Ein Analogon des Satzes von Nagell-Lutz über die Torsion einer elliptischen Kurve.
Elliptic curves with a rational point of finite order.
Elliptic curves with all quadratic twists of positive rank
Endlichkeitssätze für abelsche Varietäten über Zahlkörper.
Endlichkeitssätze für abelsche Varietäten über Zahlkörpern.
Endomorphisms of group schemes and rational points on curves
Equations diophantiennes exponentielles
Equations diophantiennes exponentielles.
Erratum : On the Hasse principle for homogeneous spaces with finite stabilizers
Erratum: Rational points of bounded height on Fano varieties. Invent. Math. 95, 421-435 (1989).
Erratum : “The rank of étale cohomology of varieties over -adic or number fields”
Existence of rational points on smooth projective varieties
Explicit Selmer groups for cyclic covers of ℙ¹
For any abelian variety J over a global field k and an isogeny ϕ: J → J, the Selmer group is a subgroup of the Galois cohomology group , defined in terms of local data. When J is the Jacobian of a cyclic cover of ℙ¹ of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the ‘fake Selmer group’, whose definition is more amenable to explicit computations. In this paper we define in the same setting the ‘explicit Selmer group’, which is isomorphic...
Failure of the Hasse principle for Châtelet surfaces in characteristic
Given any global field of characteristic , we construct a Châtelet surface over that fails to satisfy the Hasse principle. This failure is due to a Brauer-Manin obstruction. This construction extends a result of Poonen to characteristic , thereby showing that the étale-Brauer obstruction is insufficient to explain all failures of the Hasse principle over a global field of any characteristic.
Fields of definition for some Hodge cycles.
Fields of moduli of three-point -covers with cyclic -Sylow, II
We continue the examination of the stable reduction and fields of moduli of -Galois covers of the projective line over a complete discrete valuation field of mixed characteristic , where has a cyclic-Sylow subgroup of order . Suppose further that the normalizer of acts on via an involution. Under mild assumptions, if is a three-point -Galois cover defined over , then the th higher ramification groups above for the upper numbering of the (Galois closure of the) extension vanish,...
Flèches de spécialisations en cohomologie étale et applications arithmétiques
Fonctions -adiques à deux variables et -extensions
Function Fields of Quadratic Forms.
Fundamental groups and Diophantine geometry
This is a brief exposition on the uses of non-commutative fundamental groups in the study of Diophantine problems.