Geodesic
Absolutely Simple Prymians of Trigonal Curves
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Multidimensional algebraic geometry, Tome 264 (2009), pp. 212-223.

Voir la notice de l'article provenant de la source Math-Net.Ru

Using Galois theory, we explicitly construct absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations. 
@article{TM_2009_264_a20,
     author = {Yu. G. Zarhin},
     title = {Absolutely {Simple} {Prymians} of {Trigonal} {Curves}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {212--223},
     publisher = {mathdoc},
     volume = {264},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2009_264_a20/}
}
TY  - JOUR
AU  - Yu. G. Zarhin
TI  - Absolutely Simple Prymians of Trigonal Curves
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2009
SP  - 212
EP  - 223
VL  - 264
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2009_264_a20/
LA  - ru
ID  - TM_2009_264_a20
ER  - 
%0 Journal Article
%A Yu. G. Zarhin
%T Absolutely Simple Prymians of Trigonal Curves
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2009
%P 212-223
%V 264
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2009_264_a20/
%G ru
%F TM_2009_264_a20
Yu. G. Zarhin. Absolutely Simple Prymians of Trigonal Curves. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Multidimensional algebraic geometry, Tome 264 (2009), pp. 212-223. http://geodesic.mathdoc.fr/item/TM_2009_264_a20/

[1] Bosma W., Cannon J., Playoust C., “The Magma algebra system. I: The user language”, J. Symb. Comput., 24 (1997), 235–265 ; http://magma.maths.usyd.edu.au/magma/ | DOI | MR | Zbl

[2] Koo J. K., “On holomorphic differentials of some algebraic function field of one variable over $\mathbb C$”, Bull. Austral. Math. Soc., 43 (1991), 399–405 | DOI | MR | Zbl

[3] Matsumoto K., Terasoma T., “Theta constants associated to cubic threefolds”, J. Alg. Geom., 12 (2003), 741–775 | DOI | MR | Zbl

[4] Mumford D., “Prym varieties. I”, Contributions to analysis, Acad. Press, New York, 1974, 325–350 | DOI | MR

[5] Oort F., “Endomorphism algebras of abelian varieties”, Algebraic geometry and commutative algebra, V. 2, Kinokuniya, Tokyo, 1988, 469–502 | DOI | MR

[6] Passman D. S., Permutation groups, W. A. Benjamin, New York–Amsterdam, 1968 | MR | Zbl

[7] Poonen B., Schaefer E., “Explicit descent for Jacobians of cyclic covers of the projective line”, J. reine und angew. Math., 488 (1997), 141–188 | MR | Zbl

[8] Ribet K., “Galois action on division points of Abelian varieties with real multiplications”, Amer. J. Math., 98 (1976), 751–804 | DOI | MR | Zbl

[9] Schaefer E., “Computing a Selmer group of a Jacobian using functions on the curve”, Math. Ann., 310:3 (1998), 447–471 ; “Erratum”, Math. Ann., 339:1 (2007), 1 | DOI | MR | Zbl | DOI | MR

[10] Serre J.-P., Topics in Galois theory, Jones and Bartlett Publ., Boston–London, 1992 | MR | Zbl

[11] Shimura G., “On analytic families of polarized abelian varieties and automorphic functions”, Ann. Math. Ser. 2, 78 (1963), 149–192 | DOI | MR | Zbl

[12] Shimura G., Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, 1971 | MR | Zbl

[13] Shokurov V. V., “Distinguishing Prymians from Jacobians”, Invent. math., 65:2 (1981), 209–219 | DOI | MR | Zbl

[14] Shokurov V. V., “Mnogoobraziya Prima: teoriya i prilozheniya”, Izv. AN SSSR. Ser. mat., 47:4 (1983), 785–855 | MR | Zbl

[15] Towse C., “Weierstrass points on cyclic covers of the projective line”, Trans. Amer. Math. Soc., 348 (1996), 3355–3378 | DOI | MR | Zbl

[16] Zarhin Yu. G., “Cyclic covers of the projective line, their jacobians and endomorphisms”, J. reine und angew. Math., 544 (2002), 91–110 | MR | Zbl

[17] Zarhin Yu. G., “The endomorphism rings of jacobians of cyclic covers of the projective line”, Math. Proc. Cambridge Philos. Soc., 136 (2004), 257–267 | DOI | MR | Zbl

[18] Zarhin Yu. G., “Endomorphism algebras of superelliptic Jacobians”, Geometric methods in algebra and number theory, Progr. Math., 235, eds. F. Bogomolov, Yu. Tschinkel, Birkhäuser, Boston, 2005, 339–362 | DOI | MR | Zbl

[19] Zarhin Yu. G., “Endomorphisms of superelliptic jacobians”, Math. Ztschr., 261 (2009), 691–707 | DOI | MR | Zbl

[20] Zarhin Yu. G., “Cubic surfaces and cubic threefolds, jacobians and intermediate jacobians”, Algebra, arithmetic and geometry: Manin Festschrift, V. 2, Progr. Math., 270, Birkhäuser, Boston, 2009 ; arXiv: math/0610138v3 | Zbl