Number Theory Seminar

Speaker: Qixiang Wang

Title: Relative GAGA theorem and geometricity of analytic mapping stacks

Abstract: The representability of the Picard stack of a proper rigid space is a long-standing open problem in rigid analytic geometry. Classical relative GAGA theorems give an affirmative answer in the algebrizable case, but this approach does not extend to more flexible settings in modern p-adic geometry, such as v-stacks, essentially because existing relative GAGA results are only established for noetherian adic spaces. Using the Clausen–Scholze formalism of analytic geometry, we show how to overcome this obstruction in a conceptual way. Along the way, we in fact obtain a general representability result for analytic mapping stacks in the context of Gelfand stacks. In particular, for a proper algebraic source and a suitable target stack, the analytification of the algebraic mapping stack coincides with the intrinsic analytic mapping stack, providing a new approach to representability results for analytic moduli stacks in p-adic geometry.

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