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Abolfazl Mohajer

Publications and source records attributed to Abolfazl Mohajer.

17 recordsLinked to original sources
arXiv2026

Tropical covers, tropical abelian varieties and Prym varieties

We define and investigate the tropical Prym varieties associated to unramified Galois cyclic covers of tropical curves (or equivalently metric graphs) $\tildeΓ\to Γ$. Our approach here is to study the tropical Prym varieties using group actions on tropical abelian varieties induced by the cyclic Galois group of the cover of tropical curves. We also define and conider the Abel-Prym map for tropical cyclic covers extending that for double covers. As a special case we consider free $\Z_3$-covers of tropical curves and their associated tropical Prym variety and compute its volume generalizing the case of double covers.

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arXiv2024

On the Existence of Shimura curves in the Prym locus of abelian covers of projective line

Using the theory of Higgs bundles and their stabitlity properties associated to fibered surfaces and the Viehweg-Zuo characterization of Shimura curves in the moduli space of abelian varieties in terms of Higgs bundles, we prove that there does not exist any non-compact Shimura curves in the Prym locus of totally ramified $\Z_{2p}$- or $\Z_{2p}\times (\Z_{p})^{m-1}$-covers of the projective line in $A_{g}$ for $g\geq 8$, where $p\geq 5$ is a prime number.

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arXiv2023

On Shimura curves generated by families of Galois $G$-covers of curves

In this paper we prove there are no families of cyclic $\Z_n$-covers of elliptic curves which generate non-compact Shimura (special) curves that lie generically in the Torelli locus $T_g$ of abelian varieties with $g\geq 8$ when $n$ has a proper prime factor $p\geq 7$. This non-existence is also shown for families of $\Z_n$-covers of curves of any genus $s$ and when $n$ has a large enough prime number $p$ (depending on $s$). We achieve these results by applying the theory of Higgs bundles and the Viehweg-Zuo characterization of Shimura curves in the moduli space of principally polarized abelian varieties.

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arXiv2023

On Galois Coverings of curves and their Families

In this paper, we describe Galois covers of algebraic curves and their families by using local systems associated to push-forward of sheaves by the structure morphism. More precisely, if $f:C\to Y$, we consider the sheaves $f_*(\C)$. The group action by the Galois group $G$, yields a decomposition of this sheaf into irreducible local systems corresponding to irreducible representations of the group $G$. If $ρ$ is such an irreducible representation, the eigensheaf $Ł_ρ$ of $f_*(\C)$ gives rise to another useful sheaf which is related to the homology group $H_1(C,\C)$. Using this, we describe the action of the Galois group $G$ on the homology group. As a particular example, we study the Dihedral covers of $¶^1$ in some detail.

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arXiv2022

Families of Prym varieties of abelian coverings and Shimura varieties

Under the condition that the Prym map is injective in characteristic $p$, we prove that the special subvarieties in the moduli space of abelian varieties of dimension $l$ and polarization type $D$, $A_{l,D}$, arising from families of abelian covers of $¶^1$ are of a very restrictive nature. In other words, if the family is one-dimensional or if it contains an eigenspace of certain type for the group action on the cohomology of fibers, then the Shimura varieties arising from such families can only be constructed by the group action of the family.

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arXiv2021

Shimura subvarieties in the Prym locus of ramified Galois coverings

We study Shimura (special) subvarieties in the moduli space $A_{p,D}$ of complex abelian varieties of dimension $p$ and polarization type $D$. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $\mathbb{P}^1$. We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

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arXiv2021

Twists of Albanese varieties over function fields with large ranks

In this paper we construct abelian varieties of large Mordell-Weil rank over function fields. We achieve this by using a generalization of the notion of Prym variety to higher dimensions and a structure theorem for the Mordell-Weil group of abelian varieties over function fields proven in our previous works. We consider abelian and dihedral covers of the projective space and apply the above results to the twists of their Albanese varieties.

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arXiv2021

Higher dimensional Shimura varieties in the Prym loci of ramified double covers

In this paper we construct Shimura subvarieties of dimension bigger than one of the moduli space of polarised abelian varieties of a given dimension, which are generically contained in the Pym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely to use families of Galois covers of the projective line. The case of abelian covers is treated in detail, since in this case it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield a Shimura subvariety of the space polarised abelian varieties of a given dimension.

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arXiv2020

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 quasi-projective varieties, we prove a structure theorem for the Mordell-Weil group of the abelian varieties over function fields that are twists of Abelian varieties by Galois covers of irreducible quasi-projective varieties. In particular, the resutls we obtain contribute in the construction of Jacobians (of covers of the projective line) of high rank.

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arXiv2020

On the Prym map of Galois covering

In this paper we consider the Prym variety $P(\widetilde{C}/C)$ associated to a Galois coverings of curves $f:\widetilde{C}\to C$ branched at $r$ points. We discuss some properties and equivalent definitions and then consider the Prym map $\mathcal{P}=\mathcal{P}(G,g,r):R(G,g,r)\to A_{p,δ}$ with $δ$ the type of the polarization. For Galois coverings whose Galois group is abelian and metabelian (non-abelian) we show that the differential of this map at certain points is injective. We also consider the Abel-Prym map $u:\widetilde{C}\to P(\widetilde{C}/C)$ and prove some results for its injectivity. In particular we show that in contrast to the classical and cyclic case, the behavior of this map here is more complicated. The theories of abelian and metabelian Galois coverings play a substantial role in our analysis and have been used extensively throughout the paper.

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arXiv2020

Shimura varieties and abelian covers of the line

We prove that under some conditions on the monodromy, families of abelian covers of the projective line do not give rise to (higher dimensional) Shimura subvarieties in $A_g$. This is achieved by a reduction to $p$ argument. We also use another method based on monodromy computations to show that two dimensional subvarieties in the above locus are not special. In particular it is shown that such families have usually large monodromy groups. Together with our earlier results, the above mentioned results contribute to classifying the special families in the moduli space of abelian varieties and partially completes the work of several authors including the author's previous work.

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arXiv2019

On metabelian galois coverings of complex algebraic varieties

Let $f:X\to Y$ be a finite ramified Galois covering of algebraic varieties defined over the complex numbers. In this paper, we prove some structure theorems for such coverings in the case that the non-abelian Galois group of the cover is metabelian (in particular metacyclic). Our results extend the previous ones, obtained by several authors, in the case of abelian and dihedral Galois coverings. In analogy with the abelian (and generalizing the dihedral) case, we find "building data" of metabelian covers from which the cover can be reconstructed. In addition to analyzing several examples, we compute invariants and eigensheaves of the group actions of the coverings under study with some applications.

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arXiv2019

Special subvarieties in Mumford-Tate varieties

Let X be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain D by a discrete subgroup. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety Y in X (of Shimura type), and formulate necessary criteria for Y to be special. Our method consists in looking at finitely many compactified special curves C_i in Y, and testing whether the inclusion of the union of all C_i in Y satisfies certain properties. One of them is the so-called relative proportionality condition. In this paper, we give a new formulation of this numerical criterion in the case of Mumford-Tate varieties X. In this way, we give necessary and sufficient criteria for a subvariety Y of X to be a special subvariety in the sense of the Andre-Oort conjecture. We discuss in detail the important case where X=A_g, the moduli space of principally polarized abelian varieties.

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arXiv2015

On Shimura subvarieties generated by families of abelian covers of $\mathbb{P}^{1}$

We study the locus of abelian Galois covers of $\mathbb{P}^{1}$ in $A_{g}$ and the problem of occurrence of Shimura (special) subvarieties generated by these covers in the Torelli locus $T_{g}$ inside $A_{g}$. We first investigate the existence of Shimura subvarieties in the mentioned locus by some computational methods based on Moonen-Oort works and then exclude many cases using both characteristic $p$ methods and monodromy computations.

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arXiv2015

Some Remarks on Ideals with large Regularity and Regularity Jumps

This paper exhibits some new examples of the behavior of the Castelnuovo-Mumford regularity of homogeneous ideals in polynomial rings. More precisely, we present new examples of homogenous ideals with large regularity compared to the generating degree. Then we consider the regularity jumps of ideals. In particular we provide an infinite family of ideals having regularity jumps at a certain power.

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