Hyperelliptic Prym Pairs in Covering Theory

• Hyperelliptic Prym pairs are geometric structures combining hyperelliptic curves with cyclic, double, and Klein coverings to produce explicitly decomposed Prym varieties.
• They utilize dihedral and Klein symmetries to establish isogenies between Jacobians and Prym varieties, addressing polarization mismatches and Torelli phenomena.
• Recent advances include moduli stack formulations and integral Chow ring computations that enhance applications in integrable systems and tropical geometry.

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 $\deg f=n$ (Lange et al., 2016). 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 (Naranjo et al., 2020, Borówka et al., 2023). 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 (Cela et al., 27 Jan 2025, Cela et al., 18 Sep 2025).

1. Foundational definitions and basic geometry

Let $H$ be a smooth hyperelliptic curve of genus $g\ge 2$0, with hyperelliptic double cover

$g\ge 2$1

branched at the set $g\ge 2$2 of $g\ge 2$3 Weierstrass points (Lange et al., 2016). If

$g\ge 2$4

is an étale cyclic covering of degree $g\ge 2$5, with Galois group

$g\ge 2$6

then the Jacobian $g\ge 2$7 has dimension

$g\ge 2$8

and the Prym variety is defined by

$g\ge 2$9

Its dimension is

$f:X\to H$0

and the induced polarization has type

$f:X\to H$1

with $f:X\to H$2 entries equal to $f:X\to H$3 and $f:X\to H$4 entries equal to $f:X\to H$5 (Lange et al., 2016).

The hyperelliptic involution $f:X\to H$6 of $f:X\to H$7 lifts to an involution

$f:X\to H$8

and $f:X\to H$9 satisfy

$P(f)=\ker(\mathrm{Nm}_f)^0$0

Hence they generate the dihedral group

$P(f)=\ker(\mathrm{Nm}_f)^0$1

of order $P(f)=\ker(\mathrm{Nm}_f)^0$2 (Lange et al., 2016). This dihedral symmetry is the basic mechanism behind the splitting results for $P(f)=\ker(\mathrm{Nm}_f)^0$3.

A closely related ramified setting begins with a double cover

$P(f)=\ker(\mathrm{Nm}_f)^0$4

ramified in $P(f)=\ker(\mathrm{Nm}_f)^0$5 points. Then

$P(f)=\ker(\mathrm{Nm}_f)^0$6

has dimension

$P(f)=\ker(\mathrm{Nm}_f)^0$7

and polarization type

$P(f)=\ker(\mathrm{Nm}_f)^0$8

with $P(f)=\ker(\mathrm{Nm}_f)^0$9 repeated $(n-1)(g-1)$0 times (Naranjo et al., 2020). When $(n-1)(g-1)$1 is hyperelliptic, the resulting locus of coverings defines the hyperelliptic Prym locus

$(n-1)(g-1)$2

with induced Prym map

$(n-1)(g-1)$3

(Naranjo et al., 2020).

A third important class arises from hyperelliptic $(n-1)(g-1)$4-coverings. If a hyperelliptic curve $(n-1)(g-1)$5 admits commuting involutions $(n-1)(g-1)$6 with

$(n-1)(g-1)$7

then the corresponding 4:1 Klein covering produces Prym varieties with strong Torelli properties (Borówka et al., 2023). 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 (Lange et al., 2016). Denote

$(n-1)(g-1)$8

when $(n-1)(g-1)$9 is even. Then $\deg f=n$0 and $\deg f=n$1 are abelian subvarieties of $\deg f=n$2, and the addition map

$\deg f=n$3

is an isogeny for all $\deg f=n$4 (Lange et al., 2016).

The quotient curves are governed by the fixed-point behavior of the involutions $\deg f=n$5 and $\deg f=n$6 above the Weierstrass points. The branch set $\deg f=n$7 decomposes into

$\deg f=n$8

$\deg f=n$9

with

$\tau:D\to C$0

and $\tau:D\to C$1 even (Lange et al., 2016). Hurwitz formulas then give the genera of the quotient curves explicitly. In particular, when $\tau:D\to C$2,

$\tau:D\to C$3

This makes the decomposition of $\tau:D\to C$4 highly explicit at the level of tangent spaces and quotient curves (Lange et al., 2016).

In the Klein setting, if a hyperelliptic curve admits a subgroup

$\tau:D\to C$5

with $\tau:D\to C$6, then parity imposes severe restrictions. If $\tau:D\to C$7 is even, there is no such Klein subgroup; if $\tau:D\to C$8, there is a unique Klein subgroup consisting of fixed-point free involutions; if $\tau:D\to C$9, there is a unique Klein subgroup consisting of involutions with fixed points (Borówka et al., 2023). These parity constraints explain why hyperelliptic $C$0-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$1 hyperelliptic of genus $C$2, one has

$C$3

and

$C$4

(Borówka et al., 2023). The addition map

$C$5

is an isogeny of degree

$C$6

with kernel contained in 2-torsion (Borówka et al., 2023). This gives a finer three-factor analogue of the dihedral two-factor decomposition.

3. Canonical isogenies, isomorphism cases, and polarization mismatch

The canonical isogeny

$C$7

is an isomorphism in some degrees and only a nontrivial isogeny in others (Lange et al., 2016). The precise trichotomy is complete.

If $C$8, Mumford’s result identifies the Prym of an étale double cover of a hyperelliptic curve with a product of two Jacobians. If $C$9 is odd, Ortega proved that

$\mathbb Z_2^2$0

is an isomorphism. If $\mathbb Z_2^2$1, Ortega proved that

$\mathbb Z_2^2$2

is an isomorphism (Lange et al., 2016).

The remaining case is

$\mathbb Z_2^2$3

Then the central theorem gives

$\mathbb Z_2^2$4

In particular, only the degree-4 case remains an isomorphism: $\mathbb Z_2^2$5 For example,

$\mathbb Z_2^2$6

Thus the canonical map ceases to be an isomorphism already at $\mathbb Z_2^2$7 (Lange et al., 2016).

This distinction is fundamentally a polarization issue. The Prym carries polarization type

$\mathbb Z_2^2$8

whereas both $\mathbb Z_2^2$9 and $H$0 are principally polarized. The isogeny $H$1 is generally not isometric with respect to the product principal polarization, and its degree measures the discrepancy (Lange et al., 2016). 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 $H$2. There the Prym map

$H$3

has positive-dimensional fibers, and the associated Prym threefold decomposes using elliptic factors arising from involution quotients of the genus-5 covering curve (Shatsila, 28 Aug 2025). In that setting the Prym is represented as a quotient of

$H$4

by a rank-2 subgroup of 2-torsion, with pullback polarization

$H$5

(Shatsila, 28 Aug 2025). 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

$H$6

is an embedding for all $H$7 and all $H$8 (Naranjo et al., 2020). Thus, for sufficiently ramified covers, the Prym variety determines the cover uniquely.

The hyperelliptic locus is precisely where low-ramification failures occur. If

$H$9

denotes the restriction to coverings of hyperelliptic curves, then the generic fibers are: $g\ge 2$00

$g\ge 2$01

(Naranjo et al., 2020). These positive-dimensional fibers are analyzed via the bigonal construction, which sends a tower

$g\ge 2$02

of double covers to a new tower

$g\ge 2$03

and exchanges Prym varieties with dual polarized Pryms (Naranjo et al., 2020). For $g\ge 2$04, the residual freedom is a $g\ge 2$05-family of double coverings of $g\ge 2$06; for $g\ge 2$07, it is an elliptic-curve family of inverse bigonal lifts (Naranjo et al., 2020).

The codifferential of the ramified Prym map is given by

$g\ge 2$08

Using Green–Lazarsfeld surjectivity, non-injectivity can only occur for $g\ge 2$09 or $g\ge 2$10, and only on curves of very low Clifford index, especially hyperelliptic curves (Naranjo et al., 2020). 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 $g\ge 2$11-coverings exhibit rigidity rather than failure of injectivity. The Prym maps

$g\ge 2$12

for $g\ge 2$13 are globally injective (Borówka et al., 2023). In the étale case,

$g\ge 2$14

is injective for all $g\ge 2$15, and analogous injectivity holds in the branched cases $g\ge 2$16 (Borówka et al., 2023). 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 $g\ge 2$17-action rigidifies the 2-torsion and polarization data (Borówka et al., 2023).

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 $g\ge 2$18 the covering curve is never hyperelliptic (Agostini, 2020). 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\ge 2$19 is a triple $g\ge 2$20, where $g\ge 2$21 is smooth of genus $g\ge 2$22, $g\ge 2$23 is nontrivial of order $g\ge 2$24, and

$g\ge 2$25

is a chosen trivialization (Cela et al., 27 Jan 2025). The moduli stack of hyperelliptic Prym pairs is

$g\ge 2$26

(Cela et al., 27 Jan 2025).

On a hyperelliptic curve $g\ge 2$27 with hyperelliptic line bundle $g\ge 2$28 and Weierstrass set $g\ge 2$29, every nontrivial 2-torsion bundle has the form

$g\ge 2$30

where $g\ge 2$31 is a reduced effective divisor of degree $g\ge 2$32 supported on $g\ge 2$33. This yields a decomposition

$g\ge 2$34

into strata indexed by the number $g\ge 2$35 of Weierstrass pairs used to define the Prym structure (Cela et al., 27 Jan 2025). For $g\ge 2$36, the correspondence $g\ge 2$37 is injective; for $g\ge 2$38 with $g\ge 2$39 odd, it is generically 2:1 (Cela et al., 27 Jan 2025).

The first part of the Chow-ring program computes the integral Chow rings of $g\ge 2$40 for all $g\ge 2$41, and of $g\ge 2$42 for odd $g\ge 2$43 and

$g\ge 2$44

(Cela et al., 27 Jan 2025). For even $g\ge 2$45,

$g\ge 2$46

(Cela et al., 27 Jan 2025). For odd $g\ge 2$47,

$g\ge 2$48

(Cela et al., 27 Jan 2025). These generators are tautological Chern classes: $g\ge 2$49 or $g\ge 2$50 arise from hyperelliptic vector bundles, while $g\ge 2$51 records the $g\ge 2$52-torsor that orders the distinguished Weierstrass points (Cela et al., 27 Jan 2025).

The third part completes the computation for all components $g\ge 2$53 when $g\ge 2$54 is even and treats the rigidified and non-rigidified stacks

$g\ge 2$55

(Cela et al., 18 Sep 2025). It also determines when the rigidification map

$g\ge 2$56

is a root gerbe, which is essential for transferring Chow-ring calculations from rigidified to non-rigidified moduli (Cela et al., 18 Sep 2025). A plausible implication is that hyperelliptic Prym pairs form one of the rare Prym-type moduli problems for which both stack geometry and integral intersection theory can be computed completely on large families of components.

6. Related constructions, examples, and broader context

Several adjacent developments illuminate the breadth of the subject. In genus 5 on a general $g\ge 2$57-polarised abelian surface, there is, up to translation, a unique smooth hyperelliptic curve $g\ge 2$58 in the polarization class, and this curve is invariant under a Klein four subgroup of translations (Borówka et al., 2017). Every étale Klein covering of a hyperelliptic curve is then hyperelliptic provided the defining subgroup of $g\ge 2$59 is non-isotropic and every element is a difference of Weierstrass points (Borówka et al., 2017). This links the existence of hyperelliptic genus-5 curves on abelian surfaces directly to hyperelliptic Klein Prym pairs.

In genus 2 and degree 4, the Prym map for cyclic covers admits an explicit coordinate description. The moduli space

$g\ge 2$60

is identified with a parameter set $g\ge 2$61 of unordered triples $g\ge 2$62, and the associated genus-2 hyperelliptic base curve is

$g\ge 2$63

(Shatsila, 28 Aug 2025). The Prym fibers are controlled by two cross-ratios

$g\ge 2$64

and, away from two exceptional fibers, each non-empty fiber is isomorphic to the intersection of an elliptic normal curve in $g\ge 2$65 with an affine space $g\ge 2$66 (Shatsila, 28 Aug 2025). This is one of the most explicit geometric descriptions of positive-dimensional Prym fibers in the literature.

Hyperelliptic Prym varieties also appear in integrable systems. For the generalized Hénon–Heiles system, the spectral curve is a genus-4 curve $g\ge 2$67 admitting an involution with two fixed points, and the Prym variety

$g\ge 2$68

is isomorphic to the Jacobian of a genus-2 hyperelliptic curve (Enolski et al., 2014). The exact discretization of the system acts as translation on this Prym variety (Enolski et al., 2014). Similarly, the general Somos-6 recurrence is linearized on a genus-2 Jacobian arising as

$g\ge 2$69

for a genus-4 spectral curve $g\ge 2$70 with involution $g\ge 2$71 and two fixed points (Fedorov et al., 2015). These examples do not primarily concern moduli of coverings, but they reinforce the same theme: hyperelliptic Prym pairs often produce explicitly computable principally polarized abelian surfaces.

The tropical analogue exhibits both parallelism and divergence. For a free double cover

$g\ge 2$72

of hyperelliptic metric graphs, the tropical Abel–Prym map has degree $g\ge 2$73, the Abel–Prym image is a hyperelliptic metric graph of genus $g\ge 2$74, and its Jacobian is isomorphic as a principally polarized tropical abelian variety to the tropical Prym variety (Capobianco et al., 2024). Contrary to the algebraic case, if the source graph is not hyperelliptic, the Abel–Prym map is often not injective (Capobianco et al., 2024). This suggests that hyperelliptic Prym rigidity is partly algebraic and partly a consequence of smooth-curve geometry.

Taken together, these results show that hyperelliptic Prym pairs form a coherent and highly structured domain within Prym theory. Their defining features are explicit 2-torsion descriptions via Weierstrass points, quotient geometries controlled by dihedral or Klein symmetries, strong Torelli-type statements in some regimes and controlled failures in others, and unusually explicit moduli and polarization data. In this sense, hyperelliptic Prym pairs constitute both a classical subtheory of Prym varieties and a modern testing ground for questions in moduli, polarization, integrable systems, and tropical geometry (Lange et al., 2016, Naranjo et al., 2020, Borówka et al., 2023, Cela et al., 27 Jan 2025, Cela et al., 18 Sep 2025).