scholarly journals Shimura subvarieties in the Prym locus of ramified Galois coverings

2021 ◽  
Author(s):  
Gian Paolo Grosselli ◽  
Abolfazl Mohajer
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Moduli Space ◽  
Abelian Varieties ◽  
Fixed Group ◽  
Galois Coverings ◽  
Base Curve ◽  
Prym Map

AbstractWe study Shimura (special) subvarieties in the moduli space $$A_{p,D}$$ 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 $${{\mathbb {P}}}^1$$ 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.

scholarly journals Shimura curves in the Prym loci of ramified double covers

2021 ◽  
Author(s):  
Paola Frediani ◽  
Gian Paolo Grosselli
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
One Dimensional ◽  
Fixed Group ◽  
Base Curve ◽  
Double Covers ◽  
Ramification Points

We study Shimura curves of PEL type in the space of polarized abelian varieties [Formula: see text] generically contained in the ramified Prym locus. We generalize to ramified double covers, the construction done in [E. Colombo, P. Frediani, A. Ghigi and M. Penegini, Shimura curves in the Prym locus, Commun. Contemp. Math. 21(2) (2019) 1850009] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is [Formula: see text]. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci.

scholarly journals Shimura curves in the Prym locus

2019 ◽  
Vol 21 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Elisabetta Colombo ◽  
Paola Frediani ◽  
Alessandro Ghigi ◽  
Matteo Penegini
Keyword(s):  
Computer Algebra ◽  
Group Action ◽  
Shimura Curves ◽  
Simple Criterion ◽  
One Dimensional ◽  
Fixed Group ◽  
The Family ◽  
Base Curve ◽  
Double Covers ◽  
Prym Map

We study Shimura curves of PEL type in [Formula: see text] generically contained in the Prym locus. We study both the unramified Prym locus, obtained using étale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases, we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is one-dimensional and the quotient of the base curve by the group is [Formula: see text]. We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples obtained in this way up to genus 28. We obtain 43 Shimura curves contained in the unramified Prym locus and 9 families contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus.

scholarly journals Generic Torelli theorem for Prym varieties of ramified coverings

2012 ◽  
Vol 148 (4) ◽  
pp. 1147-1170 ◽  
Author(s):  
Valeria Ornella Marcucci ◽  
Gian Pietro Pirola
Keyword(s):  
Moduli Space ◽  
Abelian Varieties ◽  
Branch Points ◽  
Prym Variety ◽  
Prym Varieties ◽  
Torelli Theorem ◽  
Image Position ◽  
Prym Map ◽  
Double Coverings ◽  
Ramified Coverings

AbstractWe 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.

scholarly journals A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

2021 ◽  
Author(s):  
Anna Gori ◽  
Alberto Verjovsky ◽  
Fabio Vlacci
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
Higher Dimension ◽  
Geometric Constructions ◽  
Flat Torus ◽  
Flat Tori

AbstractMotivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

scholarly journals SUPERPOTENTIAL OF THE M-THEORY CONIFOLD AND TYPE IIA STRING THEORY

2004 ◽  
Vol 19 (04) ◽  
pp. 521-555 ◽  
Author(s):  
GOTTFRIED CURIO
Keyword(s):  
String Theory ◽  
Heisenberg Group ◽  
Group Action ◽  
Moduli Space ◽  
Monodromy Group ◽  
Witten Theory ◽  
Flat Bundle ◽  
M Theory

The membrane instanton superpotential for M-theory on the G2 holonomy manifold given by the cone on S3×S3 is given by the dilogarithm and has Heisenberg monodromy group in the quantum moduli space. We compare this to a Heisenberg group action on the type IIA hypermultiplet moduli space for the universal hypermultiplet, to metric corrections from membrane instantons related to a twisted dilogarithm for the deformed conifold and to a flat bundle related to a conifold period, the Heisenberg group and the dilogarithm appearing in five-dimensional Seiberg/Witten theory.

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