Geodesic
Complexity of categorical theories with computable models
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 5 (2005) no. 2, pp. 77-85

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In the paper 2 theorems are proved: There is an uncountably categorical but not countably categorical theory of an arbitrary given arithmetical complexity with a computable model. All countable models of this theory are computable. There is a countably categorical theory of an arbitrary given arithmetical complexity with a computable model. 
@article{VNGU_2005_5_2_a4,
     author = {E. Fokina},
     title = {Complexity of categorical theories with computable models},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {77--85},
     publisher = {mathdoc},
     volume = {5},
     number = {2},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2005_5_2_a4/}
}
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E. Fokina. Complexity of categorical theories with computable models. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 5 (2005) no. 2, pp. 77-85. http://geodesic.mathdoc.fr/item/VNGU_2005_5_2_a4/