@Will Sawin I hope there is an automorphic approach to this problem, because modularity is so difficult..

Faltings topos in p-adic Simpson correspondence, the oriented product of topos for studying nearby cycle over general bases...

Thanks, I will edit the question.

Thanks for a good example by taking product, is there an easy way to see $H^1(E_s) \rightarrow H^1(E_{\eta})$ is injective (without applying Picard-Lefschetz formula)? The Galois action on $H^1(E_s)$ and $H^1(E_{\eta})$ is easy to compute using Tate curve and restriction exact sequence.

@user45878 Can you give a reference?

@DonuArapura Thank you! Do you know any example that the vanishing cycle sheaf is not pure?