The generic isogeny decomposition of the Prym Variety of a cyclic branched covering

Let $$f:S'\longrightarrow S$$ f : S ′ ⟶ S be a cyclic branched covering of smooth projective surfaces over $${\mathbb {C}}$$ C whose branch locus $$\Delta \subset S$$ Δ ⊂ S is a smooth ample divisor. Pick a very ample complete linear system $$|{\mathcal {H}}|$$ | H | on S, such that the polarized surface $$(S, |{\mathcal {H}}|)$$ ( S , | H | ) is not a scroll nor has rational hyperplane sections. For the general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | consider the $$\mu _{n}$$ μ n -equivariant isogeny decomposition of the Prym variety $${{\,\mathrm{Prym}\,}}(C'/C)$$ Prym ( C ′ / C ) of the induced covering $$f:C'{:}{=}f^{-1}(C)\longrightarrow C$$ f : C ′ : = f - 1 ( C ) ⟶ C : $$\begin{aligned} {{\,\mathrm{Prym}\,}}(C'/C)\sim \prod _{d|n,\ d\ne 1}{\mathcal {P}}_{d}(C'/C). \end{aligned}$$ Prym ( C ′ / C ) ∼ ∏ d | n , d ≠ 1 P d ( C ′ / C ) . We show that for the very general member $$[C]\in |{\mathcal {H}}|$$ [ C ] ∈ | H | the isogeny component $${\mathcal {P}}_{d}(C'/C)$$ P d ( C ′ / C ) is $$\mu _{d}$$ μ d -simple with $${{\,\mathrm{End}\,}}_{\mu _{d}}({\mathcal {P}}_{d}(C'/C))\cong {\mathbb {Z}}[\zeta _{d}]$$ End μ d ( P d ( C ′ / C ) ) ≅ Z [ ζ d ] . In addition, for the non-ample case we reformulate the result by considering the identity component of the kernel of the map $${\mathcal {P}}_{d}(C'/C)\subset {{\,\mathrm{Jac}\,}}(C')\longrightarrow {{\,\mathrm{Alb}\,}}(S')$$ P d ( C ′ / C ) ⊂ Jac ( C ′ ) ⟶ Alb ( S ′ ) .

Rational points on abelian varieties over function fields and Prym varieties

In this paper, using a generalization of the notion of Prym variety for covers of projective varieties, we prove a structure theorem for the Mordell–Weil group of abelian varieties over function fields that are twists of abelian varieties by Galois covers of smooth projective varieties. In particular, the results we obtain contribute to the construction of Jacobians of high rank.

On the Prym variety of genus 3 covers of genus 1 curves

Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and Jac(X). This construction can be seen as a degenerate case of a result by Nils Bruin. Comment: Published version

COMPACTIFICATION OF THE PRYM MAP FOR NON-CYCLIC TRIPLE COVERINGS

According to [H. Lange and A. Ortega, Prym varieties of triple coverings, Int. Math. Res. Notices2011(22) (2011) 5045–5075], the Prym variety of any non-cyclic étale triple cover f : Y → X of a smooth curve X of genus 2 is a Jacobian variety of dimension 2. This gives a map from the moduli space of such covers to the moduli space of Jacobian varieties of dimension 2. We extend this map to a proper map Pr of a moduli space [Formula: see text] of admissible S3-covers of genus 7 to the moduli space [Formula: see text] of principally polarized abelian surfaces. The main result is that [Formula: see text] is finite surjective of degree 10.

Generic Torelli theorem for Prym varieties of ramified coverings

We consider the Prym map from the space of double coverings of a curve of genus gwithrbranch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2is generically injective ifWe also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

The arithmetic of Prym varieties in genus 3

Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be mapped into the Jacobian of a curve of genus 2 over its field of definition and how this can be used to perform Chabauty- and Brauer–Manin-type calculations for curves of genus 5 with an fixed-point-free involution. As an application, we determine the rational points on a smooth plane quartic and give examples of curves of genus 3 and 5 violating the Hasse principle. The methods are, in principle, applicable to any genus 3 curve with a double cover. We also show how these constructions can be used to design smooth plane quartics with specific arithmetic properties. As an example, we give a smooth plane quartic with all 28 bitangents defined over $\mathbb {Q}(t)$. By specialization, this also gives examples over $\mathbb {Q}$.

Dimensions of Prym varieties

Given a tame Galois branched cover of curvesπ:X→Ywith any finite Galois groupGwhose representations are rational, we compute the dimension of the (generalized) Prym varietyPrymρ(X)corresponding to any irreducible representationρofG. This formula can be applied to the study of algebraic integrable systems using Lax pairs, in particular systems associated with Seiberg-Witten theory. However, the formula is much more general and its computation and proof are entirely algebraic.