TITLE: Analogue of Tate curve for $g>1$
QUESTION [12 upvotes]: Is there any analogue of the Tate curve for (principally polarized) abelian varieties of dimension $g$ ?

REPLY [21 votes]: 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