On Jacobians with group action and coverings
Tóm tắt
Let S be a compact Riemann surface and let H be a finite group. It is known that if H acts on S, then there is a H-equivariant isogeny decomposition of the Jacobian variety JS of S, called the group algebra decomposition of JS with respect to H. If $$S_1 \rightarrow S_2$$ is a regular covering map, then it is also known that the group algebra decomposition of $$JS_1$$ induces an isogeny decomposition of $$JS_2.$$ In this article we deal with the converse situation. More precisely, we prove that the group algebra decomposition can be lifted under regular covering maps, under appropriate conditions.
Tài liệu tham khảo
Barraza, P., Rojas, A.M.: The group algebra decomposition of Fermat curves of prime degree. Arch. Math. (Basel) 104(2), 145–155 (2015)
Birkenhake, C., Lange, H.: Complex Abelian Varieties. Grundl Math Wiss, vol. 302, 2nd edn. Springer, Berlin (2004)
Broughton, S.A.: Classifying finite group actions on surfaces of low genus. J. Pure Appl. Algebra 69(3), 233–270 (1991)
Carocca, A., Rodríguez, R.E.: Jacobians with group actions and rational idempotents. J. Algebra 306(2), 322–343 (2006)
Carocca, A., Recillas, S., Rodríguez, R.E.: Dihedral groups acting on Jacobians. Contemp. Math. 311, 41–77 (2011)
Carvacho, M., Hidalgo, R.A., Quispe, S.: Jacobian variety of generalized Fermat curves. Q. J. Math. 67(2), 261–284 (2016)
Curtis, C.W., Reiner, I.: Methods of Representation Theory with Applications to Finite Groups and Orders, vol. 1. Wiley, New York (1981)
Debarre, O.: Tores et variétés abéliennes complexes, Cours Spécialisés, 6. Société Mathématique de France, Paris; EDP Sciences, Les Ulis (1999)
Farkas, H., Kra, I.: Riemann Surfaces. Texts in Maths, vol. 71. Springer, Berlin (1980)
Hidalgo, R.A., Rodríguez, R.E.: A remark on the decomposition of the Jacobian variety of Fermat curves of prime degree. Arch. Math. (Basel) 105(4), 333–341 (2015)
Hidalgo, R.A., Jiménez, L., Quispe, S., Reyes-Carocca, S.: Quasiplatonic curves with symmetry group \({\mathbb{Z}}_2^2 \rtimes {\mathbb{Z}}_m\) are definable over \({\mathbb{Q}}\). Bull. Lond. Math. Soc. 49, 165–183 (2017)
Jiménez, L.: On the kernel of the group algebra decomposition of a Jacobian variety. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM 110(1), 185–199 (2016)
Kani, R., Rosen, M.: Idempotent relations and factors of Jacobians. Math. Ann. 284, 307–327 (1989)
Lange, H., Recillas, S.: Abelian varieties with group actions. J. Reine Angew. Math. 575, 135–155 (2004)
Paulhus, J.: Decomposing Jacobians of curves with extra automorphisms. Acta Arith. 132(3), 231–244 (2008)
Paulhus, J., Rojas, A.M.: Completely decomposable Jacobian varieties in new genera. Exp. Math. 26(4), 430–445 (2017)
Recillas, S., Rodríguez, R.E.: Jacobians and representations of \(S_3\), Aportaciones Mat. Investig. vol. 13, Soc. Mat. Mexicana, México (1998)
Reyes-Carocca, S., Rodríguez, R.E.: A generalisation of Kani-Rosen decomposition theorem for Jacobian varieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. https://doi.org/10.2422/2036-2145.201706-003. arXiv:1702.00484
Reyes-Carocca, S.: On the one-dimensional family of Riemann surfaces of genus \(q\) with \(4q\) automorphisms. J. Pure Appl. Algebra 223(5), 2123–2144 (2019)
Ries, J.: The Prym variety for a cyclic unramified cover of a hyperelliptic curves. J. Reine Angew. Math. 340, 59–69 (1983)
Rojas, A.M.: Group actions on Jacobian varieties. Rev. Mat. Iberoam. 23, 397–420 (2007)
Sánchez-Argáez, A.: Actions of the group \(A_5\) in Jacobian varieties, Aportaciones Mat. Comun. vol. 25, pp. 99–108. Soc. Mat. Mexicana, México (1999)
Schottky, F., Jung, H.: Neue Sätze über Symmetralfunctionen und due Abel’schen Functionen der Riemann’schen Theorie. S.B. Akad. Wiss. (Berlin) Phys. Math. Kl. 1, 282–297 (1909)
Serre, J.P.: Linear Representations of Finite Groups, Graduate Texts in Mathematics, vol. 42. Springer, New York (1977)
Suzuki, M.: Group Theory 2, Grundlehren der Mathematischen Wissenschaften, vol. 248. Springer, New York (1986)
Wirtinger, W.: Untersuchungen über Theta Funktionen. Teubner, Berlin (1895)