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Let $E$ be a modular elliptic curve over a totally real number field $F$. Weprove the weak exceptional zero conjecture which links a (higher) derivative ofthe $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attachedto the corresponding Hilbert modular form at the places above $p$ where $E$ hassplit multiplicative reduction. Under some mild restrictions on $p$ and theconductor of $E$ we deduce the exceptional zero conjecture in the strong form(i.e.\ where the automorphic $p$-adic periods are replaced by the$\cL$-invariants of $E$ defined in terms of Tate periods) from a special caseproved earlier by Mok. Crucial for our method is a new construction of the$p$-adic $L$-function of $E$ in terms of local data.

On special zeros of $p$-adic $L$-functions of Hilbert modular forms