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. ]
Best Answer
These formulas are equalities, so their content is that the left-hand side is equal to the right-hand; a tautology for sure, but an interesting one once the left-hand side and right-hand side are properly understood. So let's them examine them in turns.
The left-hand side is the height pairing of two Heegner points constructed from the choice of two vectors; one in the representation space $\pi_{A}$ and one in its contragredient (which is almost the same by essential self-duality). So the left-hand side is a bilinear form on the vectors of $\pi_{A}$, or more precisely on $\pi_A\otimes\check{\pi}_A$, globally constructed from algebraic geometry. Now the important term on the right-hand side is the bilinear form $\alpha(-,-)$, which is a product of local bilinear forms on $\pi_{v}\otimes\check{\pi}_{v}$ for all finite places $v$ of $F$. Hence, you a have an essentially global and geometric bilinear form (the intersection pairing between certain cycles of interest) and an essentially local and automorphic pairing (the convolution of local automorphic factors). The meaning of the Gross-Zagier formula in the style of Yuan-Zhang-Zhang is that these two bilinear forms are essentially the same, and insofar as they differ, it is only through the special value of some $L$-function (the most interesting term being $L'(\pi,1/2,\chi)$).
Hence, this result fits in the large family of results of the form: when two objects with the same properties are constructed for motives/galois representations/algebraic automorphic representations, they are in fact the same thing except their difference is a period/special value of $L$-function/etc.
The formula you quoted depends on a choice of an automorphic representation $\pi$ of $\mathbf{G}(\mathbb A_{\mathbb Q}^{(\infty)})$ where $\mathbf{G}(\mathbb Q)$ is either $\operatorname{GL}_{2}$ or $B^\times$ where $B$ is a quaternion algebra isomorphic to $\operatorname{GL}_{2}(\mathbb R)$ at exactly one real place of $F$ (depending on the exact setting you are considering). This choice of $\pi$ corresponds in the usual way to a choice of an abelian variety in the Jacobian of the Shimura curve $X_{U}$ attached to $\mathbf{G}$, so the statement you quoted depends indeed of a choice of $A$, if you want to see it that way. You can formulate a version for the full Jacobian of $X_U$ (so before cutting out the part on which the Hecke operators act they way they have to). I'm not sure I understand what you mean by a statement "directly on $X_U$" itself, but as ABCDveve said, in that respect, there is no difference between the work of Yuan-Zhang-Zhang and the original work of Gross-Zagier.