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

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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

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