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Jacobians with complex multiplication. (English) Zbl 1285.11089
Summary: We construct and study two series of curves whose Jacobians admit complex multiplication. The curves arise as quotients of Galois coverings of the projective line with Galois group metacyclic groups $$G_{q,3}$$ of order $$3q$$ with $$q \equiv 1 \bmod 3$$ an odd prime, and $$G_m$$ of order $$2^{m+1}$$. The complex multiplications arise as quotients of double coset algebras of the Galois groups of these coverings. We work out the CM-types and show that the Jacobians are simple abelian varieties.

##### MSC:
 11G15 Complex multiplication and moduli of abelian varieties 14K22 Complex multiplication and abelian varieties
##### Keywords:
complex multiplications; Jacobians; abelian varieties
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##### References:
 [1] S. A. Broughton, The homology and higher representations of the automorphism group of a Riemann surface, Trans. Amer. Math. Soc. 300 (1987), no. 1, 153 – 158. · Zbl 0621.30037 [2] Angel Carocca and RubíE. Rodríguez, Jacobians with group actions and rational idempotents, J. Algebra 306 (2006), no. 2, 322 – 343. · Zbl 1109.14028 · doi:10.1016/j.jalgebra.2006.07.027 · doi.org [3] C. Chevalley, A. Weil: Über das Verhalten der Integrale erster Gattung bei Automorphisman des Funktionenkörpers. Hamb. Abh. 10 (1934), 358-361. · Zbl 0009.16001 [4] Jordan S. Ellenberg, Endomorphism algebras of Jacobians, Adv. Math. 162 (2001), no. 2, 243 – 271. · Zbl 1065.14507 · doi:10.1006/aima.2001.1994 · doi.org [5] Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119 – 221 (French). · Zbl 0118.26104 [6] Solomon Lefschetz, On certain numerical invariants of algebraic varieties with application to abelian varieties (cont.), Trans. Amer. Math. Soc. 22 (1921), no. 4, 407 – 482. · JFM 48.0428.03 [7] Anita M. Rojas, Group actions on Jacobian varieties, Rev. Mat. Iberoam. 23 (2007), no. 2, 397 – 420. · Zbl 1139.14026 · doi:10.4171/RMI/500 · doi.org [8] Goro Shimura and Yutaka Taniyama, Complex multiplication of abelian varieties and its applications to number theory, Publications of the Mathematical Society of Japan, vol. 6, The Mathematical Society of Japan, Tokyo, 1961. · Zbl 0112.03502 [9] Paul van Wamelen, Examples of genus two CM curves defined over the rationals, Math. Comp. 68 (1999), no. 225, 307 – 320. · Zbl 0906.14025
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