $W$ come from the same curve, and the twist is carried by CM curves.

II.8.2 Proof of Theorem II.4

Step 0. Let $i=\pi_{\mathbb B}\circ Z_*|_{T(S)}$. It is a nonzero Hodge morphism, hence injective, since $T(S)$ is irreducible. Since $\mathbb B\cong T\otimes_EF$ and $\mathrm{End}_{\rm Hdg}(T)=E$, we have $\mathrm{Hom}_{\rm Hdg}(T,\mathbb B)=F\cdot\iota_0$ for the descent inclusion $\iota_0$. So $i$ is $E$-linear: $i(et)=e\,i(t)$, with $e$ acting on $\mathbb B$ through $\sigma^*$ on $C_a$, algebraically. By (I4), $e|_{T}=i^{-1}\circ e\circ i$ is algebraic, with $i^{-1}$ an algebraic left inverse. This gives (1): Hodge classes in $T\otimes T$ are $E$-multiples of the polarization (I1), hence algebraic, and (I2) without determinants gives $S^n$, $n<6$.

Step 1. Let $\theta=\zeta-\zeta^{-1}$. Then $i(T)\cap\theta i(T)$ is a sub-Hodge structure of the irreducible $i(T)$. It is not all of $i(T)$, since otherwise $i(T)$ would be $F$-stable and $F\subset\mathrm{End}(T)$. So $\mathbb B=i(T)\oplus\theta\,i(T)$, by dimension. The map $T\oplus T\to\mathbb B$, $(x,y)\mapsto i(x)+\theta i(y)$, is an algebraic isomorphism with algebraic inverse. Hence every Hodge class of $\mathbb B\otimes\mathbb B$ is algebraic: the Hodge classes of $T\otimes T$ form the $E$-span of the Casimir $c_T$ of the intersection form, which is the transcendental Künneth part of the diagonal.

Step 2 (the Casimir). The wedge pairing and the multiplication $\chi\otimes\chi\to\chi^{\otimes2}$ give a symmetric $F$-bilinear Hodge pairing $\mathbb B\otimes_F\mathbb B\to\Lambda:=\det_FW\otimes\chi^{\otimes2}$. $\Lambda$ has type $(2,2)$ at every embedding (A7, A10), so by Lemma II.6.2 there is an $F$-linear Hodge isomorphism $\mu\colon\Lambda\to F(-2)$. The Casimir $c$ of $\mu\circ(\text{pairing})$ is a Hodge class in $\mathbb B\otimes_F\mathbb B$, hence algebraic by Step 1. A different $\mu$ changes $c$ by a factor in $F^\times$.

Step 3 (interleaving). Consider $c^{\otimes3}\in\mathbb B^{\otimes6}\subset H^{24}(Y_a^6)(6)$. Write the $i$-th copy of $Y_a$ as $C^{(i)}\times C'^{(i)}\times X_1^{(i)}\times X_2^{(i)}$. Apply in turn $s_F$ on all $C$-slots; the permutation of factors $C'^{(i)}\mapsto C'^{(i+1)}$ ($i$ mod 6), which is algebraic up to a Koszul sign; $\pi_{\mathbb B}$ on each copy; and $a_6s_F$ onto $\bigwedge^6_F\mathbb B$. Every map is algebraic, so the result $\Psi(c)\in\bigwedge^6_F\mathbb B$ is algebraic. We check that it is nonzero at every embedding $\tau$. Over $\tau$, $\mathbb B_\tau=\bigwedge^2W_\tau\otimes\chi_\tau$, and the pairing is the wedge pairing tensored with a nonzero form on the line $\chi_\tau$. So $c_\tau=\kappa\,c_0\otimes(u\otimes u)$, with $\kappa\ne0$, $u$ a basis of $\chi_\tau$ and $c_0$ the volume tensor of $W_\tau$ (D1). The $\chi$-slots are not moved, so they contribute the nonzero factor $u^{\otimes6}$. On the $W$-slots the construction is that of [Part I, Lemma I.8.1] with $U=W_\tau$: the $i$-th $\mathbb B$-factor is a pair ($C$-slot, $C'$-slot), exactly as $B_0$-factors are pairs ($W$-slot, $W'$-slot) there. Hence $\Psi(c)_\tau=-2880\,\kappa^3(\cdots)\ne0$ (D2), while the unshifted pairing gives $0$ (D3).

Step 4. $\bigwedge^6_F\mathbb B$ is an $F$-line, spanned by the algebraic class $\Psi(c)$, and $F$ acts algebraically. So all its elements are algebraic. Via Step 1, $\bigwedge^6_F\mathbb B=\bigwedge^6_F(i(T)\otimes_EF)$. The descent lemma (I5), with constant $2^{-5}$ (D4, control D4c), and pull-back by $i^{-1}$ give (2).

Step 5. (I2) with Steps 0 and 4 gives HC for all $S^n$. Products with $S'$ follow from (I3). $\square$

Remark II.8.1 (what (KS$_a$) is). (KS$_a$) asks for one codimension-3 cycle on the 6-fold $S\times Y_a$. It is the relative, minimal analogue of the Kuga–Satake Hodge conjecture: the full Kuga–Satake variety of $S$ is replaced by the explicit eightfold $P_a$ and two CM curves. By (I3) it is invariant under rational Hodge isometries of $S$. We do not know whether it is easier than the Kuga–Satake conjecture for $S$. It is at least explicit, and has bounded complexity: genus $13$ for $n=8$.

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II.9. Why a geometric construction is hard: Proposition II.5

Proof of Proposition II.5. (1) If $g\sigma g^{-1}=\sigma^k$, then $g^*$ maps $H^{1,0}(C_a)_j$ isomorphically onto $H^{1,0}(C_a)_{k^{-1}j}$ or $H^{1,0}(C_a)_{kj}$. So the signature function $j\mapsto h^{1,0}_j$ is invariant under $j\mapsto kj$. For all ten families the values $(h^{1,0}_1,h^{1,0}_{j_0},h^{1,0}_{-j_0},h^{1,0}_{-1})=(4,2,2,0)$ have no such symmetry for $k\ne1$ (C3). (2) In [25, Prop. 4.1] and its successors, RM by $F^+$ arises from an involution inverting the cyclic deck group on a dihedral cover. By (1), $C_a$ admits none. (For instance, if a surface is the quotient of $C_a\times C'$ by the antidiagonal $\mu_n$ and its transcendental part meets a primitive eigenpiece, the residual automorphism $(\sigma,1)$ acts on the $(2,0)$-part by a primitive $n$-th root of unity; this gives a CM field, not RM.) (3) For $n=4$, $e=(1,1,1,3,3,3)$ one computes $h^{1,0}_{\pm1}=2$, $\det H=1/4$, and a $\bigwedge^2$-monodromy algebra of dimension $36$ (C2d). Descent and genericity then follow as in §§II.6–7. $\square$

So, in this setting, the $E$-structure of $T(S_a)$ is visible only after the descent of Theorem II.2: it comes from the hermitian polarization through Lemma II.6.1, not from an automorphism. This is why we formulate (KS$_a$) as a Kuga–Satake-type hypothesis; Proposition II.5 excludes only the normalizing-automorphism mechanism, not other constructions. For $E=\mathbb Q$ the analogous algebraicity was proved by geometric constructions specific to rank $6$: double planes branched along six lines [30], and generalized Kummer sixfolds [27]. We know of no analogue of these for rank $12$.

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Verification and limitations

I.9. Verification

verify.py runs 48 exact checks (Python integers, Fractions, SymPy; finite-field sums with NumPy integer arrays). It ends with FALLOS = 0 (verify_output.txt).

| Statement | Checks | |---|---| | Dickson identities, evenness, twin | A1–A3, A9 | | Fibre at $\infty$ (smooth; IV$^*$ at $a=0$), 24 I$_1$ at samples, $R_4$ rational | A4–A8 | | $j$ nonconstant; Lemma I.5.1 | A10, B1 | | Jacobian ring $\dim R_{12}=18$, rank 3; scaling control | B2, B3 | | Good reduction; two independent trace formulas agree | C0, C1, C1b | | $\rho=10$ at $(1,1,2,3)$ via $p=17$ | C2–C5 | | Control member $(1,0,1,2)$: forced factor; sign vs. Legendre symbol | C6, C7 | | Casimir = volume; interleaving $-2880$; controls $0$ | D1–D3b | | $\det\bigwedge^2g=\det g^3$; descent constants $2^{1-m}$, $m\le6$; control | D4–D5b | | Contraction; Hodge numbers of $\bigwedge^2W\otimes\chi$; control without twist | D6–D9 | | Spread table for $B_n$, $D_n$, $m\le10$; spin dimensions; $m\le4$ for rank 2 | E1–E5 | | Rank bookkeeping | F1–F3 |

The data of Remark I.5.4 at $p=41,73,89,97,113$ and the $s_4$ values come from a separate FFT-based script. They are not part of verify.py, which is exact, and are not used in any proof.

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I.10. Limitations

• The main gap. For the very general member of $\mathcal S$ we prove HC only for $S^n$, $n\le5$. HC for all powers depends on one class in $\bigwedge^6_{\mathbb Q(\sqrt2)}T$. Theorem I.4 is a conditional criterion: we know no example with $E\ne\mathbb Q$ satisfying its hypotheses. By Theorem I.3, a curve-product approach would require curves whose Jacobians contain 8-dimensional abelian varieties with Mumford–Tate group isogenous to $\mathrm{Res}_{E/\mathbb Q}\mathrm{Spin}(T)$ times a torus. These are relative Kuga–Satake varieties, and they are not known to come from curves.
• The rank of $\mathcal S$ is 12, not 10. The problem as posed suggested $m=5$ (rank 10, three moduli, maximal). $\mathcal S$ has $m=6$ and is a non-maximal three-parameter subfamily of a four-dimensional RM moduli space. The rank-10 members probably form divisors (Remark I.5.5); we did not study them.
• Theorem I.2(1)–(3) are short consequences of Varesco, Zarhin, [25] and [18]. (The dense-locus statement has been withdrawn; see Remark I.6.1.) It covers no new rational Hodge structure.
• The behaviour at $a=0$ (resolved $\mathrm{IV}^*$ fibre) is not analysed; the earlier specialization claims are withdrawn (Remark I.6.1).
• Theorem I.3 is folklore in spirit (Deligne, van Geemen). We claim only the explicit $F$-rank bound and its consequence for [18].
• Imported: [5], [6], [12], [18], [25] and Griffiths–Steenbrink. The Frobenius computation proves $\rho=10$ at one member; the very general statement uses semicontinuity.
• Mixed products of unrelated members are not treated.

II.10. Verification

verify.py (77 exact checks in Part II, including F1–F2; about a minute) ends with FALLOS = 0 (verify_output.txt).

| Statement | Checks | |---|---| | Classification; $n=5$ empty; places; genus 13 | A1–A6 | | Types of $\bigwedge^2W\otimes\chi$; control without twist; CM curves for $\chi$; $\bar\chi\cong\chi\otimes\det W(2)$ | A7–A10 | | Kita–Yoshida signatures = Chevalley–Weil (112 cases, exact); control | B1, B1c | | Skew-hermitian, signatures, discriminant, total positivity (10 families) | B2 | | Norm certificates (9 families); non-norm by tame symbol; control | B3, B4, B4b, B4c | | Lemma II.6.1 ($\phi^2=\det$); control | B5, B5c | | Period map rank 3 (symbolic $a_i$); compact factor rigid | C1, C1c | | Monodromy algebras $16$, $36$ (9 families, two orderings); genus-2 control $26$; $E=\mathbb Q$ shadow | C2–C2d | | No automorphism conjugating $\sigma$ | C3 | | Volume tensor, $-2880$, controls; descent constants, control | D1–D4c | | Bookkeeping | E1–E3 |

Use of floating point. Exact arithmetic decides every verdict. Floats are used only (i) to say which root of $x^2-px-q$ (roots at distance $\ge1$) is the value of $w$ at a given embedding, and (ii) to propose square roots in $E$, which are then verified exactly.

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II.11. Limitations

• No unconditional HC for powers. The main theorem is conditional on (KS$_a$). We give no K3 surface with $E\ne\mathbb Q$ and $m\ge5$ for which the determinant class is proved algebraic.
• The K3 surfaces are abstract. $S_a$ is obtained from the period map, not from an equation. We do not know whether the three-parameter family of $\mathbb Q(\sqrt2)$-structures of Theorem II.3 meets the explicit family of [Part I, Thm. I.1] or the four-dimensional family of van Geemen–Schütt [4].
• Imported topology. The identification of twisted homology with $W_a$, the Kita–Yoshida numbers and the Picard–Lefschetz description of adjacent twists are quoted ([32], [33], [35]). Their joint correctness is supported by the exact signature validation B1 and the invariance and determinant checks C2, not re-proved.
• The codifferential formula for the period map of cyclic covers is quoted [34].
• Genericity is "very general", not explicit. We do not exhibit one parameter $a$ with $m=6$ proved, for instance by reduction modulo $p$.
• Non-maximality. Each family has 3 moduli inside a 4-dimensional RM moduli space.
• Theorem II.4 is close to [Part I, Thm. I.4]. The new points are the explicit realization without $\gamma$ and the verification that all its Hodge-theoretic hypotheses hold.

Reproducibility

verify.py is one self-contained Python 3 script (no __file__, no files read or written). It runs the Part I checks (48) and the Part II checks (77) in separate namespaces and reports the total. Versions: Python 3.13.16, SymPy 1.14.0, NumPy 2.5.3. Output: verify_output.txt, ending FALLOS = 0.

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