Géométrie algébrique
Shimura subvarieties via endomorphisms
Comptes Rendus. Mathématique, Tome 360 (2022) no. G10, pp. 1117-1124.

We prove that there exist two families in 2 , 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 .

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DOI : 10.5802/crmath.369
Classification : 14H10, 14H30, 14H40, 14G35
Spelta, Irene 1

1 Università degli Studi di Pavia, Dipartimento di Matematica, Via Ferrata 5, 27100 Pavia, Italy
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Spelta, Irene. Shimura subvarieties via endomorphisms. Comptes Rendus. Mathématique, Tome 360 (2022) no. G10, pp. 1117-1124. doi : 10.5802/crmath.369. http://www.numdam.org/articles/10.5802/crmath.369/

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

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

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

[4] Ciliberto, Ciro; van der Geer, Gerard; Teixidor i Bigas, Montserrat 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] Colombo, Elisabetta; Frediani, Paola; Ghigi, Alessandro; Penegini, Matteo Shimura curves in the Prym locus, Commun. Contemp. Math., Volume 21 (2019) no. 2, 1850009, 34 pages | MR | Zbl

[6] Conti, Diego; Ghigi, Alessandro; Pignatelli, Roberto 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] Ellenberg, Jordan S. Endomorphism algebras of Jacobians, Adv. Math., Volume 162 (2001) no. 2, pp. 243-271 | DOI | MR | Zbl

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

[9] Frediani, Paola; Ghigi, Alessandro; Penegini, Matteo 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] Frediani, Paola; Ghigi, Alessandro; Spelta, Irene Infinitely many Shimura varieties in the Jacobian locus for g4, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 22 (2021) no. 4, pp. 1597-1619 | MR | Zbl

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

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

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

[15] Grosselli, Gian Paolo; Spelta, Irene 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] Grushevsky, Samuel; Möller, Martin 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] de Jong, Johan; Noot, Rutger Jacobians with complex multiplication, Arithmetic algebraic geometry (Progress in Mathematics), Volume 89, Birkhäuser, 1991, pp. 177-192 | DOI | MR | Zbl

[18] de Jong, Johan; Zhang, Shou-Wu 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] Lange, Herbert; Ortega, Angela Prym varieties of cyclic coverings, Geom. Dedicata, Volume 150 (2011), pp. 391-403 | DOI | MR | Zbl

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

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

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

[23] Moonen, Ben; Oort, Frans 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] Mumford, David Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, 5, Oxford University Press, 1974

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

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

[27] Ries, John 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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