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Hida theory is a vast domain of research. I am assuming that that you are in the simplest and oldest setting: Hida theory for ordinary eigencuspforms for the group $\operatorname{GL}_2$ over $\mathbb …

Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf …

Given an eigencuspform $f$ of weight $k≥2$ (so in particular 2) and $p$ a prime of ordinary reduction (in particular good ordinary reduction), there is always a Hida family passing through $f$. This …

This is an answer of rather low quality, but let me report that in several discussions about this topic with (many) experts over the span of many years, I have neither met anyone knowing of such an ex …

Assuming you are writing about Mok's Compositio 2009 article, the answer is easy: it's a question of quantifier ordering. There are two statements which you could call the interpolation property for a …

(1) The answer is yes. In fact this was done by Ohta himself essentially simultaneously as the other cases. The relevant publication is here (the proof in the case $k\equiv 2$ is around 569/570). (2) …

Like yourself, I have found a reference for these results hard to find, even in the case of Hida families. These are a few pointers. In the case of Hida families, once the Atkin-Lehner involutions ar …