I am almost sure that some paper was published in German probably in the 60's or in the 70's proving the existence of a "p-adic analytic Hilbert scheme" (or Douady space) related to a given proper rigid-analytic space, but I am not able to find it on the web. Does anybody have a reference?
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$\begingroup$ Just a comment, in Conrad's paper on relative ampleness (math.stanford.edu/~conrad/papers/amplepaper.pdf) he discusses representability of the Hilbert and Quot functor as a rigid analytic space (see e.g., Theorem 4.1.3 of the linked paper). $\endgroup$– Jackson MorrowCommented Nov 19, 2021 at 17:51
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$\begingroup$ I'm not sure if it is related with the article you are looking for but I immediately thought of Rapaport's "Compactifications de l'espace de modules de Hilbert-Blumenthal" from 1978. (I know that he proves the existence of some scheme, with the words "rigid" and "Hilbert" in it, that Deligne and Ribet used for their construction of the p-adic L-function for totally real fields). $\endgroup$– efsCommented Nov 19, 2021 at 18:36
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