Zhang's generalization of Gross-Zagier formula

Question

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. ]

Answer 1

Could you explain me what are concretely the formulas in those cases?

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.

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

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.

Answer 2

Since no one else has answered, I will make some naive comments.

On page 82, the Theorem is rewritten with some explanation. Maybe all of Section 3.2 is useful for understanding.

In any case, for question #2, do you consider ordinary Gross-Zagier to hold on $X_0(N)$, or on the parametrized curve $E$? The answer should be the same here, as the Shimura curve plays the role of $X_0(N)$ and the abelian variety that of $E$.

Namely, in the book notation, $P$ is a CM point on $X$, and "evaluating" $P$ at a morphism $f_1:X\rightarrow A$ gives a point on the abelian variety $A$, specifically by taking a sum of $A$-points (defined in the Hilbert field) over the class group for the quadratic CM extension. So in the given formula, the height is calculated on the abelian variety $A$.

I must admit I am unable to go from the above Theorem 1.2 (even in the case of trivial characters) to ordinary Gross-Zagier, which already seems a difficult exercise to the uninitiated. The $\alpha$ somehow measures the two parametrizations, the $L'(1/2,\pi_A,\chi)$ is the $L'(E,1)$ of an elliptic curve, the $L(1,\eta)^2$ is the $L$-function of the CM quadratic extension, and the $L(1,\pi_A,ad)$ is the symmetric square $L$-function of the elliptic curve evaluated at the edge of the critical strip, and so relates to the degree of the modular parametrization. The $\alpha$ remains mysterious to me, which is only defined component-wise on the preceding page 8.