Degenerating abelian varieties

The Bogomolov conjecture for totally degenerate abelian varieties

“…Proof: If ϕ : A 1 → A 2 is a homomorphism of abelian varieties over F , then the associated Raynaud extensions are homomorphic (see [BL1], §1 and use the argument before Proposition 3.5). If ϕ is an isogeny, then it is clear that the corresponding tori (resp.…”

“…We may suppose that the uniformization A an v /M is analytically defined over a subfield F of K v such that F is finite over the completion K v of K (see [BL1], Section 1). Then we have A(E) = (E × ) n /M for every algebraic subextension E/F of K v .…”

Component groups of abelian varieties and Grothendieck's duality conjecture

“…K and lift it to a V^-linearization of HK^ viewing the latter as a line bundle on TJt. Such a line bundle is trivial, as is shown in the proof of [3], 4.5. Hence, there is an isomorphism of rigid ^-spaces HK -^ Gm,j< x T^.…”

“…As E' -> B' and, hence by [8], 4.8, also E'^ -> B' 0 are epimorphisms, the above sequence of maps yields an epimorphism z*A'° -> z*B'° with kernel T" D A' 0 . But then, going through the duality theory of [3], Sect. 6, one can realize that the map coincides with the preceding one.…”

Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic

“…We also have the analytic torus T an = (G an m,K ) g associated to T rig and an open immersion T rig → T an . The key fact is that i extends uniquely to a rigid-analytic group morphism T an → J an , and its kernel is a lattice Λ ⊂ (K × ) g of rank g. This induces an isomorphism of analytic groups T an /Λ ∼ = J an ; see [1,Thm. 1.2].…”