Hirzebruch–Zagier classes and rational elliptic curves over quintic fields

Conditionally on a conjecture on the étale 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.