--- title: Hyperelliptic Prym Pairs in Covering Theory url: https://www.emergentmind.com/topics/hyperelliptic-prym-pairs type: topic --- # Hyperelliptic Prym Pairs in Covering Theory Searching arXiv for recent and foundational papers on hyperelliptic Prym pairs, hyperelliptic Prym maps, and related cyclic/double/Klein coverings. Hyperelliptic Prym pairs are geometric data in which Prym theory is constrained by hyperelliptic structure. In the most classical form, one starts with a smooth hyperelliptic curve \(H\) of genus \(g\ge 2\) and an étale cyclic covering \(f:X\to H\); the associated Prym variety is \(P(f)=\ker(\mathrm{Nm}_f)^0\), of dimension \((n-1)(g-1)\) when \(\deg f=n\) [1601.04082]. In a second, equally important form, one considers double coverings \(\tau:D\to C\) with hyperelliptic base \(C\), or hyperelliptic \(\mathbb Z_2^2\)-coverings in which both the covering and quotient curves carry additional involutions [2005.11108] [2302.13041]. Across these settings, hyperelliptic Prym pairs are distinguished by rigid quotient structures, explicit descriptions via Weierstrass points, and unusually strong relations between Prym varieties and Jacobians of quotient curves. Recent work has also promoted them to moduli-theoretic objects in their own right, with explicit quotient-stack descriptions and integral Chow ring computations [2501.16320] [2509.15186]. ## 1. Foundational definitions and basic geometry Let \(H\) be a smooth hyperelliptic curve of genus \(g\ge 2\), with hyperelliptic double cover \[ \pi:H\to\mathbb P^1 \] branched at the set \(W\) of \(2g+2\) Weierstrass points [1601.04082]. If \[ f:X\to H \] is an étale cyclic covering of degree \(n\ge 2\), with Galois group \[ \mathrm{Gal}(X/H)=\langle \sigma\rangle \cong \mathbb Z/n\mathbb Z, \] then the Jacobian \(JX\) has dimension \[ g_X=n(g-1)+1, \] and the Prym variety is defined by \[ P(f):=\ker(\mathrm{Nm}_f)^0. \] Its dimension is \[ \dim P(f)=g_X-g=(n-1)(g-1), \] and the induced polarization has type \[ (1,\dots,1,\underbrace{n,\dots,n}_{g-1\ \text{times}}) \] with \((n-2)(g-1)\) entries equal to \(1\) and \(g-1\) entries equal to \(n\) [1601.04082]. The hyperelliptic involution \(\iota\) of \(H\) lifts to an involution \[ \tau:X\to X, \] and \(\sigma,\tau\) satisfy \[ \sigma^n=\tau^2=(\sigma\tau)^2=1. \] Hence they generate the dihedral group \[ D_n=\langle \sigma,\tau\mid \sigma^n=\tau^2=(\sigma\tau)^2=1\rangle \] of order \(2n\) [1601.04082]. This dihedral symmetry is the basic mechanism behind the splitting results for \(P(f)\). A closely related ramified setting begins with a double cover \[ \tau:D\to C \] ramified in \(r>0\) points. Then \[ P(\tau)=\ker(\mathrm{Nm}_\tau)^0 \] has dimension \[ g(C)-1+\frac r2, \] and polarization type \[ \delta=(1,\ldots,1,2,\ldots,2) \] with \(2\) repeated \(g(C)\) times [2005.11108]. When \(C\) is hyperelliptic, the resulting locus of coverings defines the hyperelliptic Prym locus \[ \mathcal R^h_{g,r}\subset \mathcal R_{g,r}, \] with induced Prym map \[ \mathcal P^h_{g,r}:\mathcal R^h_{g,r}\to \mathcal A_{g-1+r/2}^{\delta} \] [2005.11108]. A third important class arises from hyperelliptic \(\mathbb Z_2^2\)-coverings. If a hyperelliptic curve \(H\) admits commuting involutions \(\sigma,\tau\) with \[ \langle \sigma,\tau\rangle\cong \mathbb Z_2^2, \] then the corresponding 4:1 Klein covering produces Prym varieties with strong Torelli properties [2302.13041]. In this context, hyperelliptic Prym pairs are configurations in which both the covering geometry and the Prym polarization are controlled by involutions descending from hyperelliptic structure. ## 2. Dihedral and Klein structures behind the Prym decomposition For étale cyclic covers of hyperelliptic curves, the decisive structural fact is that the Prym variety is always isogenous to a product of two Jacobians [1601.04082]. Denote \[ X_\tau=X/\langle \tau\rangle,\qquad X_{\tau\sigma^m}=X/\langle \tau\sigma^m\rangle \] when \(n=2m\) is even. Then \(JX_\tau\) and \(JX_{\tau\sigma^m}\) are abelian subvarieties of \(P(f)\), and the addition map \[ \alpha:JX_\tau\times JX_{\tau\sigma^m}\longrightarrow P(f),\qquad (x,y)\mapsto x+y \] is an isogeny for all \(n\) [1601.04082]. The quotient curves are governed by the fixed-point behavior of the involutions \(\tau\) and \(\tau\sigma^m\) above the Weierstrass points. The branch set \(W\) decomposes into \[ S_0=\{x\in W\mid (f\circ \pi^{-1})(x)\text{ has a fixed point of }\tau\}, \] \[ S_1=\{x\in W\mid (f\circ \pi^{-1})(x)\text{ has a fixed point of }\tau\sigma^m\}, \] with \[ s_0=|S_0|,\qquad s_1=|S_1|,\qquad s_0+s_1=2g+2, \] and \(s_0,s_1\ge 2\) even [1601.04082]. Hurwitz formulas then give the genera of the quotient curves explicitly. In particular, when \(n=2m\), \[ g(X_{\tau})=m(g-1)+1-\frac{s_0}{2},\qquad g(X_{\tau\sigma^m})=m(g-1)+1-\frac{s_1}{2}. \] This makes the decomposition of \(P(f)\) highly explicit at the level of tangent spaces and quotient curves [1601.04082]. In the Klein setting, if a hyperelliptic curve admits a subgroup \[ \langle \sigma,\tau\rangle\cong \mathbb Z_2^2 \] with \(\iota\notin \langle \sigma,\tau\rangle\), then parity imposes severe restrictions. If \(g(H)=2k\) is even, there is no such Klein subgroup; if \(g(H)=4k+1\), there is a unique Klein subgroup consisting of fixed-point free involutions; if \(g(H)=4k+3\), there is a unique Klein subgroup consisting of involutions with fixed points [2302.13041]. These parity constraints explain why hyperelliptic \(\mathbb Z_2^2\)-coverings split into étale and ramified regimes with markedly different Prym behavior. The Jacobian of the covering curve then decomposes into isotypical components under the Klein action. In the étale case, with \(C\) hyperelliptic of genus \(4g-3\), one has \[ JC=JH^*\boxplus JH_x\boxplus JH_y\boxplus JH_z, \] and \[ P(C/H)=JH_x\boxplus JH_y\boxplus JH_z \] [2302.13041]. The addition map \[ \psi:JH_x\times JH_y\times JH_z\to P(C/H) \] is an isogeny of degree \[ 4^{2(g-1)} \] with kernel contained in 2-torsion [2302.13041]. This gives a finer three-factor analogue of the dihedral two-factor decomposition. ## 3. Canonical isogenies, isomorphism cases, and polarization mismatch The canonical isogeny \[ \alpha:JX_\tau\times JX_{\tau\sigma^m}\to P(f) \] is an isomorphism in some degrees and only a nontrivial isogeny in others [1601.04082]. The precise trichotomy is complete. If \(n=2\), Mumford’s result identifies the Prym of an étale double cover of a hyperelliptic curve with a product of two Jacobians. If \(n\) is odd, Ortega proved that \[ \Phi:(JX_\tau)^2\to P(f),\qquad (x,y)\mapsto x+\sigma(y) \] is an isomorphism. If \(n\equiv 2\pmod 4\), Ortega proved that \[ \Psi:JX_\tau\times JX_{\tau\sigma^m}\to P(f),\qquad (x,y)\mapsto x+y \] is an isomorphism [1601.04082]. The remaining case is \[ n=2^r m,\qquad r\ge 2,\quad m\ \text{odd}. \] Then the central theorem gives \[ \deg \alpha = 2^{\bigl[(2^{r}-r-1)m-(r-1)\bigr](g-1)}. \] In particular, only the degree-4 case remains an isomorphism: \[ n=4 \Rightarrow \deg \alpha=1. \] For example, \[ n=8 \Rightarrow \deg \alpha=2^{2(g-1)},\qquad n=12 \Rightarrow \deg \alpha=2^{2(g-1)}. \] Thus the canonical map ceases to be an isomorphism already at \(n=8\) [1601.04082]. This distinction is fundamentally a polarization issue. The Prym carries polarization type \[ (1^{(n-2)(g-1)},n^{g-1}), \] whereas both \(JX_\tau\) and \(JX_{\tau\sigma^m}\) are principally polarized. The isogeny \(\alpha\) is generally not isometric with respect to the product principal polarization, and its degree measures the discrepancy [1601.04082]. A plausible implication is that the hyperelliptic condition forces the abelian variety underlying the Prym into a product isogeny class, while the residual 2-primary degree records the failure of the Prym polarization to split as a product polarization. A different but analogous phenomenon appears for genus-2 cyclic covers of degree \(4\). There the Prym map \[ \mathcal P_4:\mathcal R_2^4\to \mathcal A_3^{(1,1,4)} \] has positive-dimensional fibers, and the associated Prym threefold decomposes using elliptic factors arising from involution quotients of the genus-5 covering curve [2508.20838]. In that setting the Prym is represented as a quotient of \[ E_T\times E_T\times F_T \] by a rank-2 subgroup of 2-torsion, with pullback polarization \[ \mathcal O_{E_T}(2)\boxtimes \mathcal O_{E_T}(2)\boxtimes \mathcal O_{F_T}(4) \] [2508.20838]. This is another manifestation of polarization mismatch encoded by quotient structure rather than direct-product structure. ## 4. Prym maps, Torelli phenomena, and hyperelliptic exceptional loci Hyperelliptic Prym pairs occupy a special position in Prym–Torelli theory. For ramified double covers, the global theorem is that \[ \mathcal P_{g,r} \] is an embedding for all \(r\ge 6\) and all \(g>0\) [2005.11108]. Thus, for sufficiently ramified covers, the Prym variety determines the cover uniquely. The hyperelliptic locus is precisely where low-ramification failures occur. If \[ \mathcal P^h_{g,r}:\mathcal R^h_{g,r}\to \mathcal A_{g-1+r/2}^{\delta} \] denotes the restriction to coverings of hyperelliptic curves, then the generic fibers are: \[ \text{generic fiber of }\mathcal P^h_{g,2}\ \text{birational to }\mathbb P^2, \] \[ \text{generic fiber of }\mathcal P^h_{g,4}\ \text{birational to an elliptic curve} \] [2005.11108]. These positive-dimensional fibers are analyzed via the bigonal construction, which sends a tower \[ D\xrightarrow{\tau} C\xrightarrow{f}\mathbb P^1 \] of double covers to a new tower \[ D'\xrightarrow{\tau'} C'\xrightarrow{f'}\mathbb P^1 \] and exchanges Prym varieties with dual polarized Pryms [2005.11108]. For \(r=2\), the residual freedom is a \(\mathbb P^2\)-family of double coverings of \(\mathbb P^1\); for \(r=4\), it is an elliptic-curve family of inverse bigonal lifts [2005.11108]. The codifferential of the ramified Prym map is given by \[ d\mathcal P^*_{g,r}(C,\eta,B): \mathrm{Sym}^2 H^0(C,\omega_C\otimes \eta)\to H^0(C,\omega_C^{\otimes 2}(B)). \] Using Green–Lazarsfeld surjectivity, non-injectivity can only occur for \(r=2\) or \(4\), and only on curves of very low Clifford index, especially hyperelliptic curves [2005.11108]. Thus hyperelliptic Prym pairs are not merely examples of exceptional fibers; they are the entire geometric source of low-ramification non-Torelli behavior in this range. In contrast, hyperelliptic \(\mathbb Z_2^2\)-coverings exhibit rigidity rather than failure of injectivity. The Prym maps \[ \mathcal{RH}_{g,b}\to \mathcal A^\delta \] for \(b\in\{0,4,8,12\}\) are globally injective [2302.13041]. In the étale case, \[ \mathrm{Pr}_{4g-3,g}^H:\mathcal{RH}_{g,0}\to \mathcal A^\delta_{3g-3} \] is injective for all \(g\ge 2\), and analogous injectivity holds in the branched cases \(b=4,8,12\) [2302.13041]. This contrast is sharp: hyperelliptic double Prym maps are never injective in the corresponding families, while hyperelliptic Klein Prym maps recover injectivity because the full \(\mathbb Z_2^2\)-action rigidifies the 2-torsion and polarization data [2302.13041]. A further genus-2 cyclic-cover analogue appears in the study of cyclic étale covers of genus-2 curves. There the Prym map is ramified precisely on the bielliptic locus, and for degree \(d\ge 7\) the covering curve is never hyperelliptic [2001.06264]. This does not define hyperelliptic Prym pairs in the base-hyperelliptic sense of the double-cover theory, but it shows that extra involutions continue to control Prym-map pathologies. ## 5. Moduli of hyperelliptic Prym pairs and intersection-theoretic structure Recent work has formalized hyperelliptic Prym pairs as moduli-stack objects. A Prym curve of genus \(g\) is a triple \((C,\eta,\beta)\), where \(C\) is smooth of genus \(g\), \(\eta\in \operatorname{Pic}(C)\) is nontrivial of order \(2\), and \[ \beta:\eta^{\otimes 2}\xrightarrow{\sim}\mathcal O_C \] is a chosen trivialization [2501.16320]. The moduli stack of hyperelliptic Prym pairs is \[ RH_g := R_g\times_{\mathcal M_g}\mathcal H_g \] [2501.16320]. On a hyperelliptic curve \(C\) with hyperelliptic line bundle \(H=q^*\mathcal O_{\mathbb P^1}(1)\) and Weierstrass set \(W\), every nontrivial 2-torsion bundle has the form \[ \eta_e = H^{\otimes n}\otimes \mathcal O_C(-e), \] where \(e\) is a reduced effective divisor of degree \(2n\) supported on \(W\). This yields a decomposition \[ RH_g = \bigsqcup_{1\le n\le (g+1)/2} RH_g^n \] into strata indexed by the number \(n\) of Weierstrass pairs used to define the Prym structure [2501.16320]. For \(n\le \lfloor g/2\rfloor\), the correspondence \(e\mapsto \eta_e\) is injective; for \(n=(g+1)/2\) with \(g\) odd, it is generically 2:1 [2501.16320]. The first part of the Chow-ring program computes the integral Chow rings of \(RH_g^1\) for all \(g\), and of \(RH_g^n\) for odd \(g\) and \[ 1