I have some naive questions about Zhang's generalization of Gross-Zagier formula stated in Theorem 1.2 of Yuan-Zhang-Zhang book The Gross-Zagier formula on Shimura Curves. I am mostly interested by the cases of non split cartan modular curves and Shimura curves X_D(N). I understand that the "incoherent Shimura curves" setting of Zhang contains those cases but

1. could you explain me what are concretely the formulas in those cases ?

2. does this formula holds directly on the curve X_U instead of on an abelian variety parametrized by X_U ?

Thank you very much.

[edit : I copy here the statement of Theorem 1.2 (but without detailing the notations) :

Assume $\omega_A.\chi|_{A_F^\times}=1$ [which ensures that $P_\chi(f) \neq 0$]. For any $f_1\in\pi_A, $ and $f_2\in\pi_{\check A}$, $$ \langle P_\chi(f_1),P_\chi(f_2)\rangle _L= \frac{\zeta_F(2) L'(1/2,\pi_A,\chi)}{4L(1,\eta)^2L(1,\pi_A,ad)}\alpha(f_1,f_2)$$ as an identity in $L\otimes_{\mathbb Q} \mathbb C$. Here $\langle,\rangle_L$ is the $L$-linear Neron-Tate height pairing. ]

I have some naive questions about Zhang's generalization of Gross-Zagier formula stated in Theorem 1.2 of Yuan-Zhang-Zhang book The Gross-Zagier formula on Shimura Curves. I am mostly interested by the cases of non split cartan modular curves and Shimura curves $X_D(N)$. I understand that the "incoherent Shimura curves" setting of Zhang contains those cases but

1. could you explain me what are concretely the formulas in those cases ?

2. does this formula holds directly on the curve $X_U$ instead of on an abelian variety parametrized by $X_U$?

Thank you very much.

[edit : I copy here the statement of Theorem 1.2 (but without detailing the notations) :

Assume $\omega_A.\chi|_{A_F^\times}=1$ [which ensures that $P_\chi(f) \neq 0$]. For any $f_1\in\pi_A, $ and $f_2\in\pi_{\check A}$, $$ \langle P_\chi(f_1),P_\chi(f_2)\rangle _L= \frac{\zeta_F(2) L'(1/2,\pi_A,\chi)}{4L(1,\eta)^2L(1,\pi_A,ad)}\alpha(f_1,f_2)$$ as an identity in $L\otimes_{\mathbb Q} \mathbb C$. Here $\langle,\rangle_L$ is the $L$-linear Neron-Tate height pairing. ]

I have some naive questions about Zhang's generalization of Gross-Zagier formula stated in Theorem 1.2 of Yuan-Zhang-Zhang book The Gross-Zagier formula on Shimura Curves. I am mostly interested by the cases of non split cartan modular curves and Shimura curves X_D(N). I understand that the "incoherent Shimura curves" setting of Zhang contains those cases but

1. could you explain me what are concretely the formulas in those cases ?

2. does this formula holds directly on the curve X_U instead of on an abelian variety parametrized by X_U ?

Thank you very much.

I have some naive questions about Zhang's generalization of Gross-Zagier formula stated in Theorem 1.2 of Yuan-Zhang-Zhang book The Gross-Zagier formula on Shimura Curves. I am mostly interested by the cases of non split cartan modular curves and Shimura curves X_D(N). I understand that the "incoherent Shimura curves" setting of Zhang contains those cases but

1. could you explain me what are concretely the formulas in those cases ?

2. does this formula holds directly on the curve X_U instead of on an abelian variety parametrized by X_U ?

Thank you very much.

[edit : I copy here the statement of Theorem 1.2 (but without detailing the notations) :

Assume $\omega_A.\chi|_{A_F^\times}=1$ [which ensures that $P_\chi(f) \neq 0$]. For any $f_1\in\pi_A, $ and $f_2\in\pi_{\check A}$, $$ \langle P_\chi(f_1),P_\chi(f_2)\rangle _L= \frac{\zeta_F(2) L'(1/2,\pi_A,\chi)}{4L(1,\eta)^2L(1,\pi_A,ad)}\alpha(f_1,f_2)$$ as an identity in $L\otimes_{\mathbb Q} \mathbb C$. Here $\langle,\rangle_L$ is the $L$-linear Neron-Tate height pairing. ]