Geodesic
Algebraic varieties over fields with differentiation
Sbornik. Mathematics, Tome 9 (1969) no. 3, pp. 389-413 
Yu. R. Vainberg. Algebraic varieties over fields with differentiation. Sbornik. Mathematics, Tome 9 (1969) no. 3, pp. 389-413. http://geodesic.mathdoc.fr/item/SM_1969_9_3_a7/
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     author = {Yu. R. Vainberg},
     title = {Algebraic varieties over fields with differentiation},
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     volume = {9},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1969_9_3_a7/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is known that there do not exist algebraic homomorphisms of the multiplicative group of a field $K^*$ into the additive group $K^+$. However, if the field $K$ has a nontrivial differentiation $\alpha$, then the logarithmic derivative gives a homomorphism $K^*\to K^+$, $x\to\frac{\alpha x}x$. Yu. I. Manin observed that for abelian varieties $X$ over a field $K$ with a nontrivial differentiation it is possible to construct homothetic homomorphisms of the group of points $X(K)$ into $K$. The study of such homomorphisms (in particular, the computation of the intersection of their kernels) for varieties over function fields permitted Manin to prove the function field analog of Mordell's conjecture. In this paper we introduce and systematically study a class of functions ($\mathscr D$-functions) encountered in the definition of Manin's map $\mu$. We study the map $\mu$ in the case of varieties over a field of formal power series. Bibliography: 10 titles.

[1] Yu. I. Malin, “Algebraicheskie krivye nad polyami s differentsirovaniem”, Izv. AN SSSR, seriya matem., 22 (1958), 737–756

[2] Yu. I. Manin, “Ratsionalnye tochki algebraicheskikh krivykh nad funktsionalnymi polyami”, Izv. AN SSSR, seriya matem., 27 (1963), 1397–1442 | MR

[3] M. Greenberg, “Schemata over local rings, I”, Ann. Math., 73 (1961), 624–648 | DOI | MR | Zbl

[4] M. Greenberg, “Schemata over local rings, II”, Ann. Math., 78 (1963), 256–266 | DOI | MR | Zbl

[5] J. Igusa, “Analytics groupes over complet fields”, Proc. Nat. Acad. Sci. USA, 42:8 (1956), 540–541 | DOI | MR | Zbl

[6] E. Lutz, “Sur l'équation $y^2=x^3-Ax-B$ dans les corp $p$-adiques”, J. Reine und Angew. Math., 177:4 (1937), 238–247 | Zbl

[7] Mattuck, “Abelian varieties over $p$-adic ground field”, Ann. Math., 62:1 (1955), 92–119 | DOI | MR | Zbl

[8] V. P. Palamodov, “Differentsialnye operatory v kogerentnykh analiticheskikh puchkakh”, Matem. sb., 77(119) (1968), 390–422 | MR | Zbl

[9] J. Serre, Sur les groupes algèbriques et théorie des corps des class, Paris, 1959

[10] J. Serre, Groupes proalgèbriques, Inst. Hautes Études Sci. Publ. Math., 7, 1960 | MR | Zbl