Balanced Cyclic Prym Families and the Generalized Hodge Conjecture for All Powers

Hongmin Li (Institute of Science Tokyo; The University of Tokyo). Developed with OpenAI Codex and GPT Pro under the author's direction. Submitted through the rStation account lihongmin.

1. Introduction

Exceptional Hodge classes on an Abelian variety often arise from an action by a CM field. If $F$ acts on $H^1(B,\mathbf Q)$ and this vector space has $F$-rank $n$, the determinant space

$$ \bigwedge_F^n H^1(B,\mathbf Q) $$

is a natural source of Hodge classes which need not be generated by divisors [10]. Their algebraicity is the essential extra input in many Weil-type instances of the Hodge conjecture. The problem has two logically independent parts: one must construct cycles representing the determinant classes, and one must control the Hodge group well enough to show that divisors and determinants generate every Hodge class on every self-product.

We first isolate a criterion which separates the cycle construction from the monodromy argument, and then apply it both to a multi-standard class of ramified cyclic Prym families and to a $2$-standard contraction class. These applications include families of unbounded dimension and unbounded moduli dimension even at fixed conductor. To introduce the first basic subfamily, fix an odd prime $p\geq5$, let $M_{0,p+1}$ be the moduli space of $p+1$ ordered distinct points on $\mathbf P^1_{\mathbf C}$, and, for $d\geq3$, put

$$ m=pd,\qquad n=p-1,\qquad q=\frac{p-1}{2}, \qquad F=\mathbf Q(\zeta_m), \qquad \mathbf a_{p,d}=(1,1+d,\ldots,1+(p-1)d,m-p). \tag{1.1} $$

For $\mathbf b=(b_0,\ldots,b_p)\in M_{0,p+1}(\mathbf C)$, choose linear forms $\ell_i$ vanishing at $b_i$, and let $C_{p,d,\mathbf b}$ be the smooth normalization of

$$ y^m=\prod_{i=0}^p\ell_i(s,t)^{a_i(p,d)}. \tag{1.2} $$

Let $e_m\in\mathbf Q[\mu_m]$ be the central idempotent cutting out the exact cyclotomic factor. If $N_me_m\in\mathbf Z[\mu_m]$, define

$$ B_{p,d,\mathbf b} :=\operatorname{Im}\bigl(N_me_m:J(C_{p,d,\mathbf b}) \longrightarrow J(C_{p,d,\mathbf b})\bigr). \tag{1.3} $$

Its isogeny class is independent of $N_m$. We call it the exact-conductor Prym factor of (1.2).

The main theorem

To state the criterion, let $n=2q\geq4$, let $m\geq3$, and let

$$ \mathbf a=(a_0,\ldots,a_{n+1}), \qquad 0<a_i<m, \qquad \sum_i a_i=sm. $$

We impose three conditions:

• (E) Exactness: $\gcd(m,a_0,\ldots,a_{n+1})=1$.
• (B) Balance: for every $t\in(\mathbf Z/m\mathbf Z)^\times$,$$ \sum_i\left\langle\frac{ta_i}{m}\right\rangle=q+1. $$
• (S) Source algebraicity: on the degree-$m$ Fermat $n$-fold, the primitive middle character line $V(t\mathbf a)$ is algebraic for every $t\in(\mathbf Z/m\mathbf Z)^\times$.

For ordered distinct branch points $\mathbf b\in M_{0,n+2}$, let $C_{\mathbf a,\mathbf b}$ be the corresponding cyclic cover and let $B_{\mathbf a,\mathbf b}=e_mJ(C_{\mathbf a,\mathbf b})$ denote its exact-conductor factor up to isogeny.

Theorem 1.1 (balanced exact cyclic-Prym criterion). Assume $(E),(B),(S)$.

1. For every $\mathbf b$,

   $$ \dim_FH^1(B_{\mathbf a,\mathbf b},\mathbf Q)=n, \qquad \operatorname{sig}_\sigma H^1(B_{\mathbf a,\mathbf b})=(q,q), \tag{1.4} $$

   and

   $$ W_F(B_{\mathbf a,\mathbf b}) :=\bigwedge_F^nH^1(B_{\mathbf a,\mathbf b},\mathbf Q) \subset \operatorname{cl}CH^q(B_{\mathbf a,\mathbf b})_{\mathbf Q}. \tag{1.5} $$

2. Outside a countable union $\Sigma_{\mathbf a}$ of proper closed algebraic subsets of $M_{0,n+2}$,

   $$ \operatorname{End}^0(B_{\mathbf a,\mathbf b})=F, \qquad Hg(B_{\mathbf a,\mathbf b})= \operatorname{Res}_{F^+/\mathbf Q}SU_F \bigl(H^1(B_{\mathbf a,\mathbf b},\mathbf Q),h\bigr), \tag{1.6} $$

   and, for every $r\geq1$, both the rational Hodge conjecture and the generalized Hodge conjecture hold for $B_{\mathbf a,\mathbf b}^r$.

The fibrewise assertion has no genericity hypothesis. Special fibres may have extra endomorphisms, and we make no all-powers assertion for every special fibre.

Theorem 1.2 (standard-prime family). For every odd prime $p\geq5$ and every $d\geq3$, the tuple $\mathbf a_{p,d}$ in (1.1) satisfies $(E),(B),(S)$. Consequently Theorem 1.1 applies. In particular,

$$ \dim B_{p,d,\mathbf b}=q\varphi(pd), $$

and the rank, Abelian dimension, and family dimension $p-2$ are unbounded. The member of smallest conductor occurs at $p=5,d=3$; it has conductor $15$ and dimension $16$.

Theorem 1.3 (a non-single-prime-standard juxtaposed family). Let $\ell\geq5$ be a prime with $\ell\equiv2\pmod3$, let $d\geq1$, and put

$$ m=3\ell d, \qquad n=\ell+3, \qquad q=\frac{\ell+3}{2}. $$

The juxtaposed tuple

$$ \mathbf a_{\ell,d} =( 1,1+\ell d,1+2\ell d,m-3; 1,1+3d,\ldots,1+3(\ell-1)d,m-\ell ) \tag{1.7} $$

satisfies $(E),(B),(S)$, so Theorem 1.1 applies. These tuples are not single-prime-standard, even after permutation and multiplication by a unit. Their rank $\ell+3$ and family dimension $\ell+2$ are unbounded. Indeed, Dirichlet's theorem gives infinitely many primes $\ell\equiv2\pmod3$.

The two preceding applications belong to a larger class. Let $*$ denote juxtaposition of residue tuples.

Theorem 1.4 (finite multi-standard families). Let $k\geq1$, and let $p_1,\ldots,p_k$ be odd primes, with repetitions allowed, such that $p_j\mid m$. Put

$$ d_j=\frac{m}{p_j}>2, \qquad \alpha_{p_j,m} =\bigl(1,1+d_j,\ldots,1+(p_j-1)d_j,m-p_j\bigr), $$

and

$$ \mathbf a=\alpha_{p_1,m}*\cdots *\alpha_{p_k,m}, \qquad n=\sum_{j=1}^k(p_j+1)-2=2q. $$

Assume $n\geq4$. Then $\mathbf a$ satisfies $(E),(B),(S)$, so Theorem 1.1 applies. No distinctness assumption on the primes is required.

Theorem 1.5 (two-standard contraction families). Let $d>2$, put $m=2d$, and choose $1\leq i<d$ with $\gcd(i,d)=1$. With entries read modulo $m$, set

$$ \eta_{d,i}=(i,i,i+d,i+d,-2i,-2i). \tag{1.8} $$

For every $k\geq1$, the $k$-fold juxtaposition $\eta_{d,i}^{*k}$ satisfies $(E),(B),(S)$, with

$$ n=6k-2,\qquad q=3k-1. $$

Consequently Theorem 1.1 applies. The length-six tuple $\eta_{d,i}$ is not one of the multi-standard tuples in Theorem 1.4. In the smallest case $d=3,i=1$,

$$ m=6,\qquad \eta_{3,1}^{*k}=(1^{2k},4^{4k}), \tag{1.9} $$

and no block allowed in Theorem 1.4 exists at conductor six. Thus the entire family (1.9) is genuinely outside the odd-prime multi-standard class, while

$$ \dim_FH^1(B_k)=6k-2,\qquad \dim B_k=6k-2,\qquad \dim M_{0,6k}=6k-3 $$

are simultaneously unbounded with $F=\mathbf Q(\zeta_6)=\mathbf Q(\sqrt{-3})$ fixed.

Corollary 1.6 (odd-prime fixed-conductor families). Fix an odd prime $p\mid m$ with $m/p>2$, and juxtapose $k$ copies of $\alpha_{p,m}$. For every positive integer $k$ such that

$$ n_k=k(p+1)-2\geq4, $$

the resulting exact Prym family satisfies Theorem 1.1, while

$$ \dim_FH^1(B_k)=n_k, \qquad \dim B_k=\frac{n_k}{2}\varphi(m), \qquad \dim M_{0,n_k+2}=n_k-1. $$

Thus rank, Abelian dimension, and Hurwitz-base dimension are simultaneously unbounded with both the conductor $m$ and the cyclotomic field $F=\mathbf Q(\zeta_m)$ fixed.

Theorem 1.7 (the $A_1$ boundary). For $p=3,d\geq3$, the fibrewise conclusion of Theorem 1.2 holds with $n=2,q=1$, and every power of a very general exact-conductor factor satisfies the rational and generalized Hodge conjectures. The generic endomorphism algebra is not $F$. If $d\ne4$, it is a possibly split totally indefinite quaternion algebra over $F^+$ with $F$ as a maximal subfield. If $d=4$, the two real monodromy factors form one graph and the generic commutant is a degree-four central simple algebra over $\mathbf Q$. In both cases the simple isogeny factors have Albert type I or II.

Corollary 1.8 (prime-power conductors). Let $m$ be a prime power and let $n=2q\geq4$. Every residue tuple $\mathbf a$ satisfying $(E),(B)$ satisfies all conclusions of Theorem 1.1; no separate source-algebraicity hypothesis is required.

Structure of the proof

The criterion has two branches. The algebraic-cycle branch is fibrewise. For a general exact cyclic cover with $n+2$ branch points, we construct a common finite quotient

$$ C^n/(N\rtimes\mathfrak S_n)\simeq Z_{\mathbf c}/K_{\mathbf a}, \tag{1.10} $$

where $Z_{\mathbf c}$ is a Delsarte $n$-fold. The top-dimensional fibre product gives a Chow correspondence with a normalized left inverse on the determinant block. Hypothesis $(S)$ makes the whole rational source block algebraic. The Poincaré divisor on $C\times B$, its $n$-th external power, and the small diagonal of $B$ then carry these classes to $W_F(B)$.

The group-theoretic branch is generic. Moving the branch point of exponent $a_0$ produces rank-$n$ cyclotomic Jordan--Pochhammer representations. Rohde's theorem gives $SU(q,q)$ on every real factor. A characteristic-zero Goursat argument reduces independence to projective equivalence of two Jordan--Pochhammer systems. A nonresonant generator has a distinguished multiplicity-$(n-1)$ eigenspace, while a resonant generator is a nontrivial unipotent pseudoreflection; in either case its normalized lift removes every projective scalar. The ordered generator product and exactness then recover the full residue tuple, excluding a graph between distinct real factors. The determinant Hodge classes then squeeze the generic Hodge group to $\operatorname{Res}SU$. The first fundamental theorem for $SL_n$ shows that contractions and determinant tensors generate the Hodge classes on every power. Finally, Abdulali's theorem upgrades the all-powers usual Hodge conjecture to the generalized Hodge conjecture.

For each multi-standard block, Aoki's standard-cycle theorem verifies the source condition, and the Shioda--Ran juxtaposition theorem recalled by Aoki may be iterated finitely. Theorem 1.5 starts from Aoki's explicit $2$-standard surface cycle, juxtaposes two copies, and applies the Shioda--Ran deletion theorem to remove a self-opposite pair before iterating. For $p=3$, the self-duality of the $SL_2$ standard representation instead yields a quaternionic commutant; the $SL_2$ first fundamental theorem and Hazama's result close that boundary separately.

Relation to earlier work

Schoen's simple-tuple method [13] isolates a selected determinant subspace $U$ in self-products of varieties with cyclic automorphisms; it is not a statement that the whole Hodge ring is algebraic. The tuples (1.1) are never simple: their entries cannot be paired with their negatives. The juxtaposed tuples (1.7) lie in the semigroup generated by two Aoki-standard blocks but are not themselves single-prime-standard. Their source classes come from Aoki's standard-cycle theorem and the Shioda--Ran juxtaposition theorem as recalled by Aoki [2]. Finite iteration of juxtaposition is already built into Aoki's standard semigroup, and his later work proves all-self-product Hodge-conjecture results for CM Abelian varieties of Fermat type [3] and generalized Catalan Jacobians [21]. We therefore make no novelty claim for finite standard juxtaposition, the standard-semigroup mechanism, or all-powers Hodge results in those fixed CM settings. Those results do not apply directly to our moving exact Prym factors, whose branch points are arbitrary and whose $F$-ranks are unbounded.

Schoen also studied cyclic covers of projective space branched along $v+2$ hyperplanes and applications to the generalized Hodge conjecture [14]. His Theorem 3.4 proves the generalized Hodge conjecture for a selected determinant substructure $U'$ of a generalized Prym under the quasi-simple branching-data condition, by relating it to a Fermat hypersurface. We therefore make no novelty claim for the primitive-factor/Fermat-correspondence mechanism by itself.

For the covers in this paper, Schoen's reduction is closer than the quasi-simple case alone suggests. In his notation, if $q_X$ is the genus of the quotient curve, $s$ is the number of branch points, and $h=2q_X-2+s$, then one may take $r=1$ when $X=\mathbf P^1$. There are then no preliminary inverse pairs, the remaining $v+2=s$ exponents generate the unit ideal precisely by condition $(E)$, and

$$ v=h-(2q_X+2r-2)=s-2=n. $$

Condition $(B)$ makes every primitive eigenspace multiplicity equal to $n/2$, so the determinant Hodge width is $w=0$. Schoen's Remark 3.5 states that, when $w<v$, it suffices to prove the corresponding conjecture for the degree-$m$ Fermat $v$-fold. In the present width-zero situation, condition $(S)$ supplies exactly the required algebraic primitive orbit. Consequently the fibrewise determinant conclusion of Theorem 5.1 is already contained in Schoen's reduction. Sections 3--5 give a self-contained exact-conductor Chow-correspondence proof, including the rational-orbit projector, but we make no novelty claim for that conclusion.

The published chapter is not available as a complete open-text copy. The comparison above uses the official Google Books preview and searchable text for print pp. 137, 139--142, and 144--147, together with Schoen's author-written review; it is therefore limited to the theorem, formula, and remark-level statements just cited. Gordon's survey [19, Section 10.3] describes a resulting application to Weil fourfolds with $K=\mathbf Q(i)$ and signature $(3,1)$; Abdulali [1, Introduction and Section 4.2] records this and related consequences. To the best of our knowledge, no previous result combines Schoen's fibrewise reduction with a repeated-factor monodromy-independence argument for moving exact Pryms, determines the generic Hodge group and endomorphism algebra in these balanced families, and proves the Hodge and generalized Hodge conjectures for every self-power. Our new claims are the repeated-factor independence theorem, the generic group and commutant, its application to the moving multi-standard and contraction families, and the resulting all-powers conclusions. We make no novelty claim for the Fermat correspondence, the fibrewise determinant transfer, Aoki's source cycles or semigroup, the Shioda--Ran deletion operation, or the PEL-type upgrade in isolation.

Kang [23, Theorem 3.1 and Corollary 3.2] gives a published claim of the generalized Hodge conjecture for Fermat varieties of dimensions three and four in arbitrary degree. We do not use that result as an input, and we do not claim priority for an all-degree Fermat theorem. The independent status of Kang's filtration/product estimate remains a separate question; the present paper's claims concern moving ramified exact Prym factors and their all-self-power transfer.

Generic Prym results of Biswas--Paranjape [5] and Biswas [4] concern loci on which the Hodge ring is generated by divisors. Our cyclic-cover loci carry a fixed cyclotomic action and have additional determinant classes. Van Geemen--Verra used quaternionic covers [16] to obtain algebraic Hodge classes on a family of Abelian eightfolds. In the higher-rank families, the generic endomorphism algebra is the relevant cyclotomic field, and the dimensions are unbounded. The separate $p=3$ boundary explains exactly where quaternionic endomorphisms reappear. Recent geometric constructions of Markman [9] and of Floccari--Fu [7] concern other Weil-type regimes. General constraints on Hodge groups of balanced Hodge structures are studied by Flapan [6]; the equality needed here instead comes from the cyclic-cover monodromy. Patel--Zhang [20] construct cycles for generalized Pryms of finite Abelian étale covers over bases of genus at least two. Their theorems do not address ramified covers, the full Hodge ring, or all powers. Rains [11] identifies the one-parameter Jordan--Pochhammer systems and develops arithmetic largeness and pairwise-surjectivity results. Athira E. V. and Haridas [22] give a geometric mixed-braid description for arbitrary ramification partitions and, when $t^{k_\infty}\ne1$, identify the reduced representation with an evaluated multivariate Burau representation. Neither work supplies the characteristic-zero projective graph-exclusion across the repeated $SU(q,q)$ factors used here, or connects that calculation with algebraic Hodge classes and the generic Hodge group. Rohde [12] supplies factorwise monodromy, and Abdulali [1] supplies the final generalized-Hodge implication once the all-powers usual Hodge conjecture and the PEL condition have been established.

Recent work on rStation [24] proves all-powers Hodge-conjecture results for very general quaternionic and Weil-type Pryms using unramified non-Abelian covers of curves of genus at least two. The present covers are ramified cyclic covers of $\mathbf P^1$, and the main new group-theoretic claim concerns independence of repeated primitive unitary factors in their moving branch families. The invariant-theoretic passage from algebraic determinant classes to all powers is shared background. Our earlier fixed-model paper [25] concerns a particular quartic-field Prym eightfold and retains its quotient hypotheses H1--H4. Here the theorem concerns the exact cyclotomic factor of a family, with all-powers assertions only on the very-general locus. No assertion in this paper removes the fixed-model hypotheses of [25]. The literature comparison was refreshed on October 22, 2026.

The paper is organized as follows. Section 2 defines the exact factors and computes their signatures. Section 3 proves the universal common-quotient theorem. Section 4 uses Aoki's cycles to establish source algebraicity. Section 5 transfers the determinant classes to the Prym. Section 6 proves independence of the monodromy factors and derives the very-general Hodge group. Section 7 proves the all-powers Hodge theorem, and Section 8 applies Abdulali's criterion. Section 9 treats $p=3$, and Section 10 records the scope and the first examples.

2. Exact-conductor Pryms and balanced signatures

Fix $p\ge5,d\geq3$, put $m=pd$, and abbreviate $C=C_{p,d,\mathbf b}$, $B=B_{p,d,\mathbf b}$. Since the first entry of $\mathbf a_{p,d}$ is $1$,

$$ \gcd(m,a_0(p,d),\ldots,a_p(p,d))=1. $$

Thus (1.2) is a connected exact degree-$m$ cover. The exact central idempotent in $\mathbf Q[\mu_m]$ cuts out the direct sum of the eigenspaces indexed by $t\in(\mathbf Z/m\mathbf Z)^\times$. The image construction (1.3) gives an Abelian subvariety up to isogeny, and

$$ H^1(B,\mathbf Q)=e_mH^1(C,\mathbf Q). \tag{2.1} $$

This equality means the projector-induced isomorphism of rational Hodge structures; it is not an identification obtained from an arbitrary abstract isomorphism of vector spaces.

More generally, let $\mathbf a$ have length $n+2=2q+2$ and satisfy $(E),(B)$. For every primitive $t$, the Chevalley--Weil formula gives

$$ h_t^{0,1} =-1+\sum_i\left\langle\frac{ta_i}{m}\right\rangle=q. \tag{2.2} $$

Since no $ta_i$ is zero modulo $m$, the corresponding sum for $-t$ is $(n+2)-(q+1)=q+1$. Hence $h_t^{1,0}=q$ as well. Thus $(B)$ is exactly the balanced-signature condition needed in Theorem 1.1.

Proposition 2.1. The $F$-vector space $H^1(B,\mathbf Q)$ has rank $n=p-1$. Every primitive embedding has signature $(q,q)$.

Proof. Let $t\in(\mathbf Z/m\mathbf Z)^\times$, represented by $1\leq t<m$, and let $r\in\{1,\ldots,d-1\}$ be its least positive residue modulo $d$. Multiplication by $t$ permutes the $p$ congruence classes modulo $p$, so the least positive residues of

$$ t(1+jd),\qquad 0\leq j\leq p-1, $$

are, in some order,

$$ r,r+d,\ldots,r+(p-1)d. $$

The least positive residue of $t(pd-p)$ is $p(d-r)$. Therefore

$$ \sum_{i=0}^p\langle ta_i(p,d)\rangle_m =pr+d\frac{p(p-1)}2+p(d-r) =m\frac{p+1}{2}. \tag{2.3} $$

The Chevalley--Weil formula gives

$$ h_t^{0,1} =-1+\sum_i\left\langle\frac{ta_i(p,d)}m\right\rangle=q, \qquad h_t^{1,0}=q. \tag{2.4} $$

The two Hodge indices may be interchanged by changing the deck-character convention. In either convention, the $t$-eigenspace has complex dimension $n=2q$ and signature $(q,q)$. Running through the primitive embeddings of $F$ proves the assertion. $\square$

It follows that

$$ \dim_{\mathbf Q}H^1(B,\mathbf Q)=n\varphi(m), \qquad \dim B=q\varphi(m). \tag{2.5} $$

The canonical polarization of $J(C)$ restricts, up to isogeny, to a polarization on $B$. The Rosati involution sends the deck transformation to its inverse; hence its restriction to $F$ is complex conjugation. If $F^+$ is the maximal real subfield, the polarization is encoded by a nondegenerate $F/F^+$-anti-Hermitian form $h$ on $H^1(B,\mathbf Q)$.

Finally, since each determinant line has Hodge type $(q,q)$,

$$ W_F(B)=\bigwedge_F^nH^1(B,\mathbf Q) \subset H^n(B,\mathbf Q) \tag{2.6} $$

is a rational Hodge substructure consisting entirely of Hodge classes.

The second family has the same balance for a different reason.

Proposition 2.2. Let $\ell,d,m,n,q$ and $\mathbf a_{\ell,d}$ be as in Theorem 1.3. The exact factor $B_{\ell,d,\mathbf b}$ has $F$-rank $n=\ell+3$, and every primitive embedding has signature $(q,q)$.

Proof. Let $t\in(\mathbf Z/m\mathbf Z)^\times$. If $r_3$ is the least positive residue of $t$ modulo $\ell d$, multiplication by $t$ turns the first block, up to permutation, into

$$ (r_3,r_3+\ell d,r_3+2\ell d,-3r_3), $$

whose least-positive-residue sum is $2m$. If $r_\ell$ is the least positive residue of $t$ modulo $3d$, the second block becomes

$$ (r_\ell,r_\ell+3d,\ldots,r_\ell+3(\ell-1)d,-\ell r_\ell), $$

whose residue sum is

$$ \ell r_\ell+3d\frac{\ell(\ell-1)}2 +\ell(3d-r_\ell)=\frac{\ell+1}{2}m. $$

The total normalized sum is therefore

$$ 2+\frac{\ell+1}{2}=\frac{\ell+5}{2}=q+1. $$

Condition $(B)$ and the preceding general calculation give the result. The entry $1$ gives $(E)$. $\square$

The same calculation is additive under an arbitrary finite juxtaposition.

Proposition 2.3. Let $p_1,\ldots,p_k$, $m$, $d_j$, $\alpha_{p_j,m}$, and $\mathbf a$ be as in Theorem 1.4. Then $\mathbf a$ satisfies $(E),(B)$. Its exact factor has $F$-rank

$$ n=\sum_{j=1}^k(p_j+1)-2, $$

and every primitive embedding has signature $(n/2,n/2)$.

Proof. Each block contains the entry $1$, so exactness is immediate. Let $t\in(\mathbf Z/m\mathbf Z)^\times$, and let $r_j\in\{1,\ldots,d_j-1\}$ be its least positive residue modulo $d_j$. Write $t=r_j+c_jd_j$. Modulo $p_j$, the coefficient of $d_j$ in $t(1+s d_j)$ is $c_j+ts$; because $t$ is a unit modulo $p_j$, this permutes the $p_j$ progression terms of the $j$-th block. Their least positive residues are

$$ r_j,r_j+d_j,\ldots,r_j+(p_j-1)d_j, $$

and the final entry has least positive residue $p_j(d_j-r_j)$. Hence the residue sum of this block is

$$ p_jr_j+d_j\frac{p_j(p_j-1)}2+p_j(d_j-r_j) =m\frac{p_j+1}{2}. $$

Summing over the blocks gives

$$ \sum_i\left\langle\frac{ta_i}{m}\right\rangle =\sum_{j=1}^k\frac{p_j+1}{2} =\frac{n+2}{2}=\frac n2+1, $$

which is $(B)$. The signature statement follows from the general Chevalley--Weil calculation preceding Proposition 2.1.

Since every $p_j$ is odd, $n$ is even. $\square$

The contraction family has a different balance calculation.

Proposition 2.4. Let $d,i,m,\eta_{d,i}$ be as in Theorem 1.5. For every $k\geq1$, the tuple $\eta_{d,i}^{*k}$ satisfies $(E),(B)$. Its exact factor has $F$-rank $6k-2$, and every primitive embedding has signature $(3k-1,3k-1)$. The tuple $\eta_{d,i}$ is not contained in Theorem 1.4; for $m=6$, neither is any $\eta_{3,1}^{*k}$.

Proof. Introduce the four-term tuple

$$ Q_{d,i}=(i,i+d,-2i,d)\pmod {2d}. \tag{2.7} $$

There is an equality of residue multisets

$$ Q_{d,i}\mathbin{\uplus}Q_{d,i} =\eta_{d,i}\mathbin{\uplus}(d,d). \tag{2.8} $$

Let $t\in(\mathbf Z/2d\mathbf Z)^\times$, write $\operatorname{res}_{2d}$ for least positive residue modulo $2d$, and let $r$ be the least positive residue of $ti$ modulo $d$. The integer $t$ is odd, so the least positive residues of $ti$ and $t(i+d)$ are $r$ and $r+d$, in some order; those of $-2ti$ and $td$ are $2(d-r)$ and $d$. Consequently

$$ \sum_{a\in Q_{d,i}}\operatorname{res}_{2d}(ta)=4d=2m. $$

Using (2.8), the residue sum of $t\eta_{d,i}$ is therefore

$$ 2(2m)-2d=3m. $$

For $k$ copies the normalized sum is $3k$, which is $q+1$ for $q=3k-1$. This proves $(B)$, and the general Chevalley--Weil calculation gives the rank and signature.

Exactness follows from $\gcd(i,d)=1$.

A multi-standard tuple of length six would have to consist of one $5$-standard block. Its five progression entries are distinct, whereas $\eta_{d,i}$ has at most three distinct residues, so no unit multiple and permutation can identify the two. Finally, at $m=6$ the only odd prime divisor is $3$, but $m/3=2$, so Theorem 1.4 permits no block at all. $\square$

3. A universal Kummer--Delsarte common quotient

We prove a quotient theorem in arbitrary dimension. Let $n\geq1$, let $m\geq2$, and choose distinct points $b_0,\ldots,b_{n+1}\in\mathbf P^1$. Suppose

$$ 0<a_i<m, \qquad \sum_{i=0}^{n+1}a_i=sm,\qquad s\in\mathbf Z, \qquad \gcd(m,a_0,\ldots,a_{n+1})=1. \tag{3.1} $$

Let $C$ be the normalization of

$$ y^m=\prod_{i=0}^{n+1}\ell_i(s,t)^{a_i}. \tag{3.2} $$

Set

$$ N=\{(r_1,\ldots,r_n)\in(\mathbf Z/m)^n:\sum_jr_j=0\}, \qquad G=N\rtimes\mathfrak S_n. \tag{3.3} $$

The symmetric product $\operatorname{Sym}^n\mathbf P^1$ is $\mathbf P^n$. For an effective divisor $D=p_1+\cdots+p_n$, put

$$ L_i(D)=\prod_{j=1}^n\ell_i(p_j). \tag{3.4} $$

Any $n+1$ of the $L_i$ are linearly independent: a nonzero binary form of degree $n$ cannot vanish at $n+1$ distinct points. Hence there is a unique relation up to scale,

$$ \sum_{i=0}^{n+1}c_iL_i=0, \qquad c_i\ne0. \tag{3.5} $$

Define the smooth Delsarte hypersurface

$$ Z_{\mathbf c}:\quad \sum_{i=0}^{n+1}c_ix_i^m=0\subset\mathbf P^{n+1}. \tag{3.6} $$

Let

$$ A=(\mu_m)^{n+2}/\mu_m^{\mathrm{diag}}, \qquad \chi_{\mathbf a}([\epsilon_i])=\prod_i\epsilon_i^{a_i}, \qquad K_{\mathbf a}=\ker\chi_{\mathbf a}. \tag{3.7} $$

The character is well defined because $\sum a_i\equiv0\pmod m$, and it is surjective by the exactness assumption.

Theorem 3.1. After choosing compatible Kummer generators, there is an isomorphism of finite normal covers of the hyperplane $S=\{\sum c_iZ_i=0\}\simeq\mathbf P^n$:

$$ C^n/G\simeq Z_{\mathbf c}/K_{\mathbf a}. \tag{3.8} $$

Let $Y$ denote the common quotient and let

$$ \Gamma=[C^n\times_YZ_{\mathbf c}]_{\mathrm{top}} \in CH^n(C^n\times Z_{\mathbf c})_{\mathbf Q}. \tag{3.9} $$

Then

$$ {}^t\Gamma\circ\Gamma =|K_{\mathbf a}|\sum_{g\in G}\Gamma_g, \qquad \Gamma\circ{}^t\Gamma =|G|\sum_{k\in K_{\mathbf a}}\Gamma_k. \tag{3.10} $$

Consequently,

$$ R=\frac1{|G||K_{\mathbf a}|}{}^t\Gamma \quad\Longrightarrow\quad R_*\Gamma_*|_U=\operatorname{id}_U \tag{3.11} $$

for every rational cohomology subspace $U$ fixed pointwise by $G$.

Proof. Write $y_j$ for the Kummer coordinate on the $j$-th factor of $C^n$, and put $Y_C=\prod_jy_j$. The subgroup $N$ is precisely the kernel of the product character on $(\mu_m)^n$. Therefore

$$ \mathbf C(C^n)^N =\mathbf C((\mathbf P^1)^n)(Y_C), \qquad Y_C^m=\prod_{i=0}^{n+1}L_i(D)^{a_i}. \tag{3.12} $$

The expression is symmetric in the $p_j$. Taking $\mathfrak S_n$-fixed fields gives

$$ \mathbf C(C^n)^G=\mathbf C(S)(w_C), \qquad w_C^m=\prod_i(L_i/L)^{a_i}, \tag{3.13} $$

where $L$ is any base linear form and $w_C=Y_C/L^s$. The right-hand side is not a $p$-th power for any prime $p\mid m$, because some $a_i$ is not divisible by $p$. Thus the extension in (3.13) has degree $m$.

On $Z_{\mathbf c}$, set $Z_i=x_i^m$ and

$$ w_Z=\frac{\prod_ix_i^{a_i}}{L(Z)^s}. \tag{3.14} $$

This function has $A$-character $\chi_{\mathbf a}$, and

$$ \mathbf C(Z_{\mathbf c})^{K_{\mathbf a}} =\mathbf C(S)(w_Z), \qquad w_Z^m=\prod_i(L_i/L)^{a_i}. \tag{3.15} $$

The normalization of $S$ in this common function field is unique, proving (3.8). The qualification concerning Kummer generators records the harmless constant introduced if the evaluation forms in (3.4) are rescaled.

Let $p:C^n\to Y$ and $q:Z_{\mathbf c}\to Y$ be the quotient maps. Their generic fibres are $G$- and $K_{\mathbf a}$-torsors. The top-dimensional cycles of the two self-fibre products are

$$ [C^n\times_YC^n]_{\mathrm{top}}=\sum_{g\in G}\Gamma_g, \qquad [Z_{\mathbf c}\times_YZ_{\mathbf c}]_{\mathrm{top}} =\sum_{k\in K_{\mathbf a}}\Gamma_k. $$

In the triple fibre products defining the two compositions, forgetting the middle factor has generic degree $|K_{\mathbf a}|$ or $|G|$, respectively. This proves (3.10). A component over a proper closed subset of $Y$ has dimension less than $n$, so ramification creates no extra top-dimensional term. Since

$$ |G|=m^{n-1}n!, \qquad |K_{\mathbf a}|=m^n, $$

(3.11) follows from (3.10). On the whole cohomology of $C^n$, the operator $R_*\Gamma_*$ is the average $|G|^{-1}\sum_gg_*$, not the identity. $\square$

We shall use Theorem 3.1 on a determinant subspace. Suppose that $H\subset H^1(C,\mathbf Q)$ has rank $n$ over a number field $F$, and let

$$ U=\bigwedge_F^nH\subset H^n(C^n,\mathbf Q) \tag{3.16} $$

be the rational realization obtained by the separability projector for $F/\mathbf Q$ followed by normalized alternation. After complexification, each determinant tensor lies in an $n$-fold tensor product with one fixed deck character. The subgroup $N$ acts by $\zeta_m^{t\sum r_j}=1$. A geometric permutation of the curve factors contributes the Koszul sign on degree-one cohomology, while determinant alternation contributes the same sign. Their product is $+1$. Hence $G$ fixes $U$ pointwise, and (3.11) applies.

4. Fermat source classes

We first specialize Theorem 3.1 to $n=p-1$, $m=pd$, and the tuple (1.1). The integer in (3.1) is then

$$ s=\frac{p+1}{2}=q+1. $$

Put

$$ H=e_mH^1(C_{p,d,\mathbf b},\mathbf Q), \qquad U_{p,d}=\bigwedge_F^nH\subset H^n(C_{p,d,\mathbf b}^n,\mathbf Q). \tag{4.1} $$

The preceding section supplies

$$ C_{p,d,\mathbf b}^n/(N\rtimes\mathfrak S_n) \simeq Z_{\mathbf c}/K_{\mathbf a_{p,d}} \tag{4.2} $$

and a split injection $\Gamma_*:U_{p,d}\to H^n(Z_{\mathbf c},\mathbf Q)$.

Theorem 4.1. For every odd prime $p\ge5$, every $d\geq3$, and every branch configuration $\mathbf b$,

$$ \Gamma_*U_{p,d}\subset \operatorname{cl}CH^q(Z_{\mathbf c})_{\mathbf Q}. \tag{4.3} $$

Proof. After adjoining finitely many constants, $Z_{\mathbf c}$ becomes the Fermat $n$-fold of degree $m$. For $t\in(\mathbf Z/m\mathbf Z)^\times$, the $t$-component

$$ (U_{p,d})_t=\bigwedge^nH^1(C)_t $$

is one-dimensional. The common quotient carries it nontrivially to the primitive middle character line

$$ V(\varepsilon t\mathbf a_{p,d})\subset H^n(Z_{\mathbf c},\mathbf C), \tag{4.4} $$

where $\varepsilon\in\{\pm1\}$ depends only on the pullback convention. The character containment and the asserted nonvanishing are precisely the character-compatible quotient lemma proved below (Lemma 5.0), applied with $n=p-1$. Thus the map onto this one-dimensional line is surjective.

The tuple $\mathbf a_{p,d}$ is Aoki's $p$-standard character with $i=1$ and $m=pd$:

$$ (i,i+d,\ldots,i+(p-1)d,-pi). \tag{4.5} $$

Every unit multiple is standard again. Indeed, if $r$ is the least positive residue of $t$ modulo $d$, then the first $p$ entries of $t\mathbf a_{p,d}$, after permutation, are

$$ r,r+d,\ldots,r+(p-1)d, $$

and the final entry is $-pr$ modulo $pd$. Moreover $\gcd(r,d)=1$, so $d/\gcd(r,d)=d>2$, as required in Aoki's theorem. Aoki's standard-cycle theorem [2, Theorem 2-1] therefore gives an algebraic codimension-$q$ cycle with nonzero projection to each line in (4.4). Replacing $t$ by $-t$ absorbs the global sign $\varepsilon$.

Let $A_{\mathbf Q}$ be the image of $CH^q(Z_{\mathbf c})_{\mathbf Q}$ in rational cohomology. The preceding argument gives a cycle with nonzero projection to each character line. The space $A_{\mathbf Q}\otimes\mathbf C$ is stable under the diagonal automorphism group, since those automorphisms carry algebraic cycles to algebraic cycles. This finite-group action is semisimple. Applying the complex character idempotent to a class with nonzero projection onto the one-dimensional isotypic line therefore contains that whole line and gives

$$ V(\varepsilon t\mathbf a_{p,d}) \subset A_{\mathbf Q}\otimes_{\mathbf Q}\mathbf C \qquad(t\in(\mathbf Z/m\mathbf Z)^\times). $$

On the other hand,

$$ (\Gamma_*U_{p,d})\otimes\mathbf C =\bigoplus_{t\in(\mathbf Z/m\mathbf Z)^\times} V(\varepsilon t\mathbf a_{p,d}). \tag{4.6} $$

Both sides descend to rational subspaces. Faithful flatness of $\mathbf Q\to\mathbf C$ gives (4.3). Notice that the conclusion concerns the full Galois orbit; an individual cyclotomic character projector need not be rational. $\square$

Proposition 4.2. The tuple $\mathbf a_{p,d}$ is not simple in the earlier simple-tuple sense used by Schoen for any $p\ge5,d\geq3$. This statement is separate from the quasi-simple branching condition in Schoen's 1989 theorem.

Proof. The first $p$ entries of $\mathbf a_{p,d}$ are congruent to $1$ modulo $d$. The sum of any two of them is congruent to $2$ modulo $d$, hence cannot be $0$ because $d\geq3$. Thus no two of these $p$ entries are opposite modulo $pd$. The last entry can be opposite to at most one of them, leaving $p-1\ge4$ entries which cannot be paired. $\square$

Theorem 4.1 is therefore not an application of Schoen's simple-tuple theorem.

We now verify the source condition for the juxtaposed family. Write

$$ \alpha_3=(1,1+\ell d,1+2\ell d,m-3), $$$$ \alpha_\ell=(1,1+3d,\ldots,1+3(\ell-1)d,m-\ell), $$

so that $\mathbf a_{\ell,d}=\alpha_3*\alpha_\ell$.

Theorem 4.3. For every prime $\ell\geq5$ with $\ell\equiv2\pmod3$, every $d\geq1$, every primitive $t$, and every branch configuration, the line

$$ V(t\mathbf a_{\ell,d}) $$

on the corresponding degree-$m$ Fermat $n$-fold is algebraic. Hence the whole rational source block in Theorem 3.1 is algebraic.

Proof. The first block is $3$-standard with $m=3(\ell d)$; the second is $\ell$-standard with $m=\ell(3d)$. For primitive $t$, let $r_3$ and $r_\ell$ be as in Proposition 2.2. Up to permutation,

$$ t\alpha_3=\alpha_{3,r_3}, \qquad t\alpha_\ell=\alpha_{\ell,r_\ell}. $$

Moreover

$$ \gcd(r_3,\ell d)=1, \qquad \ell d>2, \qquad \gcd(r_\ell,3d)=1, \qquad 3d>2. $$

Aoki's Theorem 2-1 [2] therefore proves algebraicity of both standard character lines. The two blocks belong to Fermat dimensions $2$ and $\ell-1$, both even, and

$$ n=2+(\ell-1)+2. $$

The Shioda--Ran juxtaposition theorem, as recalled by Aoki [2, Theorem 1-4(i)], therefore applies and proves algebraicity of

$$ V(t(\alpha_3*\alpha_\ell)). $$

The rational-orbit descent is the same finite-group idempotent argument as in the proof of Theorem 4.1. $\square$

Proposition 4.4. The tuple $\mathbf a_{\ell,d}$ is not a single-prime-standard character, even after permutation and multiplication by a unit modulo $m$.

Proof. A prime-standard character associated with a prime $r$ has length $r+1$. Equality with the length $\ell+5$ would force $r=\ell+4$. But $\ell\equiv2\pmod3$ implies $3\mid(\ell+4)$, and $\ell+4>3$. Thus $r$ is not prime. $\square$

The preceding source argument admits arbitrary finite iteration.

Theorem 4.5. Let $\mathbf a$ be any multi-standard tuple in Theorem 1.4. For every $t\in(\mathbf Z/m\mathbf Z)^\times$, the primitive middle character line

$$ V(t\mathbf a) $$

on the degree-$m$ Fermat $n$-fold is algebraic. Consequently the whole rational source block in Theorem 3.1 is algebraic for every branch configuration.

Proof. Fix $t$, and let $r_j$ be its least positive residue modulo $d_j=m/p_j$. Up to a permutation within the block,

$$ t\alpha_{p_j,m} =\bigl(r_j,r_j+d_j,\ldots,r_j+(p_j-1)d_j,m-p_jr_j\bigr). $$

Since $t$ is a unit modulo $m$,

$$ \gcd(r_j,d_j)=1, \qquad d_j>2. $$

Thus Aoki's standard-cycle theorem [2, Theorem 2-1] proves algebraicity of the character line attached to every translated block. Its Fermat dimension is $p_j-1$, which is even.

The Shioda--Ran juxtaposition theorem, as recalled by Aoki [2, Theorem 1-4(i)], says that juxtaposing two algebraic even-dimensional character lines produces an algebraic character line in dimension equal to the sum of the two dimensions plus two. This new dimension remains even. Iterating the theorem therefore proves algebraicity of

$$ V\bigl(t(\alpha_{p_1,m}*\cdots *\alpha_{p_k,m})\bigr). $$

No distinctness or non-isomorphism hypothesis is present in the Shioda--Ran juxtaposition theorem: the two source character lines may come from the same standard block, and the argument therefore permits repeated primes and repeated translated blocks.

Juxtaposition is associative up to permutation, and the character-line claim is permutation invariant, so the induction is unambiguous. Finally, the finite-group idempotent and rational-orbit descent used in Theorem 4.1 show that the complete rational source block, rather than merely each complex line separately, is algebraic. $\square$

The contraction family uses both directions of the Shioda--Ran induction.

Theorem 4.6. Let $d,i,m$ and $\eta_{d,i}$ be as in Theorem 1.5. For every $k\geq1$ and every $t\in(\mathbf Z/m\mathbf Z)^\times$, the primitive middle character line

$$ V(t\eta_{d,i}^{*k}) $$

on the degree-$m$ Fermat $(6k-2)$-fold is algebraic. Consequently the complete rational source block is algebraic for every branch configuration.

Proof. Begin with the surface character $Q_{d,i}$ in (2.7). Aoki's explicit $p=2$ curve [2, Remark 2-2] represents this character. If $t$ is primitive modulo $2d$, then $td\equiv d\pmod {2d}$ and

$$ tQ_{d,i}=Q_{d,ti} $$

with entries read modulo $2d$. Moreover $d/\gcd(ti,d)=d>2$, so the same remark represents every primitive translate.

Apply the Shioda--Ran juxtaposition theorem [2, Theorem 1-4(i)] to two copies. It makes the character line of $tQ_{d,i}*tQ_{d,i}$ algebraic. Since $td=d$, the translated multiset identity (2.8) reads

$$ tQ_{d,i}\mathbin{\uplus}tQ_{d,i} =t\eta_{d,i}\mathbin{\uplus}(d,d). $$

The pair $(d,d)=(d,-d)$ is an inverse pair. The deletion statement [2, Theorem 1-4(ii)] therefore removes it and proves algebraicity of $V(t\eta_{d,i})$. Iterating Theorem 1-4(i) proves the assertion for $\eta_{d,i}^{*k}$. The finite-group idempotent argument of Theorem 4.1 then descends the complete primitive orbit to a rational algebraic source block. $\square$

The complete source condition is automatic at prime-power conductor.

Theorem 4.7 (Aoki's prime-power source theorem). Let $m$ be a prime power and let $n=2q$ be even. Every balanced residue tuple $\mathbf a$ at level $m$ satisfies $(S)$.

Proof. Balance says that every primitive translate $t\mathbf a$ is a middle Hodge character of the degree-$m$ Fermat $n$-fold. Aoki's Corollary 2-3 [2] proves the Hodge conjecture for the degree-$m$ Fermat variety in every dimension when $m$ is a prime power. Hence the rational Galois block generated by the lines $V(t\mathbf a)$ is spanned by algebraic cycle classes, which is precisely $(S)$. Notice that exactness is not used here. $\square$

Corollary 1.8 follows by combining Theorem 4.7 with Theorems 5.1, 6.5, 7.1, and 8.1.

5. Transfer to the exact Prym factor

We now pass from the curve product to the exact factor $B=B_{\mathbf a,\mathbf b}$. Put

$$ U_{\mathbf a}=\bigwedge_F^ne_mH^1(C,\mathbf Q) \subset H^n(C^n,\mathbf Q). $$

Let $j:B\hookrightarrow J(C)$ be the inclusion of the projector image, understood up to isogeny, and let $\mathcal P_C$ be the normalized Poincaré line bundle on $C\times J(C)$. Define

$$ \mathfrak p=c_1\bigl((\operatorname{id}_C\times j)^*\mathcal P_C\bigr) \in CH^1(C\times B)_{\mathbf Q}. \tag{5.1} $$

Under the Abel--Jacobi identification, the induced map on degree-one cohomology is the projector-induced map

$$ \mathfrak p_*|_{e_mH^1(C,\mathbf Q)}: e_mH^1(C,\mathbf Q)\xrightarrow{\sim}H^1(B,\mathbf Q). \tag{5.2} $$

It is $F$-linear. This isomorphism follows from the actual isogeny-factor inclusion and the Poincaré correspondence; an abstract isomorphism between the two Hodge structures would not suffice for the tensor construction below.

Work over $\mathbf C$ throughout this construction. Choose a degree-one zero-cycle $z\in CH^1(C)_{\mathbf Q}$ (a complex point is sufficient) and use the same $z$ throughout. Thus

$$ \pi_0^C=[z\times C],\qquad \pi_2^C=[C\times z],\qquad \pi_1^C=\Delta_C-\pi_0^C-\pi_2^C \tag{5.2a} $$

are Chow--Künneth projectors over $\mathbf C$, with $\pi_1^C$ acting as the identity on $H^1(C)$ and as zero on $H^0(C)\oplus H^2(C)$. All correspondences below are therefore over $\mathbf C$. If the branch data are algebraic, the same construction can, after one finite extension, be defined over a common number field. The rational-orbit averages used below refer only to finite deck/diagonal idempotents with $\mathbf Q$-coefficients, not to an arithmetic descent of an arbitrary complex fibre. Put

$$ P=(\mathfrak p\circ\pi_1^C)^{\boxtimes n} \in CH^n(C^n\times B^n)_{\mathbf Q}. \tag{5.3} $$

Let $\Delta_B:B\hookrightarrow B^n$ be the small diagonal. The correspondence

$$ T={}^t\Gamma_{\Delta_B}\circ P \in CH^n(C^n\times B)_{\mathbf Q} \tag{5.4} $$

has codimension $n+n\dim B-\dim B^n=n$, and acts on the $(1,\ldots,1)$-Künneth component as

$$ T_*=\Delta_B^*\circ \mathfrak p_*^{\otimes n} =\operatorname{cup}\circ \mathfrak p_*^{\otimes n}. \tag{5.5} $$

After normalized determinant alternation, (5.2) gives an isomorphism

$$ T_*|_{U_{\mathbf a}}: U_{\mathbf a}\xrightarrow{\sim}W_F(B). \tag{5.6} $$

The projector $(\pi_1^C)^{\boxtimes n}$ removes every other Künneth component. Thus no auxiliary determinant projector on $B$ is needed.

We record the character compatibility needed to identify the source image on the Fermat side. This is stronger than merely observing that the two quotients have the same Kummer equation.

Lemma 5.0 (character-compatible common quotient). Choose the Kummer generators in the common-quotient construction so that the functions $w_C$ and $w_Z$ agree. Put

$$ \widetilde G_C=(\mu_m)^n\rtimes\mathfrak S_n, \qquad D_C=(\mu_m)^n/N\simeq\mu_m, $$

and, on the Fermat side,

$$ A=(\mu_m)^{n+2}/\mu_m^{\mathrm{diag}}, \qquad D_Z=A/K_{\mathbf a}\simeq\mu_m. $$

Then the common-quotient isomorphism identifies $D_C$ with $D_Z$, and the top-dimensional fibre-product correspondence $\Gamma$ satisfies

$$ (\delta_C(\zeta)\times\delta_Z(\zeta))_*\Gamma=\Gamma, \qquad (1\times k)_*\Gamma=\Gamma\quad(k\in K_{\mathbf a}). \tag{5.7} $$

If $U_t$ is the determinant summand attached to the primitive deck character $t$, then

$$ \Gamma_*U_t=V(\varepsilon t\mathbf a), \tag{5.8} $$

where $\varepsilon\in\{\pm1\}$ is fixed by the push--pull convention.

Proof. On the open locus away from the branch divisor, $C^n\to C^n/G$ and $Z\to Z/K_{\mathbf a}$ are torsors. The class of $(\zeta,1,\ldots,1)$ acts on $w_C$ by $\zeta$, while an element of $A$ acts on $w_Z$ by $\chi_{\mathbf a}$. Since $w_C=w_Z$, these actions identify $D_C$ and $D_Z$; normality extends the equality from the open locus to the common quotient. The fibre product is therefore stable under the paired residual action, and it is stable under $K_{\mathbf a}$ on the second factor.

On $H_t^{\otimes n}$, an element $(\epsilon_1,\ldots,\epsilon_n)\in N$ acts by $(\prod_j\epsilon_j)^t=1$; the permutation sign from determinant alternation cancels the Koszul sign. Hence $U_t$ descends through $G$, and the residual $D_C$-character on it is $\zeta^{\varepsilon t}$. Equivariance of $\Gamma$ and (5.7) imply that $\Gamma_*U_t$ is $K_{\mathbf a}$-fixed with $A$-character $\chi_{\mathbf a}^{\varepsilon t}$. The annihilator of $K_{\mathbf a}$ in the character group of $A$ is generated by $\chi_{\mathbf a}$, so the image is contained in the corresponding Fermat middle-character line $V(\varepsilon t\mathbf a)$. Exactness makes every entry of $\varepsilon t\mathbf a$ nonzero, and that line is one-dimensional. Finally, the split identity $R_*\Gamma_*|_U=\mathrm{id}_U$ makes $\Gamma_*|_{U_t}$ nonzero. Thus the containment is equality. $\square$

Theorem 5.1. Suppose $\mathbf a$ satisfies $(E),(B),(S)$. For every branch configuration $\mathbf b$, the determinant Hodge space $W_F(B_{\mathbf a,\mathbf b})$ is spanned by codimension-$q$ algebraic cycle classes.

Proof. After scaling its coordinates over $\mathbf C$, $Z_{\mathbf c}$ is the degree-$m$ Fermat $n$-fold. For every primitive $t$, Lemma 5.0 identifies the one-dimensional summand $\Gamma_*(U_t)$ with $V(\varepsilon t\mathbf a)$. Since $t\mapsto\varepsilon t$ permutes the primitive orbit, condition $(S)$ algebraizes the whole image of $\Gamma_*U_{\mathbf a}$ after complexification. If

$$ A_{\mathbf Q}:=\operatorname{im}\bigl(\operatorname{cl}:CH^q(Z_{\mathbf c})_{\mathbf Q} \longrightarrow H^{2q}(Z_{\mathbf c},\mathbf Q)\bigr), $$

then the finite rational-orbit construction gives $\Gamma_*U_{\mathbf a}\subset A_{\mathbf Q}\otimes_{\mathbf Q}\mathbf C$. Faithful flatness of $\mathbf C/\mathbf Q$ gives $\Gamma_*U_{\mathbf a}\subset A_{\mathbf Q}$. Hence, for $u\in U_{\mathbf a}$, the split identity (3.11) and the rational-orbit descent in the proof of Theorem 4.1 give $z_Z\in CH^q(Z_{\mathbf c})_{\mathbf Q}$ with $\operatorname{cl}(z_Z)=\Gamma_*u$. With $R$ as in (3.11), put

$$ z_C=R_*z_Z\in CH^q(C^n)_{\mathbf Q}. $$

Then $\operatorname{cl}(z_C)=u$. Finally define

$$ z_B=T_*z_C\in CH^q(B)_{\mathbf Q}. $$

Equation (5.6) gives $\operatorname{cl}(z_B)=T_*u$. Since (5.6) is an isomorphism, such classes span $W_F(B)$. $\square$

Theorem 4.5 verifies $(S)$ for every multi-standard family in Theorem 1.4, including the families of Theorems 1.2 and 1.3, while Theorem 4.6 verifies it for Theorem 1.5. Thus Theorem 5.1 proves all of their fibrewise determinant assertions.

Lemma 5.2 (projector image and restricted Poincare correspondence). Let $e=e_m\in\mathbf Q[\mu_m]\subset\operatorname{End}^0(J(C))$ be the exact cyclotomic idempotent. Choose $N>0$ such that $u=Ne$ and $v=N(1-e)$ are integral endomorphisms, and set $B=\operatorname{Im}(u)$, $B'=\operatorname{Im}(v)$. If $j:B\hookrightarrow J(C)$ and $j':B'\hookrightarrow J(C)$ are the inclusions, then

$$ \alpha:B\times B'\to J(C),\ (b,b')\mapsto j(b)+j'(b'), \qquad \beta:J(C)\to B\times B',\ x\mapsto(u(x),v(x)) $$

are isogenies with $\alpha\beta=[N]$ and $\beta\alpha=[N]$. Hence $j^*:eH^1(J(C),\mathbf Q)\to H^1(B,\mathbf Q)$ is an $F$-linear isomorphism. With the normalized Poincare class on $C\times J(C)$, its pullback to $C\times B$ induces an $F$-linear isomorphism

$$ \phi: eH^1(C,\mathbf Q)\xrightarrow{\sim}H^1(B,\mathbf Q). $$

Proof. The identities $u^2=Nu$, $v^2=Nv$, and $uv=vu=0$ give the two displayed composition formulas. After applying $\alpha^*$, the idempotent $e$ is projection onto the first summand, proving the assertion for $j^*$; centrality of $e$ gives $F$-linearity. We use the standard Abel--Poincare naturality explicitly: for a finite morphism of curves $q:C\to D$, with norm $\operatorname{Nm}_q:J(C)\to J(D)$, the based Abel--Jacobi maps satisfy

$$ \operatorname{Nm}_q\circ AJ_C=AJ_D\circ q_* \tag{5.10} $$

on degree-zero divisor classes (and hence on $H^1$); equivalently, after dualizing by the principal polarizations, the pullback map on Jacobians and the Poincare correspondence commute with $q_*$. Applying (5.10) to the projector factor and then restricting along $j:B\hookrightarrow J(C)$ gives $\phi=j^*\circ(p_J)_*$. The Abel--Poincare identification is $\mu_m$-equivariant, so restriction to the $e$-summand gives the claimed isomorphism. $\square$

Lemma 5.3 (determinant-transfer normalization). Let $H=eH^1(C,\mathbf Q)$, $V=H^1(B,\mathbf Q)$, and suppose $\operatorname{rank}_F H=n$. Normalize the separability projector and determinant alternator by

$$ A_n=\frac1{n!}\sum_{\sigma\in\mathfrak S_n}\operatorname{sgn}(\sigma)P_\sigma. $$

Let $\kappa_H:\bigwedge_F^nH\hookrightarrow H^{\otimes n}$ and $\kappa_V:\bigwedge_F^nV\hookrightarrow V^{\otimes n}$ denote the separability-projector embeddings followed by $A_n$, with the orbitwise normalization fixed once and for all. Then the $F$-linear map $\phi=\mathfrak p_*|_H$ gives the commutative square

$$ \begin{array}{ccc} \bigwedge_F^nH &\xrightarrow{\bigwedge_F^n\phi}& \bigwedge_F^nV\\ \kappa_H\downarrow && \downarrow\kappa_V\\ H^{\otimes n} &\xrightarrow{\phi^{\otimes n}}& V^{\otimes n}. \end{array} \tag{5.9} $$

The normalization is chosen so that the bottom tensor map followed by the small-diagonal cup product is exactly the determinant map; if a different descent normalization is used, the result is multiplied by a single nonzero $\lambda\in\mathbf Q^\times$, which does not affect algebraicity or spanning. In particular no unaccounted $[F:\mathbf Q]$ or orbit-size factor occurs.

For the correspondence $P=((\mathfrak p\circ\pi_1^C)^{\boxtimes n})$ and $T={}^t\Gamma_{\Delta_B}\circ P$, one has

$$ T_*|_{\bigwedge_F^nH}=\bigwedge_F^n\phi: \bigwedge_F^nH\xrightarrow{\sim}\bigwedge_F^nV. $$

With the unnormalized alternator, the right-hand side is multiplied by $n!\neq0$. Thus the determinant transfer is an isomorphism over $\mathbf Q$, up to a fixed nonzero rational normalization.

Proof. The map $\phi^{\otimes n}$ commutes with the separability projector and with $A_n$. On the degree-one Kunneth component the transpose of the small diagonal acts by cup product. For $v_1,\ldots,v_n\in H^1(B)$, graded commutativity gives

$$ \Delta_B^*A_n(v_1\otimes\cdots\otimes v_n)=v_1\cup\cdots\cup v_n. $$

Hence the induced map is precisely the exterior determinant map; replacing $A_n$ by the unnormalized alternator multiplies it by $n!$. $\square$

6. Characteristic-zero monodromy and the very-general Hodge group

Let $H=e_mR^1f_*\mathbf Q$ be the exact-conductor local system over the ordered configuration space. Its polarization and $F$-action put connected monodromy in the restriction of scalars of the unitary group. We determine the connected group directly in characteristic zero.

Lemma 6.1 (from an adjoint graph to projective representations). Let $n>2$, and let $\rho_1,\rho_2$ be two complex $n$-dimensional representations of the same group $\Pi$, with connected projective algebraic monodromy $PGL_n$ in each factor. If

$$ \overline{(\mathbf P\rho_1,\mathbf P\rho_2)(\Pi)}^{\,\circ} \subset PGL_n\times PGL_n $$

is the graph of an isomorphism, then

$$ \mathbf P\rho_2\simeq\mathbf P\rho_1 \quad\text{or}\quad \mathbf P\rho_2\simeq\mathbf P\rho_1^\vee. \tag{6.1} $$

Proof. The full algebraic monodromy normalizes its identity component. Let the graph be defined by $\phi:PGL_n\to PGL_n$. Then for every $\gamma\in\Pi$,

$$ \operatorname{Ad}(\mathbf P\rho_2(\gamma))\circ\phi =\phi\circ\operatorname{Ad}(\mathbf P\rho_1(\gamma)). $$

Comparing this with $\operatorname{Ad}(\phi(\mathbf P\rho_1(\gamma)))\circ\phi$, and using that $PGL_n$ has trivial center, gives $\mathbf P\rho_2=\phi\circ\mathbf P\rho_1$. Every algebraic-group automorphism of $PGL_n$ is inner or inner followed by $g\mapsto(g^{-1})^{\mathsf T}$, which gives (6.1). $\square$

Lemma 6.2 (projective rigidity of Jordan--Pochhammer parameters). Let $n>2$. Consider two rank-$n$ Jordan--Pochhammer systems with ordered parameters

$$ (\lambda_0;\lambda_1,\ldots,\lambda_{n+1}), \qquad (\lambda'_0;\lambda'_1,\ldots,\lambda'_{n+1}), $$

and suppose

$$ \lambda_i,\lambda'_i\ne1\quad(0\le i\le n+1). $$

Here labelled projective equivalence preserves, for every $1\leq i\leq n+1$, the distinguished generator with label $i$; it does not allow a permutation of punctures or a braid precomposition. If the two labelled systems are projectively equivalent, then their parameter tuples are equal.

Proof. After one common conjugation, their $i$-th generators satisfy $g'_i=c_ig_i$. Put $\delta_i=\lambda_0\lambda_i$ and $\delta'_i=\lambda'_0\lambda'_i$. The spectrum of $g_i$ is

$$ 1^{\oplus(n-1)}\oplus\{\delta_i\} \qquad(1\le i\le n+1). $$

If $\delta_i\ne1$, then $c_ig_i$ has the two eigenvalues $c_i$ and $c_i\delta_i$. A normalized nonresonant pseudoreflection has eigenvalue $1$ with multiplicity $n-1>1$, so comparison with $g'_i$ gives $c_i=1$; it cannot instead be conjugate to a resonant generator, which has only the eigenvalue $1$. If $\delta_i=1$, the irreducibility criterion following Rains's Lemma 2.7 [11] says that $g_i=I+N_i$, where $N_i\ne0$ has rank one. Thus $c_ig_i$ has the single eigenvalue $c_i$ and the nonzero nilpotent part $c_iN_i$. It cannot be conjugate to a nonresonant generator; if $\delta'_i=1$, comparison with the normalized unipotent generator $g'_i$ again forces $c_i=1$. Hence, after the one common conjugation, all labelled generators agree. The ordered Jordan--Pochhammer product relations $g_1\cdots g_{n+1}=\lambda_0I_n$ and $g'_1\cdots g'_{n+1}=\lambda'_0I_n$ give $\lambda'_0=\lambda_0$. Comparing the remaining eigenvalues gives $\lambda'_i=\lambda_i$ for every $i$. $\square$

Lemma 6.2a (configuration-space and one-moving bridge). Assume $(E)$, and put $N=n-1$, so that Rohde's $P_N$ is the configuration space of the $n+2$ ordered branch points used here. For every primitive eigenspace, the connected Zariski closure of monodromy over the present base $M_{0,n+2}$ is the same as the connected Zariski closure over Rohde's $P_N$, and also over the subspace $P_N(a_{N+3})$ in which one labelled branch point is fixed at infinity. Moreover, after fixing all but one of the remaining labelled branch points, the restricted local system is the labelled irreducible Jordan--Pochhammer system with parameters

$$ \lambda_i(t)=\zeta_m^{t a_i}\quad(0\leq i\leq n+1). $$

In particular, a projective graph relation between two primitive factors over $M_{0,n+2}$ restricts to a projective equivalence of these labelled Jordan--Pochhammer systems, with no permutation of the labels.

Proof. Rohde's Remarks 5.1.5--5.1.7 [12] give holomorphic product decompositions

$$ P_N\simeq PGL_2(\mathbf C)\times M_N, \qquad P_N(a_{N+3})\simeq GT\times M_N, $$

for normalized slices $M_N\simeq M_{0,n+2}$, and conclude that the three monodromy groups are commensurable. Commensurable subgroups have the same identity component of their real, and hence complex, Zariski closures. The normalization $N=n-1$ gives exactly $n+2$ ordered branch labels.

Condition $(E)$ is the connected-cover hypothesis in this one-moving family. Fix the branch points with labels $1,\ldots,n+1$, and let the point with label $0$ move. After one fixed Möbius change of coordinate, choose an affine coordinate whose point at infinity is unramified and, for $1\leq i\leq n+1$, let $\gamma_i$ be the positively oriented based loop in which the moving point goes once around the fixed point with label $i$. Rains's Proposition 2.5 [11] identifies $g_i(t)=\rho_t(\gamma_i)$ as the normalized pseudoreflection satisfying

$$ \det g_i(t)=\zeta_m^{t(a_0+a_i)}, \qquad g_1(t)\cdots g_{n+1}(t)=\zeta_m^{ta_0}I_n. $$

Thus, in the convention of (2.19), the ordered Jordan--Pochhammer parameters are exactly $\lambda_i(t)=\zeta_m^{ta_i}$ for $0\leq i\leq n+1$. Their product is $1$ because $\sum_i a_i\equiv0\pmod m$, and none is $1$ because $t$ is primitive and $0<a_i<m$. The irreducibility criterion following Rains's Lemma 2.7 [11] therefore applies. Since the full configuration space and the loops $\gamma_i$ are ordered, restriction uses the same labelled elements of the fundamental group in both factors: there is no puncture permutation or braid precomposition. A projective graph identity restricts along this inclusion of fundamental groups. If $\lambda_0(t)\lambda_i(t)=1$, the corresponding generator is still a nontrivial rank-one unipotent pseudoreflection, so the resonant projective scalar is fixed exactly as in Lemma 6.2. This proves the last assertion. $\square$

Proposition 6.3. Suppose $\mathbf a$ satisfies $(E),(B)$. Then

$$ \operatorname{Mon}^{\circ}(H) =\operatorname{Res}_{F^+/\mathbf Q}SU_F(H,h). \tag{6.2} $$

Proof. For $t\in(\mathbf Z/m\mathbf Z)^\times$, let $H_t$ be the primitive $\zeta_m^t$-eigenspace. Condition $(B)$ and (2.2) give $h^{1,0}(H_t)=h^{0,1}(H_t)=q$. Rohde's Theorem 5.1.1 [12, p. 53] says that the connected real algebraic monodromy on an eigenspace of type $(a,b)$, with $a,b\geq1$ and $t\ne m/2$, is $SU(a,b)$. A primitive $t$ is never $m/2$, since $m>2$. Therefore every real-factor projection is surjective:

$$ \operatorname{pr}_{\{t,-t\}} \operatorname{Mon}^{\circ}(H)_{\mathbf R}=SU(H_t,h_t)=SU(q,q). \tag{6.3} $$

As (6.3) holds in every coordinate, the full connected group is contained in the product of these special-unitary factors,

$$ \operatorname{Mon}^{\circ}(H)_{\mathbf R} \subseteq \prod_{\{t,-t\}\subset(\mathbf Z/m\mathbf Z)^\times}SU(H_t,h_t). \tag{6.4} $$

If this subdirect product were proper, the Lie-algebra form of Goursat's lemma would produce two distinct real embeddings $\tau=\{t,-t\}$ and $\upsilon=\{u,-u\}$ for which the projection to the two adjoint factors is the graph of an isomorphism

$$ PSU(H_t,h_t)\xrightarrow{\sim}PSU(H_u,h_u). \tag{6.5} $$

Lemma 6.1 and the polarization identification $\rho_t^\vee\simeq\rho_{-t}$ imply

$$ \mathbf P\rho_u\simeq\mathbf P\rho_t \quad\text{or}\quad \mathbf P\rho_u\simeq\mathbf P\rho_{-t}. \tag{6.6} $$

Restrict to the curve on which the branch point of exponent $a_0$ moves. A graph relation obtained on the full ordered configuration space restricts to the same labelled one-point monodromy representations, so the resulting projective equivalence persists on this curve. Rains's Proposition 2.5, together with (2.19) and the irreducibility criterion following Lemma 2.7 [11], identifies $\rho_t$ with the rank-$n$ Jordan--Pochhammer representation having parameters

$$ \lambda_i(t)=\zeta_m^{ta_i}\qquad(0\leq i\leq n+1). \tag{6.7} $$

Because the branch-configuration space is ordered, the distinguished loops in this description have the same labels for every primitive $t$.

They satisfy

$$ \prod_i\lambda_i(t)=1, \qquad \lambda_i(t)\ne1\quad(0\leq i\leq n+1), \tag{6.8} $$

because $0<a_i<m$ and $t$ is primitive. Lemma 6.2 applied to the first possibility in (6.6) gives

$$ \zeta_m^{(u-t)a_i}=1\qquad(0\leq i\leq n+1). $$

Condition $(E)$ then gives $u\equiv t\pmod m$. For the second possibility, dualize first; the same argument gives $u\equiv-t\pmod m$. Both contradict the assumption that $\tau\ne\upsilon$. No diagonal block exists. Goursat's lemma makes (6.4) an equality, and equality after base change to $\mathbf R$ implies equality of the connected $\mathbf Q$-groups. This proves (6.2). $\square$

Remark 6.4 (the even-$d$ sign concern). For the standard-prime family, if $d$ is even, all $a_i$ are odd. The class $t'=t+m/2$ is primitive exactly when $4\mid d$; only then does it define $\zeta_m\mapsto-\zeta_m$. Its parameters satisfy $\lambda_i(t')=-\lambda_i(t)$. Although the pseudoreflection eigenvalue ratios $\lambda_0\lambda_i$ agree, the projective-rigidity argument above excludes projective equivalence because $t'\not\equiv\pm t\pmod m$ for $m>4$. If $d\equiv2\pmod4$, $t+m/2$ is not primitive; for odd $d$ there is no integral half-period. Hence no parity restriction on $d$ is needed.

Theorem 6.5. Suppose $\mathbf a$ satisfies $(E),(B)$. For a very general branch configuration,

$$ Hg(B)=\operatorname{Res}_{F^+/\mathbf Q}SU_F(H^1(B),h), \qquad \operatorname{End}^0(B)=F. \tag{6.10} $$

Proof. General Mumford--Tate theory and Proposition 6.3 give

$$ \operatorname{Res}_{F^+/\mathbf Q}SU_F(H^1(B),h) =\operatorname{Mon}^{\circ} \subseteq Hg(B)_{\mathrm{gen}}. \tag{6.11} $$

The deck action consists of Hodge endomorphisms, and Rosati restricts to complex conjugation on $F$. Hence the generic Hodge group centralizes $F$ inside the symplectic group and

$$ Hg(B)_{\mathrm{gen}} \subseteq\operatorname{Res}_{F^+/\mathbf Q}U_F(H^1(B),h). \tag{6.12} $$

The whole rational space $W_F(B)$ is of type $(q,q)$, so the Hodge group fixes it pointwise. On each complex factor the unitary group acts on the determinant line through its determinant character. Thus (6.12) refines to

$$ Hg(B)_{\mathrm{gen}} \subseteq\operatorname{Res}_{F^+/\mathbf Q}SU_F(H^1(B),h). \tag{6.13} $$

The two inclusions give equality. The centralizer of the standard rational representation is $F$, hence

$$ \operatorname{End}_{\mathbf Q\text{-HS}}H^1(B,\mathbf Q)=F. $$

Full faithfulness of the weight-one Hodge realization up to isogeny yields $\operatorname{End}^0(B)=F$. The locus where the Hodge group differs from the generic Hodge group is a countable union of proper closed algebraic subsets. $\square$

Propositions 2.3--2.4 verify exactness and balance for the explicit families, while Theorems 4.5--4.6 verify their source algebraicity. Together with Theorem 6.5 these give the generic Hodge-group and endomorphism assertions in Theorems 1.2--1.5 and Corollary 1.6.

When there is more than one real primitive factor, the no-repeated-factor criterion of Spelta--Tamborini [15] does not apply because all these factors have the repeated signature $SU(q,q)$; the projective-rigidity argument above supplies their independence. Venkataramana's arithmeticity theorem [17] has different large-branch-number and coprimality hypotheses and is not used.

7. Hodge classes on all powers

We use the following invariant-theoretic statement.

Theorem 7.1. Let $B$ be a complex Abelian variety, let $F/F^+$ be a CM field with a specified embedding

$$ F\hookrightarrow \operatorname{End}^0(B) $$

whose Rosati involution is the CM conjugation $a\mapsto\bar a$, and suppose

$$ \dim_FH^1(B,\mathbf Q)=n=2q>2, \qquad Hg(B)=\operatorname{Res}_{F^+/\mathbf Q}SU_F(H^1(B),h). \tag{7.1} $$

Assume in addition that every real embedding of $F^+$ gives $h$ signature $(q,q)$ (equivalently, $W_F(B)$ is a Hodge substructure of type $(q,q)$), and that

$$ W_F(B)=\bigwedge_F^nH^1(B,\mathbf Q) \subset \operatorname{cl}\,CH^q(B)_{\mathbf Q}. \tag{7.2} $$

Then the rational Hodge conjecture holds for $B^r$ for every $r\geq1$.

Proof. Put $V=H^1(B,\mathbf Q)$. After complexification, the group in (7.1) is a product of copies of $SL_n$, indexed by the real embeddings of $F^+$. For a conjugate pair $\sigma,\bar\sigma$, the two summands $V_\sigma$ and $V_{\bar\sigma}$ are the standard representation and its dual. The cohomology of $B^r$ is the exterior algebra of $r$ copies of these representations.

The first fundamental theorem for $SL_n$ [8] says that the invariant tensors are generated by contractions between the standard representation and its dual, together with the volume tensors in $\bigwedge^nV_\sigma$ and $\bigwedge^nV_{\bar\sigma}$. Since $SL_n$ is reductive in characteristic zero, the invariants functor is exact; passing to the exterior-algebra quotient therefore introduces no new generators.

The contraction tensors span a $\mathbf Q$-defined invariant subspace of $H^2(B^r,\mathbf Q)$ of Hodge type $(1,1)$. More explicitly, the copy-to-copy contractions are realized by the Rosati-symmetric part of the natural $F$-linear subalgebra $M_r(F)\hookrightarrow\operatorname{End}^0(B^r)$, under

$$ \operatorname{NS}(B^r)_{\mathbf Q} \supseteq M_r(F)^{\dagger=1}, $$

so they are divisor classes by the Lefschetz $(1,1)$ theorem. The volume tensors with all $n$ labels on one copy are the complex components of $W_F(B)$, hence are algebraic by hypothesis. Mixed copy labels are obtained by polarization. Indeed, for $a=(a_1,\ldots,a_r)\in F^r$, the rational homomorphism

$$ f_a\in\operatorname{Hom}^0(B^r,B), \qquad (x_1,\ldots,x_r)\longmapsto\sum_i[a_i]x_i \tag{7.2a} $$

is represented by an actual homomorphism after clearing a common denominator. If $D>0$ is chosen so that $Df_a\in\operatorname{Hom}(B^r,B)$, then for $w\in W_F(B)$,

$$ (Df_a)^*w=w\otimes D^n(a_1x_1+\cdots+a_rx_r)^n. \tag{7.3} $$

After dividing by $D^n$ in the rational Chow group, pure $n$-th powers span $\operatorname{Sym}_F^n(F^r)$ in characteristic zero. Thus the algebraic classes (7.3) span every mixed determinant tensor. Products of divisor and determinant classes span all complexified Hodge classes. Since both the Hodge space and the image of rational cycle classes are defined over $\mathbf Q$, faithful flatness descends the spanning statement to $\mathbf Q$. $\square$

Combining Theorems 5.1, 6.5, and 7.1 (with balance supplying the signature and the codimension-$q$ statement in (7.2)) proves the all-powers rational Hodge assertion of Theorem 1.1.

Lemma 7.2 (rational mixed determinant module). Let $V=H^1(B,\mathbf Q)$, assume that $\operatorname{End}^0(B)=F$ and $\dim_FV=n$, and put $W_F(B)=\bigwedge_F^nV$. For every $r\geq1$, the degree-$n$ $SL_F(V)$-invariant summand is canonically

$$ \left(\bigwedge_F^n(V\otimes_F F^r)\right)^{SL_F(V)} \simeq W_F(B)\otimes_F\operatorname{Sym}_F^n(F^r). $$

Here both sides are viewed in rational cohomology through the standard separability-projector realization of the $F$-exterior power. Suppose that $W_F(B)$ is spanned over $\mathbf Q$ by algebraic cycle classes. For such a class $w=\operatorname{cl}(z_w)$, with $z_w\in CH^{n/2}(B)_{\mathbf Q}$, and $a=(a_1,\ldots,a_r)\in F^r$, choose $D\in\mathbf Z_{>0}$ with $Da_i\in\operatorname{End}(B)$ for every $i$. The homomorphism

$$ f_{Da}:B^r\longrightarrow B, \qquad (x_1,\ldots,x_r)\longmapsto\sum_i[Da_i]x_i $$

then satisfies, under the displayed identification,

$$ \operatorname{cl}(f_{Da}^{*}z_w) =D^n w\otimes(a_1X_1+\cdots+a_rX_r)^n, $$

where the $X_i$ denote the copy variables. Consequently all mixed determinant invariants are algebraic after division by $D^n$ in $CH^*(B^r)_{\mathbf Q}$.

Proof. The Cauchy decomposition for $\bigwedge_F^n(V\otimes_F F^r)$ shows that its unique $SL_F(V)$-trivial summand is the determinant Schur functor $\bigwedge_F^nV\otimes_F\operatorname{Sym}_F^n(F^r)$. The same statement holds simultaneously at every conjugate embedding and therefore descends to the displayed rational $F$-module. Clearing denominators makes $f_{Da}$ an actual homomorphism, so its pullback preserves algebraic cycle classes, and the formula follows from functoriality on $H^1$. Pure $n$-th powers $(a_1X_1+\cdots+a_rX_r)^n$ span $\operatorname{Sym}_F^n(F^r)$ in characteristic zero by polarization. This proves the final assertion. $\square$

8. The generalized Hodge conjecture

We now apply Abdulali's criterion. We use the convention that the generalized Hodge conjecture is the equality between the geometric and Hodge-theoretic coniveau filtrations in every degree.

Theorem 8.1. Assume $(E),(B),(S)$, and let $\mathbf b\notin\Sigma_{\mathbf a}$. The generalized Hodge conjecture holds for $B_{\mathbf a,\mathbf b}^r$ for every $r\geq1$.

Proof. Write $B=B_{\mathbf a,\mathbf b}$, fix $r$, and set $X=B^r$. By Theorem 6.5,

$$ \operatorname{End}^0(B)=F, \qquad Hg(B)=\operatorname{Res}_{F^+/\mathbf Q}SU_F(V,h), \qquad V=H^1(B,\mathbf Q). \tag{8.1} $$

In particular, $B$ is simple of Albert type IV, and

$$ \operatorname{End}^0(X)=M_r(F), \qquad H^1(X,\mathbf Q)=V\otimes_{\mathbf Q}\mathbf Q^r. \tag{8.2} $$

The Hodge cocharacter acts diagonally on the $r$ copies. Thus

$$ Hg(X)=\Delta Hg(B). \tag{8.3} $$

For a product polarization, Rosati on $M_r(F)$ is conjugate transpose. The double-centralizer identity identifies the connected Lefschetz group as

$$ L(X)^\circ=\Delta\operatorname{Res}_{F^+/\mathbf Q}U_F(V,h), $$

and hence

$$ (L(X)^\circ)^{\mathrm{der}} =\Delta\operatorname{Res}_{F^+/\mathbf Q}SU_F(V,h) =Hg(X)=Hg(X)^{\mathrm{der}}. \tag{8.4} $$

Therefore $X$ is of PEL type in Abdulali's sense, and its Hodge group is semisimple. Its only simple isogeny factor is the type-IV variety $B$, so Abdulali's additional odd-relative-dimension condition on type-III factors is vacuous.

The hypotheses of Abdulali's Theorem 5.1 are matched as follows.

| Hypothesis in Abdulali's Theorem 5.1 | Verification for $X=B^r$ | |---|---| | $X$ is of PEL type | Equation (8.4) gives equality of the derived connected Lefschetz and Hodge groups. | | $Hg(X)$ is semisimple | Equations (8.3)--(8.4) identify it with the diagonal copy of $\operatorname{Res}_{F^+/\mathbf Q}SU_F(V,h)$. | | Every type-III simple factor has odd relative dimension | There is no type-III factor: the only simple isogeny factor is the type-IV variety $B$. | | The usual Hodge conjecture holds for every power of $X$ | For every $k\geq1$, $X^k=B^{rk}$, which is covered by Theorem 7.1. |

Thus all hypotheses used in the implication from the usual to the generalized Hodge conjecture are satisfied. For every $k\geq1$,

$$ X^k=B^{rk}. $$

The usual rational Hodge conjecture for this variety follows from Theorem 7.1. Abdulali's Theorem 5.1 [1] now implies the generalized Hodge conjecture for $X$. Since $r$ was arbitrary, the result follows. $\square$

The phrase ``any sufficiently large power'' in Abdulali's domination conclusion is not an extra uniform hypothesis here. The theorem is applied to the fixed variety $X=B^r$, and every usual-Hodge instance required by its argument is a power of $X$, hence a power of $B$, already covered by Theorem 7.1.

Theorems 5.1, 6.5, 7.1, and 8.1 prove Theorem 1.1. The residue calculations of Section 2 and the source theorems of Section 4 then prove Theorems 1.2--1.5 and Corollary 1.6.

9. The self-dual boundary prime

We now take $p=3$, so that

$$ m=3d,\qquad n=2,\qquad q=1,\qquad \mathbf a_{3,d}=(1,1+d,1+2d,m-3). \tag{9.1} $$

The residue calculation of Section 2 and the common-quotient and transfer arguments of Sections 3--5 remain valid. Thus, for every fibre,

$$ \dim_FH^1(B,\mathbf Q)=2,\qquad \operatorname{sig}_{\sigma}H^1(B)=(1,1),\qquad \dim B=\varphi(3d), \tag{9.2} $$

and

$$ \bigwedge_F^2H^1(B,\mathbf Q) \subset\operatorname{cl}CH^1(B)_{\mathbf Q}. \tag{9.3} $$

The last assertion also follows abstractly from the Lefschetz $(1,1)$ theorem, but the common quotient gives the same explicit Aoki source and Poincaré transfer as in higher rank.

The monodromy independence argument changes because $\mathrm{Std}_{SL_2}\simeq\mathrm{Std}_{SL_2}^{\vee}$. We record the exact exception.

Lemma 9.1 (rank-two projective coupling). For the family (9.1), distinct real primitive factors are projectively equivalent if and only if $d=4$. For $d=4$, the two real factors form one graph.

Proof. Move the branch point of exponent $a_0=1$. The three projective local eigenvalue ratios are

$$ \delta_i(t)=\zeta_m^{t b_i},\qquad (b_1,b_2,b_3)=(d+2,2d+2,3d-2). \tag{9.4} $$

Suppose $\mathbf P\rho_u\simeq\mathbf P\rho_t$, and put $v=ut^{-1}\pmod m$. For a rank-two pseudoreflection, projective normalization may either preserve or interchange its two eigenvalues. If $S\subset\{1,2,3\}$ is the set of interchanged generators, the necessary and sufficient congruences are

$$ (v-1)b_i\equiv0\quad(i\notin S),\qquad (v+1)b_i\equiv0\quad(i\in S), \tag{9.5} $$$$ v-1\equiv-\sum_{i\in S}b_i\pmod{3d}. \tag{9.6} $$

The eight subsets give

$$ \begin{array}{c|c} S&\text{consequence}\\ \hline \varnothing&v=1\\ \{1\}&d=4,\ v=7\\ \{2\}&\text{no solution}\\ \{3\}&v=3\text{, not a unit}\\ \{1,2\}&v=-3\text{, not a unit}\\ \{1,3\}&d\mid4\text{, and the }d=4\text{ value is not a unit}\\ \{2,3\}&d=4,\ v=5\\ \{1,2,3\}&v=-1. \end{array} \tag{9.7} $$

The cases $v=\pm1$ are the same real factor. Thus a distinct graph occurs only for $d=4$. Conversely, if $d=4$, then

$$ m=12,\qquad\mathbf a=(1,5,9,9). $$

For $t=1,u=5$, the tuples of ratios are

$$ (-1,\zeta_{12}^{10},\zeta_{12}^{10}), \qquad (-1,\zeta_{12}^{2},\zeta_{12}^{2}). $$

If $(g_1,g_2,g_3)$ is the normalized $t=1$ tuple, set $(h_1,h_2,h_3)=(g_1,\zeta_{12}^2g_2,\zeta_{12}^2g_3)$. The eigenvalue ratios are those for $u=5$, and

$$ h_1h_2h_3=\zeta_{12}^4(\zeta_{12}I_2)=\zeta_{12}^5I_2. $$

Thus it has the correct product scalar, while its classes in $PGL_2$ agree with those of the first tuple. Equivalently, after replacing the swapped generators by $c_i=\delta_i^{-1}g_i$, one obtains the normalized rank-one Jordan--Pochhammer tuple with the same product scalar. The uniqueness statement for normalized Jordan--Pochhammer systems therefore gives $\mathbf P\rho_5\simeq\mathbf P\rho_1$. Rohde's factorwise surjectivity then makes connected monodromy the graph. $\square$

Proposition 9.2 (generic group and commutant). If $d\ne4$, then

$$ Hg(B)=\operatorname{Res}_{F^+/\mathbf Q}SU_F(1,1) \tag{9.8} $$

at a very general point, and $\operatorname{End}^0(B)$ is a totally indefinite quaternion algebra over $F^+$, containing $F$ as a maximal subfield. If $d=4$, then $Hg(B)_{\mathbf R}$ is the graph in $SU(1,1)^2$, and $\operatorname{End}^0(B)$ is a degree-four central simple algebra over $\mathbf Q$, split at the real place.

Proof. For $d\ne4$, Lemma 9.1 and Goursat give full restriction-of-scalars monodromy. The determinant divisors bound the generic Hodge group above by the same special-unitary group, proving (9.8). Over $\overline{\mathbf Q}$,

$$ H^1(B)_{\overline{\mathbf Q}} \simeq \bigoplus_{\tau:F^+\hookrightarrow\overline{\mathbf Q}} W_\tau^{\oplus2}, \tag{9.9} $$

where $W_\tau$ is the standard $SL_2$-module. Hence the complexified commutant is $\prod_\tau M_2$. Its rational form has center $F^+$, degree two, and $F$ as a maximal quadratic subfield. It is therefore a quaternion algebra. The noncompact real group $SU(1,1)$ makes the algebra split at every real place.

If $d=4$, Lemma 9.1 gives a monodromy graph in $SL_2^2$. Connected monodromy is normal in the derived generic Hodge group, while the connected normalizer of this graph in $SL_2^2$ is the graph itself. The determinant divisors again bound the Hodge group inside $SL_2^2$, so the generic Hodge group equals the graph. It acts on four copies of its standard module. The complex commutant is $M_4(\mathbf C)$; its rational form has center $\mathbf Q$, degree four, and is split over $\mathbf R$. Indeed, the nontrivial element of $\operatorname{Gal}(F^+/\mathbf Q)$ exchanges the two real factors, while complex conjugation exchanges standard and dual constituents; the graph and self-duality put all four constituents in one Galois orbit. Hence the rational commutant is central over $\mathbf Q$. The Albert classification leaves $M_4(\mathbf Q)$ or $M_2(D_0)$, where $D_0/\mathbf Q$ is an indefinite quaternion division algebra. Thus the simple isogeny factors have type I or II in all cases. $\square$

We first make explicit the contraction-to-divisor input needed in the self-dual case. Let $Y=B^s$ be any self-power and write, after complexification,

$$ H^1(Y,\mathbf C)=\bigoplus_{\tau}(W_\tau\otimes M_\tau), \qquad W_\tau=\operatorname{Std}_{SL_2}. \tag{9.10} $$

For $d\ne4$, the multiplicity spaces have the quaternionic commutant described in Proposition 9.2; for $d=4$, there is one graph factor and the same statement holds with its central-simple commutant. The double-centralizer identity

$$ \operatorname{End}_{\mathrm{HS}}H^1(Y,\mathbf Q) =\operatorname{Cent}_{\operatorname{End}_{\mathbf Q}H^1(Y,\mathbf Q)} (Hg(Y)) \tag{9.11} $$

shows that the matrix coefficients on every $M_\tau$ are type-$(0,0)$ Hodge endomorphisms (with their Galois orbit taken for the rational form). The invariant alternating form $\varepsilon_\tau\in\bigwedge^2W_\tau$ is the corresponding component of the polarization. Hence

$$ \varepsilon_\tau\otimes\operatorname{Sym}^2(M_\tau) \subset H^2(Y,\mathbf Q)\cap H^{1,1}(Y). \tag{9.12} $$

Under the Lefschetz identification $\operatorname{NS}(Y)_{\mathbf Q}\simeq \operatorname{End}^0(Y)^{\dagger=1}$, these rational classes are divisor classes. The first fundamental theorem for $SL_2$, together with exactness of invariants in characteristic zero, now shows that all $Hg(Y)$-invariants in the exterior algebra $H^\bullet(Y,\mathbf Q)$ are generated by these contraction divisors. In particular every self-power is divisor-generated, so $B$ is stably nondegenerate.

Theorem 9.3. For $p=3,d\ge3$, every power of a very general $B_{3,d,\mathbf b}$ satisfies the rational and generalized Hodge conjectures.

Proof. After complexification, each simple Hodge-group factor is $SL_2$, acting on several copies of its standard representation. The contraction-to-divisor argument above shows that the alternating contraction generators furnished by the first fundamental theorem for $SL_2$ are rational divisor classes. Consequently the Hodge ring of every self-power is generated by divisors: $B$ is stably nondegenerate and all powers satisfy the rational Hodge conjecture.

Every simple component has Albert type I or II by Proposition 9.2. The same is true for $B^r$, which remains stably nondegenerate. Hazama's Theorem 5.1 [18] therefore gives the generalized Hodge conjecture for $B^r$, for every $r\ge1$. This proves Theorem 1.7. $\square$

10. Examples and scope

The contraction application already begins with the conductor-six tuple

$$ \eta_{3,1}=(1,1,4,4,4,4), \qquad n=4, \qquad \dim B=4. $$

Its $k$-fold juxtaposition is $(1^{2k},4^{4k})$, with

$$ n_k=6k-2, \qquad q_k=3k-1, \qquad \dim B_k=6k-2, \qquad \dim M_{0,6k}=6k-3. $$

The first values are

$$ \begin{array}{c|c|c|c|c} p&d&m&\mathbf a_{p,d}&\dim B_{p,d}\\ \hline 3&3&9&(1,4,7,6)&6\\ 3&4&12&(1,5,9,9)&4\\ 5&3&15&(1,4,7,10,13,10)&16\\ 7&3&21&(1,4,7,10,13,16,19,14)&36. \end{array} $$

The smallest juxtaposed member has $\ell=5,d=1,m=15$:

$$ \mathbf a_{5,1}^{\mathrm{jux}} =(1,6,11,12;1,4,7,10,13,10), \qquad n=8, \qquad \dim B=32. $$

The odd-prime multi-standard version of the fixed-conductor phenomenon starts at $m=15$. Put

$$ \alpha=(1,4,7,10,13,10) $$

and juxtapose $k$ copies of $\alpha$. Then

$$ n_k=6k-2, \qquad q_k=3k-1, \qquad \dim B_k=8(3k-1)=24k-8, \qquad \dim M_{0,6k}=6k-3. $$

Thus a single conductor and the single field $\mathbf Q(\zeta_{15})$ support moving examples of arbitrarily large rank and dimension.

For every displayed family, determinant algebraicity holds at every point of the branch-configuration space. The endomorphism, Hodge-group, usual-Hodge, and generalized-Hodge statements hold on the very-general locus, with the $p=3$ group and commutant interpreted as in Section 9.

The method does not prove a theorem for arbitrary ramified cyclic Pryms. At prime-power conductor it covers every exact balanced residue tuple. At general conductor it additionally requires algebraicity of the complete primitive Fermat character orbit. That source condition may come from Aoki's standard cycles, the Shioda--Ran juxtaposition and deletion theorems recalled by Aoki, or another source theorem; it is not known for every balanced tuple at general conductor. The generic group theorem uses Rohde's factorwise monodromy theorem together with the exact Jordan--Pochhammer coefficients. It does not give explicit equations for the transferred Chow cycles on $B$; it gives them as images of Aoki cycles under explicit algebraic correspondences. Nor does it decide the all-powers conjectures at every special Hodge locus, where the endomorphism algebra and Hodge group may jump.

The citation-dependent representation inputs are Rains's Jordan--Pochhammer identification and Rohde's equality on each real factor. The Goursat reduction, projective rigidity, even-$d$ sign separation, and passage back to the connected $\mathbf Q$-group are proved in Sections 6 and 9.

Reproducibility

The evidence for the main theorem is the complete natural-language argument in Sections 2--9. Section 3 defines the common quotient and the finite fibre-product correspondence; Section 4 identifies the algebraic Fermat source orbit; Section 5 gives the Poincare and small-diagonal operator with its codimension and normalization; Section 6 proves monodromy independence; Sections 7--9 give the invariant-theoretic and generalized-Hodge consequences. The explicit tuples in Section 10 can be checked directly with the residue-sum formulas of Section 2. No numerical period approximation, finite-level census, private calculation output, or unprovided code is an input to the proof. No computational supplement is required.

The manuscript is submitted for mathematical review. AI assistance is disclosed above; no independent human verification of the complete proof is asserted. Copyright remains with the author, subject to the publication and review license in rStation's terms.

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