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I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families.

Suppose we have an algebraic family of varieties over a number field, and Galois representations arising from $l$-adic cohomology of one of the varieties in the family are known to be automorphic. What can one say about automorphy of (Galois representations attached to $l$-adic cohomology of) other varieties in the family?

Alternatively, and more specifically, on the moduli space of algebraic curves (of a fixed genus) over $\mathbb{Q}$, take a small/formal neighborhood of a curve which is automorphic in the above sense. What can be said about automorphy of curves in the neighborhood?

One can also ask a related global question: suppose one somehow knows that the universal curve over the moduli space of algebraic curves of genus $g$ is automorphic. Would it follow that all algebraic curves of genus $g$ are automorphic?

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    $\begingroup$ As a non-expert, I expect that very little can be said. Galois representations vary in a rather erratic way in families, as can be seen already with non-CM elliptic curves specialising to CM ones. $\endgroup$ – R. van Dobben de Bruyn Aug 14 '18 at 16:29
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    $\begingroup$ @R. van Dobben de Bruyn, good point, although erraticism of variation of Galois reps. does not seem to preclude automorphy of families of varieties, though there may be "singularities" or other obstructions. Perhaps the argument against it needs to be spelled out? $\endgroup$ – user127738 Aug 28 '18 at 3:53

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