Let $\pi$ be a global automorphic representation of some reductive group over a number field, and let $L(\pi,s)$ denote its L-function. Assume $L(\pi,s)$ extends meromorphically to the complex plane and satisfies a functional equation of the form
$$
L(\pi,s)= \varepsilon(\pi,s) L(\pi^\star,1-s),
$$
where $\pi^\star$ denotes the contragredient dual of $\pi$.
Assume $L(\pi,1/2)=0$ and $L'(\pi,1/2)\ne 0$.
Question 1:
Under what circumstances do we expect the existence of an algebraic null-homologous cycle $D$ on some variety $V$ for which its height $h(D)$ is meaningful and well-defined, and the equality (up to a non-zero, well-understood, fudge factor)
$$
L'(\pi,1/2) \overset{\cdot}{=} h(D)
$$
holds?
Question 2: Assume the answer to the first question is expected to be "yes" for a given $\pi$, and assume we are also given a good, conjectural, candidate for $D$. Is it possible to axiomatize what one needs to show about the L-function $L(\pi,s)$, the height pairing $h$ and the cycle $D$ in order to prove the above Gross-Zagier formula?
My feeling is that the answer to Q1 should be yes at least when $\pi$ is self-dual, that is to say, $\pi \simeq \pi^\star$. But I ignore do not know whether such a formula is to be expected also for non self-dual $\pi$.
Let me also clarify that I am not asking about the difficulties of constructing a suitable candidate for $D$. Not because it is an uninteresting question, but rather to focus the discussion. In the first question I just ask for whether there exists a cycle satisfying the Gross-Zagier formula, but I am not asking who $D$ is. In the second question I am assuming $D$ is given, and I am asking what properties it should satisfy, but I again don't care who $D$ is.
Example 1: Let me explain the basic scenario I have in mind. Let $E/\mathbb{Q}$ be an elliptic curve and $K$ an imaginary quadratic field. If the pair $(E,K)$ satisfies the Heegner hypothesis, then the order of vanishing of the (self-dual) L-function $L(E/K,s)$ at its central critical value $s=1$ is odd, and Gross-Zagier proved that there exists a certain (Heegner) point $P_K\in E(K)$ such that
$$
L'(E/K,1) \overset{\cdot}{=} h(P_K).
$$
Here $P_K$ plays the role of $D$ in the general question. And we are evaluating the L-function at $s=1$ instead of $s=1/2$ just because we re-normalized it so that the functonal equation relates the values at $s$ and $2-s$.
And let me explain now some examples in which I do not know the answers.
Example 2: Let $E/\mathbb{Q}$ be an elliptic curve of prime conductor $p$ and $K$ a real quadratic field in which $p$ remains inert. Then the order of vanishing of the (self-dual) L-function $L(E/K,s)$ at its central critical value $s=1$ is odd. Henri Darmon has constructed a point $P_K\in E(K_p)$, rational over the completion of $K$ at $p$, which he conjectures to be actually rational over $K$. I am not asking how to prove this statement here, but rather: assume as a black box that $P_K$ indeed lies in $E(K)$. What one would need to know about $L(E/K,s)$ and $P_K$ in order to prove that $L'(E/K,1) \overset{\cdot}{=} h(P_K)$?
Example 3: Let $E/\mathbb{Q}$ be an elliptic curve. Let $\chi$ be a Dirichlet character and $\Q_\chi$ \mathbb{Q}_\chi$ be the abelian extension of $\Q$ \mathbb{Q}$ cut out by $\chi$. Assume $L(E,\chi,1)=0$ and $L'(E,\chi,1) \ne 0$. The conjecture of Birch and Swinneton-Dyer predicts the existence of a non-zero point $P_\chi \in E(\Q_\chi)\otimes E(\mathbb{Q}_\chi)\otimes \C$ mathbb{C}$ such that $P_\chi^\sigma = \chi(\sigma) P_\chi$ for all $\sigma \in \Gal(\Q_\chi/\Q)$. operatorname{Gal}(\mathbb{Q}\chi/\mathbb{Q})$. Do we expect the equality $L'(E,\chi,1) \overset{\cdot}{=} h(P_\chi)$ to hold up to some conceptually well-understood fudge factor? (Note that if we assume both sides to be non-zero, the formula obviously holds by setting the fudge factor to be $L'(E,\chi,1)/h(P_\chi)$!) L'(E,\chi,1)/h(P\chi)$!)
Example 4: Let $f\in S_2(N,\chi)$ be a (cuspidal, normalized) newform of weight $2$, level $N$ and nebentype character $\chi$. Then the field $\mathbb{Q}_f$ generated by the fourier coefficents of $f$ is a finite extension of $\mathbb{Q}$, say of degree $d$. The Eichler-Shimura construction yields an abelian variety $A_f/\mathbb{Q}$.
On the geometric side, we again have a natural construction of Heegner points: $A$ is a quotient of the jacobian $J_1(N)$ of $X_1(N)$. Given an imaginary quadratic field $K$, the theory of complex multiplication allows us to construct Heegner points $P$ on $X_1(N)$ which are rational over a suitable abelian extension $H/K$. This has been extensively studied for $X_0(N)$, in which case $H$ is a ring class field. But is also well-known for $X_1(N)$, where $H$ is no longer anticyclotomic; it contains for instance the abelian extension of $\mathbb{Q}$ cut out by $\chi$.
In any case, one can construct a Heegner point $P_K\in A(K)$ by tracing down $P$ from $H$ to $K$. And if $\psi$ is a character of $\mathrm{Gal}(H/K)$, one can also define
$$
P_\psi = \sum_{\tau\in \mathrm{Gal}(H/K)} \psi^{-1}(\tau)P^\tau \in E(H)\otimes \mathbb{C},
$$
which lies in the $\chi$-eigenpart of $E(H)\otimes \mathbb{C}$.
On the L-function side, $L(A/\mathbb{Q},s)$ factors as
$$
L(A/\mathbb{Q},s) = \prod L(f^\sigma,s)
$$
where $\sigma$ ranges over the $d$ different embeddings of $\mathbb{Q}_g$ into $\mathbb{C}$.
While $L(A/\mathbb{Q},s)$ is self-dual, each of the individual factors $L(f^\sigma,s)$ is self-dual if and only if $\chi$ is the trivial character. If $f^\star$ denotes the modular form obtained from $f$ by complex conjugating its fourier coefficients, then the functional equation of $L(f,s)$ relates it to $L(f^*,2-s)$.
A similar discussion holds for the base change of $A$ to $H$. The L-function of $A\times H$ is self-dual, but it factors as the product of L-series of the type $L(f/K,\psi,s)$ where $\psi$ ranges over the characters of $\mathrm{Gal}(H/K)$. Each of the individual L-functions are not always self-dual (regarding $\chi$ and $\psi$ adelically over $\mathbb{Q}$ and $K$ respectively, $L(f/K,\psi,s)$ is self-dual if and only if the restriction of $\psi$ to the ideles of $\mathbb{Q}$ is the inverse of $\chi$.)
Gross-Zagier formulas are proved in the self-dual setting by Zhang and his collaborators, and also by Howard. And Olivier reminded us that such a formula is not to be expected if we insist to use the point $P_\psi$. So the question is: for arbitrary pairs $(\chi,\psi)$, does there exist a point $P\in (E(H)\otimes \mathbb{C})^{\psi}$ for which $L'(f/K,\psi,1)\overset{\cdot}{=} h(P)$ up to a well-understood non-zero fudge factor?
