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21
Abelian varieties without homotheties
, 2006
"... Abstract. A celebrated theorem of Bogomolov asserts that the ℓadic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic p: a “counterexample ” is provided by an ordinary elliptic curve ..."
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Abstract. A celebrated theorem of Bogomolov asserts that the ℓadic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic p: a “counterexample ” is provided by an ordinary elliptic curve defined over a finite field. In this note we discuss (and explicitly construct) more interesting examples of “nonconstant ” absolutely simple abelian varieties (without homotheties) over global fields in characteristic p. 1.
Analytic variation of Tate–Shafarevich groups
, 2015
"... Let K be a number field. For a prime p, we study the inductive limit of the pordinary part of the TateShafarevich groups and the Selmer groups (over K) of modular Jacobians of level Npr as r → ∞ for a fixed integer N prime to p. We prove control theorems of the pordinary pprimary part of WK(AP ..."
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Let K be a number field. For a prime p, we study the inductive limit of the pordinary part of the TateShafarevich groups and the Selmer groups (over K) of modular Jacobians of level Npr as r → ∞ for a fixed integer N prime to p. We prove control theorems of the pordinary pprimary part of WK(AP) over padic analytic family of abelian varieties AP. In particular, under mild conditions, we show that if WK (AP0) ord p is finite for one member AP0 of the analytic family and the Mordell–Weil rank of AP0 is ≤ 1 over its Hecke field, thenWK(AP) ord p is finite for almost all members AP.
Detecting linear dependence on an abelian variety via reduction maps
 Comment. Math. Helv
"... Abstract. Let A be a geometrically simple abelian variety over a number field k, let X be a subgroup of A(k) and let P ∈ A(k) be a rational point. We prove that if P belongs to X modulo almost all primes of k then P already belongs to X. ..."
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Abstract. Let A be a geometrically simple abelian variety over a number field k, let X be a subgroup of A(k) and let P ∈ A(k) be a rational point. We prove that if P belongs to X modulo almost all primes of k then P already belongs to X.
On the Kernels of the Prol Outer Galois Representations Associated to Hyperbolic Curves over Number Fields
, 2013
"... In the present paper, we discuss the relationship between the Galois extension corresponding to the kernel of the prol outer Galois representation associated to a hyperbolic curve over a number field and lmoderate points of the hyperbolic curve. In particular, we prove that, for a certain hyperb ..."
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In the present paper, we discuss the relationship between the Galois extension corresponding to the kernel of the prol outer Galois representation associated to a hyperbolic curve over a number field and lmoderate points of the hyperbolic curve. In particular, we prove that, for a certain hyperbolic curve, the Galois extension under consideration is generated by the coordinates of the lmoderate points of the hyperbolic curve. This may be regarded as an analogue of the fact that the Galois extension corresponding to the kernel of the ladic Galois representation associated to an abelian variety is generated by the
MORDELL EXCEPTIONAL LOCUS FOR SUBVARIETIES OF THE ADDITIVE GROUP
"... Abstract. We define the Mordell exceptional locus Z(V) for affine varieties V ⊂ G g a with respect to the action of a product of Drinfeld modules on the coordinates of G g a. We show that Z(V) is a closed subset of V. We also show that there are finitely many maximal algebraic φmodules whose transl ..."
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Abstract. We define the Mordell exceptional locus Z(V) for affine varieties V ⊂ G g a with respect to the action of a product of Drinfeld modules on the coordinates of G g a. We show that Z(V) is a closed subset of V. We also show that there are finitely many maximal algebraic φmodules whose translates lie in V. Our results are motivated by DenisMordellLang conjecture for Drinfeld modules. 1.
Contents
, 1995
"... Abstract. We study monodromy action on abelian varieties satisfying certain bad reduction conditions. These conditions allow us to get some control over the Galois image. As a consequence we verify the Mumford–Tate conjecture for such abelian varieties. ..."
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Abstract. We study monodromy action on abelian varieties satisfying certain bad reduction conditions. These conditions allow us to get some control over the Galois image. As a consequence we verify the Mumford–Tate conjecture for such abelian varieties.