April 2023

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12 Reads

Journal of Pure and Applied Algebra

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April 2023

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12 Reads

Journal of Pure and Applied Algebra

November 2022

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2 Reads

September 2022

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18 Reads

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1 Citation

Transformation Groups

In this work, we study the monodromy group of covers φ ∘ ψ of curves Y→ψ$\mathcal {Y}\xrightarrow {\quad {\psi }}$X→φℙ1$\mathcal {X} \xrightarrow {\quad \varphi } \mathbb {P}^{1}$, where ψ is a q-fold cyclic étale cover and φ is a totally ramified p-fold cover, with p and q different prime numbers with p odd. We show that the Galois group G$\mathcal {G}$ of the Galois closure Z$\mathcal {Z}$ of φ ∘ ψ is of the form G=ℤqs⋊U$\mathcal {G} = \mathbb {Z}_{q}^{s} \rtimes \mathcal {U}$, where 0 ≤ s ≤ p − 1 and U$\mathcal {U}$ is a simple transitive permutation group of degree p. Since the simple transitive permutation group of prime degree p are known, and we construct examples of such covers with these Galois groups, the result is very different from the previously known case when the cover φ was assumed to be cyclic, in which case the Galois group is of the form G=ℤqs⋊ℤp$\mathcal {G} = \mathbb {Z}_{q}^{s} \rtimes \mathbb {Z}_{p}$. Furthermore, we are able to characterize the subgroups H${\mathscr{H}}$ and N$\mathcal {N}$ of G$\mathcal {G}$ such that Y=Z/N$\mathcal {Y} = \mathcal {Z}/\mathcal {N}$ and X=Z/H$X = \mathcal {Z}/{\mathscr{H}}$.

September 2022

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71 Reads

Let Y, X, T be closed Riemann surfaces, P : Y → X be an unbranched abelian Galois covering and let π : X → T be a (possible branched) covering. We provide a description of the structure of the Galois cover group of the associated Galois closure of π • P : X → T .

June 2022

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4 Reads

June 2022

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6 Reads

June 2022

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4 Reads

This chapter deals with the decomposition of the Jacobians of curves with actions of three series of groups, namely, cyclic groups of order n, dihedral groups of order 2n, and the semidirect product of the group of order 3 by an arbitrary self-product of the Klein group of order 4. For large n we obtain in each case examples of components which are neither Jacobians nor Prym varieties of subcovers. The importance of the last case lies in the fact that it gives examples of curves whose Jacobian is isogenous to a product of an arbitrary number of Jacobians of the same genus.

June 2022

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3 Reads

The second chapter gives the definition of Prym varieties and derives the properties which are needed in the subsequent chapters. For any finite cover of smooth projective curves f : C˜ → C the Prym variety P(f) = P(C˜∕C) is defined to be the complementary abelian subvariety of f ∗ J in $\widetilde J$ with respect to the canonical polarization of $\widetilde J$. Here J and $\widetilde J$ denote the Jacobian varieties of C and $\widetilde C$ respectively.So our notion of Prym variety is more general than the original notion introduced by Mumford. We call his more special Prym varieties principally polarized Prym varieties. Hence for us, Prym varieties are not necessarily principally polarized.

June 2022

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4 Reads

This chapter covers the investigation of Prym varieties of covers f′ : C′→ C of degree 4. It is the main chapter of the book. As mentioned also in the introduction, most of the results of this chapter appeared already in an unpublished paper by Recillas and the second author of this book. Our contribution consists in several improvements of the proofs. In particular we use more results on the representation theory of the corresponding groups.There are five possibilities for f′: Either it is a Galois cover. Then it is either cyclic of degree 4 or given by the action of the Klein group of order 4. Or f′ is non-Galois. If f : C˜ → C is its Galois closure, then the corresponding Galois group G is either the dihedral group D 4 of order 8, the alternating group A4 of order 12, or the symmetric group S4 of order 24. In each case we compute the isotypical decomposition as well as a group algebra decomposition of the Jacobian JC˜ with respect to the Galois group. In all cases the factors are given either by Jacobians or Prym varieties of subcovers. We will also compute the degree of the map giving the isotypical decomposition. As an application we obtain new proofs of the bigonal construction as well as the trigonal construction.

January 2022

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4 Reads

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6 Citations

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... . We refer to [20] for more details on decomposition of Jacobians by Prym varieties. ...

- Citing Article
January 2022

... Then P is a Galois covering with deck group A A , and U = Y × (P,Q) S (the fiber product of P and Q). In this particular situation, in [7] it is asserted that G K ⋊ L (even if q is not relatively prime to the order of L). ...

- Citing Article
- Publisher preview available
September 2022

Transformation Groups

... For example, in [RCR20], Reyes-Carocca and Rodriguez prove that if π : X → Y is a regular covering map the group algebra decomposition of J(X) can be lifted to J(Y ) in an equivariant way. Another example, In [CLR21], Carocca, Lange and Rodriguez study a weaker question and prove that given any positive integer g, there exists a smooth projective curve X whose jacobian variety J(X) is isogenous to the product of m ≥ g jacobian varieties of the same dimension. In [CLZ17], Chen, Lu and Zuo reformulate the Ekedahl and Serre's question to a Coleman-Oort problem. ...

- Citing Article
January 2021

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

... In many cases, G K ⋊ L ( [5,8,10,14,21,25,28]). As a consequence of Theorem 1, we have the following. ...

- Citing Article
- Full-text available
January 2021

Journal of Algebra

... It is not difficult to find other examples of splitting Jacobians in genus 3 and 4. For example, the Jacobians of Klein's quartic [18] in genus 3 and Bring's curve in genus 4 [19] also split as complex tori. Throughout our discussion we have stressed that only splitness at the level of complex tori is required for isotriviality, as opposed to the reducibility of the polarized abelian variety in question. ...

- Citing Article
October 1999

Proceedings of the American Mathematical Society

... In this paper we consider an easier question, namely: can a Jacobian be isogenous to the product of arbitrary many Jacobians of the same genus (not necessarily equal to one)? In [4] we gave examples of Jacobians which are isogenous to an arbitrary number of Prym varieties of the same dimension. The main result of this paper is the following theorem (see Corollary 5.3). ...

Reference:

Decomposable Jacobians

- Citing Article
January 2021

Journal of Algebra

... Proposition 10 We have that K ≤ R and G ≤ C S pq ( ) (the centralizer of in the symmetric group S pq ). ...

Reference:

The Monodromy Group of pq-Covers

- Citing Article
July 2019

Journal of Algebra

... where N acts on each canonically defined abelian subvariety A W of P( X/X) by (an appropriate multiple of) the representation W. Since N Z 2g q is an abelian group, a precise description of each A W is given as follows. According to [8] Corollary 4.2, each non trivial complex irreducible linear character V of N is defined over the field Q[w q ], with w q a primitive q-th root of unity, and each non trivial rational irreducible representation W of N is given by ...

Reference:

q-étale covers of cyclic p-gonal covers

- Citing Article
April 2019

Archiv der Mathematik

... Finally, we should mention the survey article by Rodríguez [14] in which the author review part of the theory of abelian varieties with group actions and the decomposition of them up to isogeny. Somewhere else we will try to describe our decomposition results for generalized Fermat curves in their language. ...

- Citing Article
October 2018

Archiv der Mathematik

... The question of Ekedahl and Serre remains elusive and many mathematicians have being involved into it. For example, in [RCR20], Reyes-Carocca and Rodriguez prove that if π : X → Y is a regular covering map the group algebra decomposition of J(X) can be lifted to J(Y ) in an equivariant way. Another example, In [CLR21], Carocca, Lange and Rodriguez study a weaker question and prove that given any positive integer g, there exists a smooth projective curve X whose jacobian variety J(X) is isogenous to the product of m ≥ g jacobian varieties of the same dimension. ...

- Citing Article
- Full-text available
February 2020

Mathematische Zeitschrift