$p$-adic Asai $L$-functions for finite slope quadratic Hilbert eigenforms

Abstract

We define a two-variable $p$-adic Asai $L$-function for a finite-slope family of Hilbert modular forms over a real quadratic field (with one component of the weight, and the cyclotomic twist variable, varying independently). This generalizes a construction due to Grossi, Zerbes and the second author for ordinary families. Our construction relies on a ``nearly-overconvergent'' version of higher Coleman theory for Hilbert modular surfaces.

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