Geodesic
Minimal models of curves of genus~2 and homomorphisms of abelian varieties defined over a~field of finite characteristic
Izvestiya. Mathematics , Tome 6 (1972) no. 1, pp. 65-108.

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In this article, we prove a finiteness theorem for isogenous abelian varieties of dimension 2 defined over a field of algebraic functions of one variable whose characteristic $\ne2$. By means of this result, we prove Tate's conjecture on homomorphisms of abelian varieties of dimension 1 defined over the same field. 
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A. N. Parshin. Minimal models of curves of genus~2 and homomorphisms of abelian varieties defined over a~field of finite characteristic. Izvestiya. Mathematics , Tome 6 (1972) no. 1, pp. 65-108. http://geodesic.mathdoc.fr/item/IM2_1972_6_1_a2/

[1] Borevich Z. I., Shafarevich I. R., Teoriya chisel, Nauka, M., 1964 | MR

[2] Shafarevich I. R., “Polya algebraicheskikh chisel”, Rros. Internat. Congr. Math., Stockholm, 1962, 163–176

[3] Shafarevich I. R., Lectures on minimal models, Bombay, 1966 | Zbl

[4] Deligne P., Mumford D., “The irreducibility of the space of curves of given genus”, Publ. Math. IHES, 36 (1969), 75–109 | MR | Zbl

[5] Deligne P., Grothendieck A., Seminaire de Geometric Algebrique, preprint, 1968–1970 | Zbl

[6] Donovan P., “The Lefscheitz–Riemann–Roch fomula”, Bull. Soc. Math. France, 97 (1969), 257–273 | MR | Zbl

[7] Grothendieck A., “Elements de Geometrie Aligebrique. IV. Étude locale des schémas et des morphismes de schémas IV”, Inst. Hautes Études Sci. Publ. Math., 1967, no. 32, 361 | MR | Zbl

[8] Grothendieck A., Fondements de la Geometrie Algebrique, Paris, 1962

[9] Grothendieck A., “Le groupe de Brauer. I”, Séminaire Bourbaki, 9, no. 290, Soc. Math. France, Paris, 1965, 199–219 | MR

[10] Igusa J., “Fibre systems of Jacobian varieties. III”, Amer. J. Math., 81 (1959), 453–476 | DOI | MR | Zbl

[11] Lang S., “Sur les Séries L d'une variété algébrique”, Bull. Soc. Math. France, 84 (1956), 385–407 | MR | Zbl

[12] Lang S., Diophantine geometry, New York, 1962 | MR

[13] Mumford D., “Enriques' classification of surfaces in ${\rm char}\ p$. I”, Global Analysis, Univ. Tokyo Press, Tokyo, 1969, 325–339 | MR

[14] Ogg A. P., “On pencils of curves of genus two”, Topology, 5 (1966), 355–362 | DOI | MR | Zbl

[15] Parshin A. N., “Algebraicheskie krivye nad funktsionalnymi polyami. I”, Izv. AN SSSR. Ser. matem., 32 (1968), 1191–1219 | Zbl

[16] Parshin A. N., “Quelques conjectures de finitude en Géometrie Diophantienne”, Actes du Congrès International des Mathématiciens (Nice, 1970), Gauthier-Villars, Paris, 1971, 467–471 | MR

[17] Raynoud M., “Modeles de Neron”, C. R. Acad. Sci., 262 (1966), 413–416 | MR

[18] Raynoud M., “Spécialisation du foncteur Picard”, Publ. Math. IHES, 39 (1970), 27–76 | MR

[19] Serre J.-P., Abelian $l$-adic Representations and Elliptic curves, New York, 1968 | MR

[20] Serre J.-P., Tate J., “Good reduction of abelian varieties”, Ann. Math., 88 (1968), 492–517 | DOI | MR | Zbl

[21] Tate J., “Algebraic cycles and poles of zeta functions”, Arithmetical algebraic geometry, New York, 1965, 93–110 | MR | Zbl

[22] Tate J., “Endomorphisms of abelian varieties over finite fields”, Inv. Math., 2 (1966), 134–144 | DOI | MR | Zbl

[23] Weil A., “Zum Beweis des Torellischen Satzes”, Nachr. Akad. Wiss. Göttingen, 1957 (1957), 33–53 | MR