# $p$-adic uniformisation of abelian varieties

In the paper $p$-adic L-functions and $p$-adic periods of modular forms of Greenberg and Stevens $\S3$ page $420$ they make the following statement:

Let $A$ over $\mathbf{Q}_p$ be an abelian variety with split multiplicative reduction and $B$ be its dual. Let $X,Y$ be the character groups of $B^0_{\mathbf{F}_p}$ and $A^0_{\mathbf{F}_p}$ with trivial $G_{\mathbf{Q}_p}$ action. Then from the theory of $p$-adic uniformization we get a multiplicative pairing

$$j : X \times Y \to \mathbf{Q}_p^*$$ and exact sequences of $G_{\mathbf{Q}_p}$ modules $$0 \to X \xrightarrow{\;j\;} \text{Hom}(Y,\overline{\mathbf{Q}}_p^*) \longrightarrow A(\overline{\mathbf{Q}}_p) \longrightarrow 0$$ $$0 \to Y \xrightarrow{\;j\;} \text{Hom}(X,\overline{\mathbf{Q}}_p^*) \longrightarrow B(\overline{\mathbf{Q}}_p) \longrightarrow 0$$ Furthermore the pairing $\alpha : ord_p \circ j : X \times Y \to \mathbf{Z}$ is non degenerate.

They give two references for this : McCabe's thesis at harvard "$p$-adic theta functions" which I haven't found on line and Morikawa : "On theta functions and abelian varieties over valuation fields of rank one, part I and part II" from where I haven't been able to extract the mentioned result.

My question is : does anyone know of a good reference for this or can explain to me how to obtain the result from Morikawa's paper ?

• This is about 2 things: the relationship between uniformizations of an abelian variety and its dual for split toric reduction (really duality between the two uniformization lattices), a canonical isomorphism between the uniformization lattice and the character lattice of the split torus identity component of the special fiber of the Neron model of the dual abelian variety. These are addressed by the analytic construction of the dual, explained in 6.3 of Lutkebohmert's new book "Rigid geometry of curves and their Jacobians", alas written for alg. closed ground field but applies in general. – nfdc23 Apr 16 '16 at 22:33
• Thanks a lot ! I couldn't find the book on the usual russian sites so I'll have to wait until monday (if they have it at my university) to check this out. – cannonball Apr 17 '16 at 5:11
• There is a 1991 paper by Bosch and Lutkebohmert called "Degenerating Abelian Varieties" which discusses these matters in much wider generality (abeloid spaces and their Picard functors over any rigid-analytic base), though the book is probably more accessible than the research article (does seem a pity that the book assumes the ground field to be algebraically closed in that discussion, even though the methods used are more widely applicable). The book was published only in the last month or two. – nfdc23 Apr 17 '16 at 5:42