June 2023

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4 Reads

Journal of Pure and Applied Algebra

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June 2023

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4 Reads

Journal of Pure and Applied Algebra

April 2023

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12 Reads

Journal of Pure and Applied Algebra

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 .

December 2021

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32 Reads

In this work we study the monodromy group of covers $\varphi \circ \psi$ of curves \linebreak $\mathcal{Y}\xrightarrow {\quad {\psi}} \mathcal{X} \xrightarrow {\quad \varphi} \mathbb{P}^{1}$, where $\psi$ is a $q$-fold cyclic \'etale cover and $\varphi$ is a totally ramified $p$-fold cover, with $p$ and $q$ different prime numbers with $p$ odd. We show that the Galois group $\mathcal{G}$ of the Galois closure $\mathcal{Z}$ of $\varphi \circ \psi$ is of the form $\mathcal{G} = \mathbb{Z}_q^s \rtimes \mathcal{U}$, where $0 \leq s \leq p-1$ and $\mathcal{U}$ 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 $\varphi$ was assumed to be cyclic, in which case the Galois group is of the form $\mathcal{G} = \mathbb{Z}_q^s \rtimes \mathbb{Z}_p$. Furthermore, we are able to characterize the subgroups $\mathcal{H}$ and $\mathcal{N}$ of $\mathcal{G}$ such that $\mathcal{Y} = \mathcal{Z}/\mathcal{N}$ and $X = \mathcal{Z}/\mathcal{H}$.

September 2021

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61 Reads

In this article we classify compact Riemann surfaces of genus $1+q^2$ with a group of automorphisms of order $3q^2,$ where $q$ is a prime number. We also study decompositions of the corresponding Jacobian varieties.

September 2021

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15 Reads

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1 Citation

Journal of Algebra

In this article we classify compact Riemann surfaces of genus 1+q2 with a group of automorphisms of order 3q2, where q is a prime number. We also study decompositions of the corresponding Jacobian varieties.

August 2021

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52 Reads

In this article we consider Riemann surfaces and abelian varieties endowed with a group of automorphisms isomorphic to a generalized quaternion group. We provide isogeny decompositions of each abelian variety with this action, compute dimensions of the corresponding factors and provide conditions under which this decomposition is nontrivial. We then specialize our results to the case of Jacobians and relate them to the so-called genus-zero actions on Riemann surfaces. We also give a complete classification and description of the complex one-dimensional families of Riemann surfaces and Jacobians with a generalized quaternion group action, extending known results concerning the quasiplatonic case. Finally, we construct and describe explicit families of abelian varieties with a quaternion group action and derive a period matrix for the Jacobian of the surface with full automorphism group of second largest order among the hyperellip-tic surfaces of genus four.

January 2021

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8 Reads

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3 Citations

ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

January 2021

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24 Reads

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6 Citations

Journal of Algebra

In this paper we study the Galois group of the Galois cover of the composition of a q-cyclic étale cover and a cyclic p-gonal cover for any odd prime p. Furthermore, we give properties of isogenous decompositions of certain Prym and Jacobian varieties associated to intermediate subcovers given by subgroups.

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... 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). ...

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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. ...

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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. ...

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- Full-text available
January 2021

Journal of Algebra

... 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

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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

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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

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April 2019

Archiv der Mathematik

... So consider G = a, b ≤ A 14 with a := (1, 2, 3, 4, 5, 6, 7) (8,9,10,11,12,13,14) and ...

Reference:

The Monodromy Group of pq-Covers

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December 2018

Revista Matemática Iberoamericana

... 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

... Observe that A K is the abelian subvariety of A on which K acts trivially. In addition, if K 1 ⩽ K 2 are subgroups of G then A K 2 ⊆ A K 1 and, as proved in [7], there exists an abelian subvariety P of A K 1 in such a way that ...

- Citing Article
October 2017

Journal of Pure and Applied Algebra

... We compute the degrees of the Abel-Prym maps and show that the bounds obtained in Theorems 3 and 9 are sharp. In the last two examples we consider the isotypical decomposition of a Jacobian of a smooth curve with an action of the dihedral group D p , for p an odd prime, and the quaternions group Q 8 , following Carocca et al. (2002) and Lange and Recillas (2004). We then use our results to compute or give bounds for the degrees of the Abel-Prym maps in these cases, under some hypotheses on the action. ...

- Citing Article
January 2002