On the order of vanishing of Stickelberger elements of Hilbert modular forms

We construct Stickelberger elements for Hilbert modular cusp forms of parallel weight 2 and use recent results of Dasgupta and Spie{\ss} to bound their order of vanishing from below. As a special case the vanishing part of Mazur and Tate's refined "Birch and Swinnerton-Dyer type"-conjecture for elliptic curves of rank $0$ follows.