  The main result of this text is a generalization of Perrin-Riou's p-adic
Gross-Zagier formula to the case of Shimura curves over totally real fields.
Let $F$ be a totally real field. Let $f$ be a Hilbert modular form over $F$ of
parallel weight $2$, which is a new form and is ordinary at $p$. Let $E$ be a
totally imaginary quadratic extension of $F$ of discriminant prime to $p$ and
to the conductor of $f$. We may construct a $p$-adic $L$ function that
interpolates special values of the complex $L$ functions associated to $f$, $E$
and finite order Hecke characters of $E$. The $p$-adic Gross-Zagier formula
relates the central derivative of this $p$-adic $L$ function to the $p$-adic
height of a Heegner divisor on a certain Shimura curve.
  The strategy of the proof is close to that of the original work of
Perrin-Riou. In the analytic part, we construct the analytic kernel via adelic
computations; in the geometric part, we decompose the geometric kernel into two
parts: places outside $p$ and places dividing $p$. For places outside $p$, the
$p$-adic heights are essentially intersection numbers and are computed in works
of S. Zhang, and it turns out that this part is closely related to the analytic
kernel. For places dividing $p$, we use the method in the work of J.
Nekov\'a\v{r} to show that the contribution of this part is zero.
