Is there any analogue of the Tate curve for (principally polarized) abelian varieties of dimension $g$ ?



Analogue of Tate curve for $g>1$
Yes, due to Mumford. See


Mumford, D.: An analytic construction of degenerating abelian varieties over complete rings, *Comp.. Math.*
**24**, 239-272 (1972).


Of course, there's been plenty of work done since then. See for example the survey:


W. Lutkebohmert,
From Tate's Elliptic Curve to Abeloid Varieties, *Pure and Applied Mathematics Quarterly*
Volume 5, Number 4
(Special Issue: In honor of
John Tate, Part 1 of 2)
1385-1427, 2009. <http://intlpress.com/site/pub/files/_fulltext/journals/pamq/2009/0005/0004/PAMQ-2009-0005-0004-a007.pdf>


