On some differences between number fields and function fields
Received: 20 January 2016
Accepted in revised form: 21 June 2016
Mathematics Subject Classification (2010): 14G40, 14G22, 11G50.
Keywords: Arithmetic over function fields, height theory, Lang and Vojta conjectures.
 : IRMA, UMR 7501, 7 rue René-Descartes, 67084 Strasbourg Cedex, France
Research supported by the FRIAS-USIAS.
Abstract: The analogy between the arithmetic of varieties over number fields and the arithmetic of varieties over function fields is a leading theme in arithmetic geometry. This analogy is very powerful but there are some gaps. In this note we will show how the presence of isotrivial varieties over function fields (the analogous of which does not seem to exist over number fields) breaks this analogy. Some counterexamples to a statement similar to Northcott Theorem are proposed. In positive characteristic, some explicit counterexamples to statements similar to Lang and Vojta conjectures are given.
 C. Gasbarri, The strong abc conjecture over function fields (after McQuillan and Yamanoi), Séminaire Bourbaki, Vol. 2007/2008, Astérisque No. 326 (2009), Exp. No. 989, viii, 219–256 (2010). MR2605324
 M. Kim, Geometric height inequalities and the Kodaira-Spencer map, Compositio Math. 105 (1997), no. 1, 43–54. MR1436743
 M. McQuillan, Old and new techniques in function field arithmetics, preprint, avaliable at http://www.mat.uniroma2.it/~mcquilla/files/oldnew.pdf
 A. Moriwaki, Geometric height inequality on varieties with ample cotangent bundles, J. Algebraic Geom. 4 (1995), no. 2, 385–396. MR1311357
 P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Math., 1239, Springer-Verlag, Berlin 1987. MR0883451