Geodesic
Algebraic curves over functional fields with a finite field of constants
Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 561-570 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove a theorem of finiteness for curves of genus $g>1$, defined over a functional field of finite characteristic and having fixed invariants. As an application we obtain Tate's conjecture concerning homomorphisms of elliptic curves over a field of functions. 
@article{MZM_1974_15_4_a6,
     author = {A. N. Parshin},
     title = {Algebraic curves over functional fields with a~finite field of constants},
     journal = {Matemati\v{c}eskie zametki},
     pages = {561--570},
     year = {1974},
     volume = {15},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a6/}
}
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A. N. Parshin. Algebraic curves over functional fields with a finite field of constants. Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 561-570. http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a6/