Shimura subvarieties via endomorphisms

Irene Spelta

doi:10.5802/crmath.369

Géométrie algébrique

Shimura subvarieties via endomorphisms

Irene Spelta ¹

¹ Università degli Studi di Pavia, Dipartimento di Matematica, Via Ferrata 5, 27100 Pavia, Italy

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1117-1124.

Résumé

We prove that there exist two families in $ℳ_{2}, ℳ_{3}$ of non-Galois covers of the projective line whose Jacobians trace out Shimura subvarieties of $𝒜_{2}, 𝒜_{3}$ . They provide the first two explicit examples of Shimura subvarieties obtained by means of Jacobians carrying non-trivial endomorphisms not directly induced by the automorphisms of the curves. We also obtain a new example of a positive dimensional family of special Pryms in $𝒜_{4}^{δ}$ .

Reçu le : 2021-12-03
Révisé le : 2022-02-01
Accepté le : 2022-04-22
Publié le : 2022-10-04

DOI : 10.5802/crmath.369

Classification : 14H10, 14H30, 14H40, 14G35

Affiliations des auteurs :

Irene Spelta ¹

¹ Università degli Studi di Pavia, Dipartimento di Matematica, Via Ferrata 5, 27100 Pavia, Italy

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Irene Spelta},
     title = {Shimura subvarieties via endomorphisms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1117--1124},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.369},
     language = {en},
}

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Irene Spelta. Shimura subvarieties via endomorphisms. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 1117-1124. doi : 10.5802/crmath.369. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.369/

Bibliographie
Cité par

[1] Alberto Albano; Gian Pietro Pirola Dihedral monodromy and Xiao fibrations, Ann. Mat. Pura Appl., Volume 195 (2016) no. 4, pp. 1255-1268 | DOI | MR | Zbl

[2] Matt Bainbridge; Martin Möller The locus of real multiplication and the Schottky locus, J. Reine Angew. Math., Volume 686 (2014), pp. 167-186 | MR | Zbl

[3] Christina Birkenhake; Herbert Lange Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften, 302, Springer, 2004 | DOI

[4] Ciro Ciliberto; Gerard van der Geer; Montserrat Teixidor i Bigas On the number of parameters of curves whose Jacobians possess nontrivial endomorphisms, J. Algebr. Geom., Volume 1 (1992) no. 2, pp. 215-229 | MR | Zbl

[5] Elisabetta Colombo; Paola Frediani; Alessandro Ghigi; Matteo Penegini Shimura curves in the Prym locus, Commun. Contemp. Math., Volume 21 (2019) no. 2, 1850009, 34 pages | MR | Zbl

[6] Diego Conti; Alessandro Ghigi; Roberto Pignatelli Some evidence for the Coleman–Oort Conjecture, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM, Volume 116 (2022) no. 1, 50, 19 pages | MR | Zbl

[7] Jordan S. Ellenberg Endomorphism algebras of Jacobians, Adv. Math., Volume 162 (2001) no. 2, pp. 243-271 | DOI | MR | Zbl

[8] Paola Frediani Abelian covers and Second Fundamental Form (2021) (https://arxiv.org/abs/2105.07947)

[9] Paola Frediani; Alessandro Ghigi; Matteo Penegini Shimura varieties in the Torelli locus via Galois coverings, Int. Math. Res. Not., Volume 2015 (2015) no. 20, pp. 10595-10623 | DOI | MR | Zbl

[10] Paola Frediani; Alessandro Ghigi; Irene Spelta Infinitely many Shimura varieties in the Jacobian locus for $g \leq 4$ , Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 22 (2021) no. 4, pp. 1597-1619 | MR | Zbl

[11] Paola Frediani; Gian Paolo Grosselli Shimura curves in the Prym loci of ramified double covers, Int. J. Math., Volume 32 (2021) no. 14, 2150110, 23 pages | MR | Zbl

[12] Paola Frediani; Gian Paolo Grosselli; Abolfazl Mohajer Higher dimensional Shimura varieties in the Prym loci of ramified double covers (2021) (https://arxiv.org/abs/2101.09016, to appear in Math. Nachr.)

[13] Paola Frediani; Matteo Penegini; Paola Porru Shimura varieties in the Torelli locus via Galois coverings of elliptic curves, Geom. Dedicata, Volume 181 (2016), pp. 177-192 | DOI | MR | Zbl

[14] Gian Paolo Grosselli; Abolfazl Mohajer Shimura subvarieties in the Prym locus of ramified Galois coverings (2021) (https://arxiv.org/abs/2106.05704)

[15] Gian Paolo Grosselli; Irene Spelta Explicit analysis of positive dimensional fibres of $𝒫_{g, r}$ and Xiao conjecture (2021) (https://arxiv.org/abs/2110.03938, to appear in Rend. Ist. Mat. Univ. Trieste)

[16] Samuel Grushevsky; Martin Möller Shimura curves within the locus of hyperelliptic Jacobians in genus 3, Int. Math. Res. Not., Volume 2016 (2016) no. 6, pp. 1603-1639 | DOI | MR | Zbl

[17] Johan de Jong; Rutger Noot Jacobians with complex multiplication, Arithmetic algebraic geometry (Progress in Mathematics), Volume 89, Birkhäuser, 1991, pp. 177-192 | DOI | MR | Zbl

[18] Johan de Jong; Shou-Wu Zhang Generic abelian varieties with real multiplication are not Jacobians, Diophantine geometry (Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series (Nuova Serie)), Volume 4, Edizioni della Normale, 2007, pp. 165-172 | MR | Zbl

[19] Herbert Lange; Angela Ortega Prym varieties of cyclic coverings, Geom. Dedicata, Volume 150 (2011), pp. 391-403 | DOI | MR | Zbl

[20] Abolfazl Mohajer; Kang Zuo On Shimura subvarieties generated by families of abelian covers of $ℙ^{1}$ , J. Pure Appl. Algebra, Volume 222 (2018) no. 4, pp. 931-949 | DOI | MR | Zbl

[21] Ben Moonen Linearity properties of Shimura varieties. I, J. Algebr. Geom., Volume 7 (1998) no. 3, pp. 539-567 | MR | Zbl

[22] Ben Moonen Special subvarieties arising from families of cyclic covers of the projective line, Doc. Math., Volume 15 (2010), pp. 793-819 | MR | Zbl

[23] Ben Moonen; Frans Oort The Torelli locus and special subvarieties, Handbook of moduli. Volume II (Advanced Lectures in Mathematics), Volume 25, International Press, 2013, pp. 549-594 | MR | Zbl

[24] David Mumford Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Oxford University Press, 1974

[25] Frans Oort; Joseph Steenbrink The local Torelli problem for algebraic curves, Journées de Géometrie Algébrique d’Angers, Sijthoff & Noordhoff, Alphenaan den Rijn, 1980, pp. 157-204 | Zbl

[26] Gian Pietro Pirola On a conjecture of Xiao, J. Reine Angew. Math., Volume 431 (1992), pp. 75-89 | MR | Zbl

[27] John Ries The Prym variety for a cyclic unramified cover of a hyperelliptic Riemann surface, J. Reine Angew. Math., Volume 340 (1983), pp. 59-69 | MR | Zbl

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