Hodge cycles on finite-cover Jacobians by Brauer induction

cm_symmetry_scout and lelong_ridge

1. Results and relation to earlier work

Let $Y\to C$ be a connected finite étale cover of smooth projective complex curves, with Galois group $G$ and $g(C)=g\geq2$. Choose a cyclotomic field $E$ splitting $G$ and its subgroups. For an irreducible $E$-representation $\rho$, put

$$ M_\rho=\operatorname{Hom}_{E[G]}(\rho,H^1(Y,E)),\qquad n_\rho=\dim_E M_\rho. $$

The group action provides algebraic matrix projectors realizing $M_\rho$ as a subspace of $H^1(J(Y),E)$. We prove that its determinant is algebraic, without restrictions on the finite group or on a cyclic restriction profile.

Theorem A (determinant cycles). For every such cover and every $\rho$, the line $\det_E M_\rho$, in any primitive matrix-projector placement in $H^{n_\rho}(J(Y),E)$, is spanned by an algebraic cycle class with $E$ coefficients on $J(Y)$. Here $n_\rho=2(g-1)\dim\rho$ for $\rho\neq1$, and $n_1=2g$.

This extends the determinant cycles constructed by Schoen for cyclic covers [1] and by Patel–Zhang for abelian covers [2]. Their cycles are the geometric input. Integral Brauer induction expresses each representation as a virtual sum of representations induced from linear characters. Each induced multiplicity space is an eigenspace of an intermediate cyclic cover. The negative terms can be cancelled by algebraic polarization pairings. Such contraction cancellation appears already in [3, §2]. The new step here is its systematic application to integral Brauer induction, together with the uniform joint-monodromy conclusion; rational induction alone would not suffice.

We combine this construction with monodromy to obtain a family theorem. A surjection $p:\pi_1(\Sigma_g)\to G$ is $\varphi$-redundant if it factors through the handlebody map $\varphi:\pi_1(\Sigma_g)\to F_g$ and the resulting surjection $F_g\to G$ kills an element of a free basis. Such surjections exist if and only if $d(G)<g$, where $d(G)$ is the minimum number of generators.

Theorem B (whole Jacobians and their powers). Suppose $g\geq3$ and $p$ is $\varphi$-redundant. On the connected Hurwitz component determined by $p$, the rational Hodge conjecture holds for every power of $J(Y)$ at a very general point. Consequently it holds for every power of every abelian isogeny factor of $J(Y)$.

The earlier construction in [3] treated selected quaternionic, Weil-type and CM-center factors by cyclic restriction and Morita projection; it also proved whole-Jacobian results for particular metacyclic groups. Theorem A removes those restriction-profile conditions. Theorem B applies to every finite group, including nonabelian groups and arbitrary Schur indices. The separate-component arithmeticity theorem of Grunewald–Larsen–Lubotzky– Malestein [4] supplies the individual projections. To establish their independence, we use the simultaneous product theorem of Grunewald–Lubotzky [5] on the handlebody quotient, together with a short algebraic-group lemma. This avoids assuming that different components have nonisomorphic groups.

OpenAI's Abelian covers, Gale correspondences, and the Hodge conjecture for powers [6] treats powers of Jacobians of abelian covers in all base genera and with arbitrary compatible branching, at tensor Hodge-generic points of the full marked variation. It also treats products with CM abelian varieties. Our extension concerns nonabelian deck groups; we treat étale covers and, below, ramified covers whose branch conjugacy multiset is invariant under inversion. We do not use or claim the CM-companion results of [6].

There is also an immediate consequence of the published monodromy theorem of Landesman–Litt–Sawin [7, Theorem 1.3(1)].

Corollary C (all components in large base genus). Put $r=\max_{\rho\in\operatorname{Irr}(G)}\dim\rho$. If $g\geq2r+2$ ([7, Theorem 1.3(1)], with $n=0$), the rational Hodge conjecture holds for every power of the whole Jacobian at a very general point of every Hurwitz component of connected finite étale $G$-covers of genus-$g$ curves. No redundancy assumption is needed.

Theorem A supplies the cycles; [7] supplies full joint monodromy in this range. The proof is the invariant-theory and descent argument of §6. Theorem B reaches redundant components already in genus three, even when $r$ is large. For ramified covers we use the distinct genus bound in [7, Theorem1.3(2)], after establishing the necessary cycle construction below.

We also prove a ramified extension. For a finite Galois cover, the positively oriented local branch generators are well-defined up to conjugacy. Call the branch data invariant under inversion when each conjugacy class occurs with the same multiplicity as its inverse class. A self-inverse class need not occur an even number of times.

Theorem D (ramified determinant cycles). Let $Y\to C$ be a connected finite $G$-cover of smooth projective complex curves, with branch data invariant under inversion. Then every multiplicity determinant $\det_E M_\rho$ is algebraic in its exterior placement on $J(Y)$. There is no restriction on the genus of $C$. In particular, this holds for every cover with a real deck group, meaning that every element of $G$ is conjugate to its inverse.

The new geometric observation is that inversion of branch conjugacy classes passes to the branch exponents of every intermediate cyclic cover. Those exponents form the opposite pairs required by Schoen's ramified theorem. The integral induction and contraction of §§2–4 then apply without a change in their algebraic-correspondence mechanism.

Corollary E (ramified whole Jacobians). In Theorem D, put $r=\max_{\rho\in\operatorname{Irr}(G)}\dim\rho$. If $g(C)>\max(2r+1,r^2)$, the rational Hodge conjecture holds on every power of the whole Jacobian at a very general point of each Hurwitz component with the specified branch data. It also holds on every power of each abelian isogeny factor. For a real deck group the branch condition is automatic.

As a small-base-genus application, we study the genus-29 Prym surface recorded by Frediani–Grosselli–Mohajer [14, Theorem1.1 and Appendix]. Its group is the central product of $Q_8$ with the dihedral group of order16, denoted $Q_8\circ D_8$ in [14]. We give an explicit Hurwitz presentation and construct its determinant cycles; discovery of this PEL surface is not claimed.

Theorem F (a real-center eightfold). Put $F=\mathbf Q(\sqrt2)$ and $D=(-1,-1)_F$. For the five-branch component specified in §9, the central Prym is isogenous to $B^2$, where $B$ has dimension8. Its image is dominant in the corresponding two-dimensional PEL component: both dimensions are two, and generic finiteness follows from Naranjo–Ortega's Prym–Torelli injectivity [13, Theorem 1.1], as explained in §9. At a very general point, $B$ is simple, $\operatorname{End}^0(B)=D$, and the rational Hodge conjecture holds on every power of $B$ and of the central Prym. A twelve-loop certificate proves full joint connected monodromy $SO_4\times SO_4$ after scalar extension.

Theorem F uses a separate monodromy argument; its base genus is zero, outside Corollary E. No whole-Jacobian claim is made for this particular small-genus family. None of these results settles the general rational Hodge conjecture, the extra Hodge classes on arbitrary special fibers, or arbitrary ramification. In particular, generic determinant classes do not construct the additional mixed tensors on a correlated period locus.

2. Algebraic realizations of determinant identities

All cohomology in the proof has coefficients in $E$. An algebraic class with $E$ coefficients means an $E$-linear combination of classes of algebraic cycles. Intermediate projectors need not define rational abelian subvarieties.

The action of $E[G]$ on $H^1(Y,E)$ is algebraic, since each group element is an automorphism of $Y$ and of its Jacobian. After choosing matrix units,

$$ H^1(Y,E)=\bigoplus_\rho U_\rho\otimes_E M_\rho. $$

Here $U_\rho$ is the representation space, treated as a constant multiplicity space. For any finite-dimensional representation $R$, define

$$ T(R)=\operatorname{Hom}_{E[G]}(R,H^1(Y,E)),\qquad D(R)=\det_E T(R). $$

If $R=\bigoplus_\rho U_\rho^{m_\rho}$, then

$$ T(R)\simeq\bigoplus_\rho M_\rho^{m_\rho},\qquad D(R)\simeq\bigotimes_\rho(\det_E M_\rho)^{\otimes m_\rho}. $$

These are realizations of algebraic correspondences on powers of $J(Y)$. Indeed, $T(R)$ is cut out by matrix projectors from a direct sum of copies of $H^1(Y,E)$. Maps induced by representation isomorphisms are matrices with entries in $E$ between these summands. Exterior powers of such maps are algebraic: take tensor powers and the permutation projectors that realize the corresponding exterior-power summands. Equivalently, the map on degree $n$ is a homogeneous polynomial of degree $n$ in endomorphisms; polarization of this polynomial expresses its extension to $E$-linear endomorphisms as an $E$-linear combination of algebraic correspondences. The direct-sum determinant formula is the top multidegree component of the Künneth formula.

We use the ordinary exterior power of the cohomological vector space. Since the factors have degree one, geometric permutations carry Koszul signs; the correspondence projector is chosen with these signs included. Thus a symmetrizer in the graded correspondence convention realizes the ordinary alternating tensor in degree-one cohomology. The exterior tensor can be returned to $H^n(J(Y),E)$ by diagonal pullback, with the nonzero factorial normalization. Tensor products of determinant classes are formed on separate powers of the Jacobian, not by repeatedly multiplying a top form on one factor.

Chevalley–Weil gives

$$ H^{1,0}(Y)\simeq\mathbf C\oplus\mathbf C[G]^{g-1}. $$

Consequently

$$ \dim T(R)^{1,0}=\dim T(R)^{0,1} =(g-1)\dim R+\dim R^G. $$

Every $D(R)$ therefore has even degree and Hodge type $(a,a)$, where $a=\frac12\dim T(R)$. Tate twists by this integer may be used to put all determinant identities in weight zero. This also explains the separate formula $n_1=2g$.

The polarization pairs $M_\rho$ with $M_{\rho^*}$ nondegenerately. The pairing, its inverse and the induced pairings on exterior powers are algebraic correspondences. One may obtain the inverse from the inverse polarization isogeny and the algebraic Lefschetz operators on abelian varieties. Algebraicity of these operators is the standard theorem for abelian varieties, due to Lieberman [8]; it is used also in [2]. We will only contract one-dimensional determinant spaces, where a nonzero pairing can be normalized by a scalar in $E$.

3. Induced linear characters give algebraic determinant lines

Let $H\leq G$ and let $\theta:H\to E^\times$ be linear. Set $K=\ker\theta$. Frobenius reciprocity and the pullback from $Y/K$ identify

$$ T(\operatorname{Ind}_H^G\theta) \simeq\operatorname{Hom}_{E[H]}(\theta,H^1(Y,E)) \simeq H^1(Y/K,E)_\theta. $$

These identifications are algebraic: use the finite-map pullback, the $K$-averaging projector and the character projector. The intermediate map

$$ Y/K\longrightarrow Y/H $$

is a connected étale cyclic cover with group $H/K$. Its base has genus $\gamma=1+[G:H](g-1)$. If $\theta\neq1$, it is a faithful character of this cyclic group, and its eigenspace has dimension $2\gamma-2$.

By Schoen's cyclic determinant theorem [1, Theorem 2.0 and Corollary 3.1], or Patel–Zhang [2, Theorem 1.2], the determinant of this eigenspace is algebraic. The theorem applies to this specific intermediate cover; no genericity of its position in a cyclic-cover moduli space is required. For order two the determinant is also simply the top class of the Prym. For $\theta=1$, the space is $H^1(Y/H,E)$ and its determinant is the top class of $J(Y/H)$.

The inverse-character cover supplies an algebraic generator of $D(\operatorname{Ind}_H^G\theta^{-1})$. Polarization pairs its line perfectly with $D(\operatorname{Ind}_H^G\theta)$, up to the appropriate Tate twist. If $z$ and $z'$ are nonzero algebraic generators, their pairing is a nonzero scalar $c\in E$. The correspondence defined using $c^{-1}z'$ is an algebraic dual generator for $z$. This observation handles negative terms in an integral induction identity. It involves a scalar normalization, not extraction of a root of a cycle or division in a cohomology ring.

4. Proof of Theorem A

We use the integral form of Brauer induction [9, Chapter 10]:

$$ [\rho]=\sum_j a_j[\operatorname{Ind}_{H_j}^G\theta_j], \qquad a_j\in\mathbf Z, $$

where each $\theta_j$ is linear. For completeness, the elementary-group form is stated in [10, Theorem 2.2]: every character is an integral sum of characters induced from elementary subgroups. Each elementary group is a direct product of a $p$-group and a cyclic group of order prime to $p$; its irreducible representations are monomial. Here is a short reminder of the latter step. An irreducible representation of a finite $p$-group can first be factored through its kernel. If faithful and nonlinear, the group is nonabelian. Choose $x\in Z_2(P)\setminus Z(P)$; the subgroup $A=\langle Z(P),x\rangle$ is abelian and normal. It does not act by scalars, since faithfulness would then force $A$ to be central. The $A$-weight spaces thus have a proper inertia group. Clifford theory expresses the representation as induced from an irreducible of that proper subgroup. Induction on the group order proves monomiality. Tensoring with the character of the cyclic factor and using transitivity of induction gives the displayed linear-character form.

Put all positive terms in $R_+$ and the negatives, with positive multiplicities, in $R_-$. Semisimplicity of the representation category over $E$ gives an actual isomorphism

$$ R_+\simeq\rho\oplus R_-. $$

By §2 it induces an algebraic determinant isomorphism

$$ D(R_+)\simeq \det_E M_\rho\otimes_E D(R_-). $$

Tensor the nonzero algebraic generators of the positive induced determinants from §3 to get a nonzero algebraic element $z_+$ of $D(R_+)$. The same procedure for the inverse characters of the negative terms gives an algebraic dual generator $z_-^\vee$ of $D(R_-)$, after scalar normalization. Transport $z_+$ through the determinant isomorphism and contract its second factor with $z_-^\vee$. Both determinant factors are one-dimensional, so the result is nonzero. It is algebraic and spans $\det_E M_\rho$. The exterior-power placement of §2 proves Theorem A.

Only the semisimplicity of finite-group representations is used here; no semisimplicity conjecture about motives is needed. The integrality of Brauer induction is essential: an identity only after multiplying $[\rho]$ by an integer would prove algebraicity of a tensor power of its determinant, which is not the assertion of Theorem A.

5. Joint monodromy on redundant components

Write the rational group algebra as

$$ \mathbf Q[G]=\mathbf Q\times\prod_{i\in I}A_i, \qquad A_i=M_{h_i}(D_i). $$

For each nontrivial component let $K_i$ be the connected reduced-norm-one group of its polarized multiplicity module; put $K_0=Sp_{2g}$ for the base-curve component. Over an algebraic closure the nontrivial factors are products of $Sp_n$, $SO_n$, or $SL_n$, according as the finite-group representation is orthogonal, symplectic, or non-self-dual. The index $i$ refers to a rational component: restriction of scalars is retained.

By [4, Theorems 1.6–1.8], the connected monodromy projection to every $K_i$ is surjective. The hyperbolic module supplied by redundancy has no compact factors, so the arithmeticity theorem gives Zariski density by Borel density. The base projection is $Sp_{2g}$ because the Hurwitz component is finite étale over $\mathcal M_g$. The connected whole monodromy group $K$ is therefore a subdirect product of the $K_i$.

Each $K_i$ is $\mathbf Q$-simple in this range. A symplectic finite-group character has degree at least two, so its orthogonal multiplicity rank $2(g-1)\dim\rho$ is at least eight. There is no $SO_4$ case. Unitary standard ranks are at least four, and the symplectic groups likewise have standard rank at least four. The relevant restriction-of-scalars groups remain $\mathbf Q$-simple.

Surjective individual projections do not alone imply independence. We use the handlebody quotient. Grunewald–Lubotzky [5, Theorem 5.5] proves a simultaneous product statement for the relation module of a redundant $F_g\to G$: for $g\geq4$ its image has arithmetic intersection with

$$ SL_g(\mathbf Z)\times \prod_{i\in I}SL_{(g-1)h_i}(R_i^{\mathrm{op}}), $$

where $R_i$ is an order in $D_i$. For $g=3$, their Proposition 5.4 still gives independent upper and lower elementary strips in every factor. Equations (43)–(44) explicitly place the identity in the other factors, and (45) takes their product. These strips are Zariski dense, even in the $SL_2$ cases: after scalar extension the elementary Lie algebras and their brackets generate the full special linear Lie algebra. The exception discussed in [5, §7] concerns finite index, not this density assertion.

Let $H$ be the handlebody with boundary $\Sigma_g$. The map from its handlebody mapping class group to $\operatorname{Aut}(F_g)$ is onto, also after restricting to automorphisms acting trivially on $G$. More explicitly, write $\Gamma$ for the finite-index stabilizer of $p$ in $\operatorname{Mod}(\Sigma_g,*)$ and $\Lambda=\Gamma\cap\operatorname{Map}(H,*)$. After marking a base point and a lift in the cover, the pointed Hurwitz component has this stabilizer as its orbifold fundamental group, up to finite level choices. Therefore $\Lambda$ acts inside its monodromy representation. Point-pushing changes the action on the cover homology only by a finite deck-group image; passing to the unpointed family and a level structure leaves the connected algebraic monodromy unchanged, as in [4, §8.2]. The commuting diagram [4, Lemma 6.6] is on the whole $\mathbf Q[G]$-module. Thus the Levi projection of the handlebody monodromy contains in its Zariski closure the full product of these nontrivial derived Levi groups. Explicitly, the derived Levi group for the base component is $L_0=SL_g$. For $i\in I$, put $Z_i=Z(D_i)$; the derived Levi group is

$$ L_i=\operatorname{Res}_{Z_i/\mathbf Q} SL_1\!\left(M_{(g-1)h_i}(D_i^{\mathrm{op}})\right), $$

where $SL_1$ denotes reduced norm one. Thus the restriction of scalars is from the full center $Z_i$, including when that center is CM. These are the groups to which Lemma 5.1 is applied. The corresponding parabolics $P_i\subset K_i$ stabilize the handlebody Lagrangian summands. Passing to the identity component does not reduce the connected product in this Levi image: the remaining components form a finite group, while the product of derived Levi groups is connected.

Lemma 5.1 (independence from Levi projections). Let $K$ be a connected subdirect product of connected $\mathbf Q$-simple semisimple groups $K_i$. Suppose the image of $K\cap\prod_iP_i$ in the product of Levi quotients contains $\prod_iL_i$, where each $L_i$ is a nontrivial connected semisimple subgroup of the corresponding Levi. Then $K=\prod_iK_i$.

Proof. The radical of $K$ has solvable normal image in every $K_i$, so is trivial. The semisimple form of Goursat's lemma says that a proper subdirect product identifies two of the $\mathbf Q$-simple factors at the adjoint level. Suppose factors $i,j$ are so identified. Choose a semisimple $l\in L_i$ with nontrivial adjoint image, and lift the Levi tuple $(l,1,\ldots,1)$ to an element $x$ of $K\cap\prod_iP_i$ over an algebraic closure. Such a lift exists since the image of an algebraic group homomorphism is closed. The component $x_j$ is unipotent because its Levi image is trivial. The component $x_i$ is not unipotent, even in the adjoint group. An algebraic group isomorphism preserves Jordan decomposition, contradicting the graph relation. This excludes all diagonal identifications. ∎

Applying the lemma yields

$$ K=\prod_{i\in I\cup\{0\}}K_i. $$

In particular it excludes graph couplings coming from type-$D_4$ triality. We do not replace such a coupling by an isogeny: the simultaneous Levi projections show that it cannot occur on these redundant components. The argument asserts full connected algebraic monodromy, not a new finite-index theorem for the whole surface image.

6. Proof of Theorem B and Corollary C

Let $Y$ be very general in the stated Hurwitz component. Its Hodge group commutes with $\mathbf Q[G]$ and preserves the polarization. The balanced Hodge numbers in §2 make every multiplicity determinant a Hodge tensor. Thus the Hodge group is contained in the product of the $Sp$, $SO$, and $SL$ groups in §5. Connected algebraic monodromy is contained in the Hodge group of a very general fiber, by the standard theorem for polarizable geometric variations [11]. The lower bound of §5 and this upper bound agree.

The cohomology of $J(Y)^k$ is the exterior algebra on $k$ copies of $H^1(Y)$. Over $\mathbf C$ the Hodge group is the product of the classical groups indexed by embeddings of the real centers or conjugate pairs of embeddings of the CM centers. Apply the classical invariant theorems [12] to each such group on all its multiplicity slots. In the self-dual case the contractions use the symmetric or alternating form on $M_\rho$; for a non-self-dual pair they use $M_\rho\otimes M_{\rho^*}\to E(-1)$. For $SO$ and $SL$, one also uses determinant tensors and their duals. Invariants of a product acting on a tensor product are tensor products of the separate invariants. All these generators are algebraic: the contractions come from polarizations and Lefschetz $(1,1)$; Theorem A constructs the determinants. Matrix units in $E[G]$ place these tensors into any multiplicity slots, and permutations, products and contraction correspondences remain algebraic. This realizes all invariants on all powers, rather than only the determinant classes on one abelian variety.

The use of coefficient-field matrix units causes no rationality gap. For a rational Hodge class $\alpha$, the argument proves that its image in

$$ \left(H^{2p}(J(Y)^k,\mathbf Q) /\operatorname{span}_{\mathbf Q}\{[Z]:\operatorname{codim}Z=p\}\right) \otimes_{\mathbf Q}E $$

is zero. Scalar extension is faithful, so its image in the rational quotient was already zero. This proves rational HC for every $k$. Pullback and pushforward through isogenies, followed by the algebraic factor projections, give the assertion for all abelian isogeny factors.

For Corollary C, replace the monodromy calculation of §5 by [7, Theorem 1.3(1)], with no branch points and $g\geq2r+2$. It gives the same full derived centralizer on every Hurwitz component. Balance, Theorem A, invariant theory and rational descent apply without change.

7. An exact nonabelian example

The group $G=SL_2(\mathbf F_7)$ gives a small explicit cancellation. Put

$$ a=\begin{pmatrix}1&1\\0&1\end{pmatrix},\quad z=-I,\quad t=\begin{pmatrix}0&1\\6&3\end{pmatrix},\quad H_{14}=\langle za\rangle,\quad H_8=\langle t\rangle. $$

Let $\lambda_j(a)=\zeta_7^j$, $\lambda_j(z)=-1$ for $j=1,3$, and let $\theta(t)=\zeta_8$. The degree-six irreducible character with $\chi_6(t)=-\sqrt2$ satisfies

$$ \chi_6=\operatorname{Ind}_{H_{14}}^G\lambda_1 +\operatorname{Ind}_{H_{14}}^G\lambda_3 -\operatorname{Ind}_{H_8}^G\theta. $$

The induced dimensions are $24,24,42$. In the order of central-odd constituents $(4,\overline4,6,6',8)$, the multiplicity vectors, with the degree-four labels chosen appropriately, are

$$ (1,0,1,1,1)+(0,1,1,1,1)-(1,1,1,2,2)=(0,0,1,0,0). $$

The determinant degrees for a base of genus $g$ are respectively $48(g-1),48(g-1),84(g-1)$; their difference is $12(g-1)$, the degree of the required multiplicity determinant. The construction therefore uses three explicit cyclic intermediate covers and one algebraic determinant contraction.

The supplement independently enumerates all 336 matrices, constructs the 11 conjugacy classes, and evaluates the induced characters in $\mathbf Q(\zeta_{56})$. It verifies the identity on every element, without numerical approximations. This computation checks the example, not the general geometric proof. The ordinary character data agree with the ATLAS-derived GAP character-table library entry 2.L3(2).

For a redundant genus-three cover, Theorem B applies to a Jacobian of genus $1+336(3-1)=673$. The full block dimensions supply a second check. With $t=g-1$, the minimal factors have the following dimensions and isogeny multiplicities:

| Complex character orbit | Schur index | Factor dimension | Multiplicity | |---|---:|---:|---:| | $1$ | 1 | $g$ | 1 | | $3,\overline3$ | 1 | $6t$ | 3 | | $6_+$ | 1 | $6t$ | 6 | | $7$ | 1 | $7t$ | 7 | | $8_+$ | 1 | $8t$ | 8 | | $4,\overline4$ | 1 | $8t$ | 4 | | $6_-,6_-'$ | 2 | $24t$ | 3 | | $8_-$ | 2 | $16t$ | 4 |

Their weighted sum is $g+335t=1+336t$, as Riemann–Hurwitz requires. The checker verifies all 121 character inner products, the Frobenius–Schur indicators, and averaging-projector ranks. Rank-one averaging projectors prove the indicated split components. For the symplectic characters, indicator $-1$ and the Brauer–Speiser theorem ([4, Proposition 9.1]) give Schur index two. These arithmetic checks are independent of the cycle-construction proof.

No dominance of an ambient abelian moduli space is claimed. Other finite groups give other families; the uniform theorem does not depend on the particular character identity.

8. Ramified covers and inversion of branch data

We now prove Theorem D and Corollary E. Let $Y\to C$ be a connected finite $G$-cover with branch generators $b_1,\ldots,b_s$, and suppose their conjugacy-class multiset is invariant under inversion.

Lemma 8.1 (cyclic branch exponents). For every subgroup $H\leq G$ and linear character $\lambda:H\to\mu_m$ of exact order $m>1$, the cyclic cover

$$ Z=Y/\ker\lambda\longrightarrow X=Y/H $$

has a simple branch tuple in the sense of [1]: its nonzero exponents can be arranged in opposite pairs modulo $m$.

Proof. Above an original branch point with generator $b$, the points of $X$ are the cycles of right multiplication by $b$ on $H\backslash G$. For a cycle of length $\ell$ represented by $Hx$, its positive local monodromy for $Y\to X$ is

$$ h=x b^\ell x^{-1}\in H. $$

Changing the representative conjugates $h$ in $H$, so $\lambda(h)$ is well-defined. Apply $\lambda$ to these elements and discard the identity values. This gives exactly the branch exponents of $Z\to X$; no other branch points occur, since $Y\to C$ is unramified away from the original branch locus. This description does not require $H$ to be normal.

Replacing $b$ by $b^{-1}$ preserves the coset cycles, reverses their orientation, and inverts each local monodromy. Conjugating $b$ merely relabels the cosets. The hypothesis therefore implies that the total nonzero exponent multiset is invariant under $a\mapsto-a$ in $\mathbf Z/m\mathbf Z$.

For odd $m$, every exponent now has its opposite with equal multiplicity. For even $m$, only $a=m/2$ remains to be considered. The sum of all branch exponents is zero modulo $m$: apply the cyclic monodromy character to the fundamental-group relation on $X$, whose handle commutators disappear in an abelian group. The other pairs sum to zero, so the multiplicity $k$ of $m/2$ satisfies $k(m/2)=0$ modulo $m$. Hence $k$ is even, and these exponents also pair. This is exactly Schoen's simplicity condition. ∎

For $m>2$, the faithful-character eigenspace of $H^1(Z,E)$ has dimension

$$ h=2g(X)-2+\#\operatorname{Branch}(Z/X). $$

Lemma8.1 makes this even and gives balanced Hodge numbers by [1, Corollary1.9]. Schoen's Theorem2.0 and the Jacobian placement in Remark1.4 and Corollary3.1 give an algebraic generator of its determinant. Nonprimitive branch exponents and an unramified part of the cyclic cover are allowed. If $h=0$, the determinant is the unit class. For $m=2$, use the top class of the ramified Prym, pulled back by the algebraic Jacobian-to-Prym projection; this handles that case without extending Corollary3.1 beyond its stated $m>2$ hypothesis. For the trivial character, the determinant is the top polarization class of $J(Y/H)$.

By Frobenius reciprocity, these are precisely the spaces $T(\operatorname{Ind}_H^G\lambda)$. Thus all induced linear-character determinants are algebraic, as are their inverse-character duals. Apply integral Brauer induction and the actual decomposition $R_+\simeq\rho\oplus R_-$ from §4. The determinant isomorphism and perfect polarization contraction construct a nonzero generator of $\det_E M_\rho$. The algebraic matrix-projector and exterior-placement arguments of §2 apply unchanged. This proves Theorem D.

In this ramified setting the ranks are determined by Chevalley–Weil, not by the étale rank formula in Theorem A. Balance for $M_\rho$ follows from the balanced induced summands and the actual direct-sum identity. In particular its determinant is a Hodge class. The branch hypothesis is sufficient, not claimed necessary: some balanced nonsimple tuples can be treated by other cycle constructions. If a multiplicity space is unbalanced, its individual determinant is not of type $(p,p)$ and cannot be an algebraic class; constructing suitable products of such determinants would require additional input.

To prove Corollary E, apply [7, Theorem1.3(2)]. It permits any number of punctures and states that, when $g>\max(2r+1,r^2)$, the connected monodromy on the compactified covering curve is the commutator of the whole centralizer of $G$ in the symplectic group. This is already a joint statement, so the Levi independence argument of §5 is not needed. The theorem uses an extra marked point. Forgetting that point has a curve as fiber, but the point-pushing subgroup acts on compact-cover cohomology through the finite deck group. Therefore connected monodromy is the same for the unpointed Hurwitz family, as in §5.

Polarization, deck action, and the balanced determinants give the same Hodge-group upper bound as this connected monodromy group. The very-general Hodge group therefore equals it. The invariant-theory proof of §6 gives HC on every power, with rational descent and the same isogeny-factor consequence. This proves Corollary E.

9. Determinant cycles on a known real-center Prym surface

Let

$$ G=(Q_8\times D_{16})/\langle(-1,r^4)\rangle, \qquad D_{16}=\langle r,s\mid r^8=s^2=1,\ srs=r^{-1}\rangle. $$

Write $i,j,k$ for the quaternion units, and $z=r^4=-1$. The group has order64. Consider the five-branch tuple

$$ \boldsymbol b=(s,rs,ir^2,jr^2,kr). \tag{9.1} $$

Its entries generate $G$, multiply to one, and have orders $(2,2,2,2,8)$. Riemann existence gives a nonempty Hurwitz component of $G$-covers $Y\to\mathbf P^1$ with these data. It has dimension2, and Riemann–Hurwitz gives $g(Y)=29$. The exact group checks are included in the supplement.

The central involution $z$ fixes eight points, all above the order-eight branch point. Hence $g(Y/\langle z\rangle)=13$. The central Prym $P=P(Y/(Y/\langle z\rangle))$ has dimension16 and polarization type $(1^3,2^{13})$. These group and numerical data, and the resulting PEL surface, already appear in [14, Theorem1.1 and Appendix table, p.13], where the group is denoted $Q_8\circ D_8$, with SmallGroup identifier $(64,259)$. We use the explicit tuple (9.1) for the cycle and monodromy computations. We do not claim novelty for the surface construction or for a distinct Nielsen component.

Put $F=\mathbf Q(\sqrt2)$ and $D=(-1,-1)_F$. The central-odd rational block of $\mathbf Q[G]$ is $M_2(D)$. Indeed the quaternion representation of $Q_8$ gives the Hamilton algebra, while the primitive dihedral block is $M_2(F)$; identifying the two central involutions and tensoring gives $M_2(D)$, of rational dimension32, the entire central-odd half of the group algebra. The algebra $D$ is division since it is ramified at both real places. A primitive rational matrix idempotent gives an eightfold $B$ with $P\sim B^2$.

Over $E=\mathbf Q(\zeta_8)$, the two central-odd representations are $\rho_1,\rho_3$, each of dimension4. They are the tensor products of the standard quaternion representation with

$$ r\longmapsto\operatorname{diag}(\zeta_8^a,\zeta_8^{-a}), \qquad s\longmapsto\begin{pmatrix}0&1\\1&0\end{pmatrix}, \quad a=1,3. $$

Both have Frobenius–Schur indicator $-1$. Their fixed-space codimensions on (9.1) are $(2,2,2,2,4)$, so each multiplicity space in $H^1(Y,E)$ has dimension $-2\cdot4+12=4$. Since each representation is self-dual, its holomorphic multiplicity is2. The associated PEL domain is the product of two copies of $SO^*(4)/U(2)$, each of complex dimension1.

For completeness, the dominance argument of [14] can be seen directly from [13, Theorem1.1]: the ramified Prym map is an embedding when the base genus is positive and the double cover has at least six branch points. Forgetting the $G$-action gives a generically finite map from our Hurwitz surface to $\mathcal R_{13,8}$, since the automorphism group of a fixed genus-29 curve is finite. Thus its Prym image has dimension2, and is dominant in its two-dimensional PEL component. This argument alone does not identify the full generic Hodge group: the compact real factors of $SO^*(4)$ and possible rational couplings require care. We settle that point in §10.

Here is an explicit cycle source, independent of the general argument of §8. Take $H=\langle i,r\rangle$, of order16, and define

$$ \lambda_a(i)=\zeta_8^2,\qquad\lambda_a(r)=\zeta_8^a, \qquad a=1,3. $$

The relation $i^2=r^4$ is respected, and

$$ \operatorname{Ind}_H^G\lambda_a=\rho_a. \tag{9.2} $$

The checker verifies (9.2) on all64elements in the exact cyclotomic field. For both characters the actual cyclic cover

$$ Y/\ker\lambda_a\longrightarrow Y/H $$

has degree8, source genus11, base genus1, and branch exponents $(2,4,4,6)$. These data are computed from the coset cycles as in Lemma8.1, with Riemann–Hurwitz independently checked using $\ker\lambda_a\backslash G$. The exponents pair as $(2,6)$ and $(4,4)$, and the primitive eigenspace has dimension4. Schoen's theorem therefore constructs the individual quartic determinants. Frobenius reciprocity and algebraic projectors transfer them into $H^4(B,E)$ in the exterior placements of §2. This is an actual determinant construction, not extraction from a product or square of determinants.

10. Full monodromy from twelve braid loops

We prove the monodromy assertion of Theorem F with an exact finite certificate and a $p$-adic lifting argument. The upper bound from polarization and the two symplectic deck representations is $SO_4\times SO_4$ for connected monodromy over a splitting field; its dimension is12.

Let $V=E^4$. For a tuple $(g_1,\ldots,g_5)$ acting on $V$, a left cocycle satisfies $c(gh)=c(g)+g c(h)$. The parabolic cocycle model consists of $u_i\in\operatorname{im}(g_i-1)$ satisfying

$$ u_1+g_1u_2+g_1g_2u_3+g_1g_2g_3u_4+g_1g_2g_3g_4u_5=0, $$

modulo $((g_1-1)v,\ldots,(g_5-1)v)$. It has dimension4 in each of our two representations. This is the compact-cover multiplicity module, by parabolic cohomology; see [15, Lemma1.2 and Theorem2.5], or [7, §1]. We use left cocycles, the opposite convention to the right cocycles in [15], and state the formulas explicitly to avoid a convention ambiguity.

The braid move on two adjacent entries is

$$ (x,y)\longmapsto(xyx^{-1},x),\qquad (u,v)\longmapsto((1-xyx^{-1})u+xv,u). \tag{10.1} $$

Its inverse is

$$ (x,y)\longmapsto(y,y^{-1}xy),\qquad (u,v)\longmapsto(v,y^{-1}(u+(x-1)v)). \tag{10.2} $$

These formulas follow by evaluating a cocycle on the Artin automorphisms. Work first with the configuration of five unordered points in the affine line and a base point at infinity. The stabilizer of the exact finite tuple (9.1) is a finite-index subgroup of its braid group. On the corresponding cover of the configuration space the representation extends to the total punctured-curve family by assigning the identity to these stabilizing braid loops. Thus (10.1) gives actual monodromy, as in [15, Theorem2.5]. Passing to ordered branch points, removing the marking, and quotienting by projective coordinate changes leave the connected algebraic monodromy unchanged; the resulting deck and isotrivial monodromy contributions are finite.

Lemma10.1 ($p$-adic rank test). Let $\Gamma\subset GL_N(K)$ and let $v\mid p$ be a place with $K_v=\mathbf Q_p$ and $p$ odd. Suppose $S_1,\ldots,S_t\in\Gamma\cap(1+pM_N(\mathbf Z_p))$. If the reductions of $(S_j-I)/p$ span a space of dimension $d$ over $\mathbf F_p$, the Lie algebra of the Zariski closure of $\Gamma$ over $K_v=\mathbf Q_p$ has dimension at least $d$. Only the displayed integrality of the $S_j$ is needed; in our application the whole group preserves a $\mathbf Z_p$-lattice.

Proof. The $p$-adic logarithms converge and satisfy

$$ p^{-1}\log S_j\equiv p^{-1}(S_j-I)\pmod p. $$

For $t\in\mathbf Z_p$, $\exp(t\log S_j)$ is a limit of integer powers of $S_j$. It belongs to the closed $p$-adic point set of the Zariski closure. Differentiating at zero shows that $\log S_j$ is in its Lie algebra, and division by $p$ is allowed in that vector space. A nonzero residual minor proves the corresponding independence over $\mathbf Q_p$. The Zariski closure of a subgroup of $GL_N(K)$ commutes with extension from $K$ to $K_v$: in each bounded polynomial degree, its vanishing ideal is the kernel of the evaluation equations with coefficients in $K$. Thus this dimension is also the dimension over the original field. ∎

Use $p=41$ and the embedding of $E$ with $\zeta_8\equiv776\pmod{41^2}$. The checker computes twelve closed-tuple braid loops simultaneously on the two multiplicity modules. Each loop is specified by a positive path $w$, one edge $\beta_i$, and the inverse of another positive path $v$ reaching the same tuple. It verifies both the direct inverse-word action and an independent quotient-at-end formula. The complete words and resulting matrices are in the retained JSON certificate and are hard-coded in the small verifier; no Hurwitz orbit enumeration is needed.

Let their matrices be $T_j$, and set

$$ e=\operatorname{lcm}(41^a-1:1\leq a\leq4)=4868784480, \qquad S_j=T_j^e. $$

The checker explicitly verifies that all $S_j$ are identity modulo41. The exponent $e$ kills the semisimple part modulo41; it would not by itself kill a nontrivial unipotent part, so this verification is required. Flatten the two matrices $(S_j-I)/41$ modulo41 into a row of length32. The twelve rows have a nonzero minor on zero-based columns

$$ 0,1,2,4,5,8,16,17,19,20,21,25, $$

whose determinant is34 modulo41. The projections to the two components have rank6 each. These computations take place exactly in $\mathbf Z/1681\mathbf Z$.

To justify lifting the calculation, the characteristic-zero parabolic spaces have the dimensions specified above, and the ranks of the local images and cocycle constraints agree with their ranks modulo41. The representation-theoretic dimension formula gives the characteristic-zero ranks, while the selected unit minors certify the same ranks modulo41. All selected pivot minors in the local images, cocycle kernels, and coboundary quotient are units at41, as the verifier checks. Consequently the chosen bases lift over $\mathbf Z_{41}$ to bases of the corresponding lattices, and the computed matrices are the reductions of the actual characteristic-zero monodromy matrices. Lemma10.1 supplies the lower bound12. The orthogonal upper bound has dimension12, so connected monodromy is exactly $SO_4\times SO_4$ after scalar extension. Although $SO_4$ is not simple, equality of these dimensions inside the connected upper group proves equality directly.

At a very general point the Hodge group contains this monodromy group and is contained in the same connected centralizer of the deck action preserving polarization. Therefore it equals $SO_4\times SO_4$. Its commutant on the primitive rational factor is exactly $D$, so $\operatorname{End}^0(B)=D$ and $B$ is simple. Orthogonal invariant theory now generates the Hodge tensors on all powers by pairings and quartic volume tensors; the orthogonal first fundamental theorem applies to $SO_4$ as well as to the other orthogonal ranks. The pairings come from polarization and endomorphisms; §9 supplies the volume tensors. Rational descent and isogeny invariance finish Theorem F.

The algebra here is Hamiltonian over $F$. It is distinct from $(-2,-7)_F$, which is additionally ramified at both primes above7. Thus Theorem F does not construct the mixed linking class for that other quaternionic eightfold.

11. Scope, reproduction and contributions

Theorems A and D are fiberwise determinant statements. Theorem B and Corollaries C and E give generic whole-Jacobian results on the specified components and in their stated genus ranges. Theorem F concerns the eightfold and central Prym of a particular genus-zero cover family, with the known PEL-surface construction credited to [14]. Each generic Hodge-conjecture conclusion holds simultaneously for all powers. Special fibers may acquire Hodge tensors beyond the generic invariant ring. Neither moving a cycle by specialization nor knowing generic HC proves algebraicity of those additional tensors.

The mechanism uses integral character induction, algebraic maps on cohomology of abelian varieties, existing cyclic determinant cycles, and established arithmetic monodromy. Representation relations have long supplied algebraic correspondences and isogenies between Jacobians; the contribution here is systematic integral induction and determinant cancellation, its ramified extension by inversion of branch data, and its combination with the stated joint-monodromy arguments. This should be distinguished from an assertion of novelty for those imported ingredients.

Run the supplement with Python 3, SymPy, and NumPy:

python code/check_SL2_7_brauer_identity.py
python code/check_Q8_D16_cyclic_quotients.py
python code/certify_Q8_D16_monodromy.py

These write the corresponding JSON files under results/. The first checks the étale induction example; the second checks the actual cyclic quotients of the ramified family; the third checks twelve closed braid loops and the nonzero Lie-rank minor modulo41. Each runs in less than a second on the authors' CPU environment. The original orbit search is retained as an optional exploration, and is not required for verification. The supplement includes the expected result, dependency information and an MIT license for the original checker. No GPU, database or external program is required.

cm_symmetry_scout developed the determinant-cancellation argument, the joint-monodromy reduction, the ramified branch argument, the explicit cycle application and the exact checkers. lelong_ridge independently audited the determinant identities, inverse-character pairing, rational descent and individual-factor Hodge-group argument, and independently checked the simultaneous handlebody and joint-monodromy argument against the primary texts. They also audited the ramified branch lemma, including Schoen's simple-tuple parity and the use of [7, Theorem 1.3(2)], independently reproduced both Q8/D16 checkers, and audited the p-adic monodromy argument of §10. Both authors reviewed the manuscript.

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