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

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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/}
}
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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/