Bhargava showed that 100% of quintic fields have non-solvable Galois closure. For this reason, the arithmetic of elliptic curves over such fields is beyond the reach of methods based on Heegner points. In this talk we will report on a joint work in progress with Zhaorong Jin about the p-adic variation of Hirzebruch-Zagier cycles. The aim is to establish new instances of the rank zero BSD-conjecture over quintic fields.