Hirzebruch-Zagier classes and rational elliptic curves over quintic fields

Conditionally on a conjecture on the 'etale cohomology of Hilbert modular surfaces and some minor technical assumptions, we establish new instances of the equivariant BSD-conjecture in rank $0$ with applications to the arithmetic of rational elliptic curves over quintic fields. The key ingredients are a refinement of twisted triple product $p$-adic $L$-functions, the construction of a compatible collection of Hirzebruch-Zagier cycles and an explicit reciprocity law relating the two.