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Example 3: Let $E/\mathbb{Q}$ be an elliptic curve. Let $\chi$ be a Dirichlet character and $\mathbb{Q}_\chi$ be the abelian extension of $\mathbb{Q}$ 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 $$P_{\chi} P_{\chi} \in E(\mathbb{Q}{\chi})\otimes \mathbb{C}, \quad P{\chi}^{\sigma} = \chi(\sigma) P_{\chi}$$ for all $\sigma E(\mathbb{Q}_{\chi})\otimes \mathbb{C}$ lying in \operatorname{Gal}(\mathbb{Q}{\chi}/\mathbb{Q})$. the$\chi$-eigenpart of the Modell-Weil group under the Galois action. Do we expect the equality$L'(E,\chi,1) \overset{\cdot}{=} h(P\chi)$h(P_\chi)$ 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 $L'(E,\chi,1)/h(P_\chi)$, and this is not what one would call a well-understood fudge factor!)

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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 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 $\mathbb{Q}_\chi$ be the abelian extension of $\mathbb{Q}$ 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 $$P_{\chi} P_{\chi} \in E(\mathbb{Q}{\chi})\otimes \mathbb{C}, \quad P{\chi}^{\sigma} = \chi(\sigma) P_{\chi}$$ P_{\chi}$$for all \sigma \in \operatorname{Gal}(\mathbb{Q}{\chi}/\mathbb{Q}). Do we expect the equality L'(E,\chi,1) \overset{\cdot}{=} h(P\chi) to hold up to some conceptually well-understood fudge factor? (Note that if we assume both sides to be non-zero, the formula obviously holds by setting the fudge factor to be L'(E,\chi,1)/h(P_\chi), and this is not what one would call a well-understood fudge factor!) Example 4: Let f\in S_2(N,\chi) be a (cuspidal, normalized) newform of weight 2, level N and nebentype character \chi. Then the field \mathbb{Q}_f generated by the fourier coefficents of f is a finite extension of \mathbb{Q}, say of degree d. The Eichler-Shimura construction yields an abelian variety A_f/\mathbb{Q}. On the geometric side, we again have a natural construction of Heegner points: A is a quotient of the jacobian J_1(N) of X_1(N). Given an imaginary quadratic field K, the theory of complex multiplication allows us to construct Heegner points P on X_1(N) which are rational over a suitable abelian extension H/K. This has been extensively studied for X_0(N), in which case H is a ring class field. But is also well-known for X_1(N), where H is no longer anticyclotomic; it contains for instance the abelian extension of \mathbb{Q} cut out by \chi. In any case, one can construct a Heegner point P_K\in A(K) by tracing down P from H to K. And if \psi is a character of \mathrm{Gal}(H/K), one can also define$$ 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) $$where \sigma ranges over the d different embeddings of \mathbb{Q}_g into \mathbb{C}. While L(A/\mathbb{Q},s) is self-dual, each of the individual factors L(f^\sigma,s) is self-dual if and only if \chi is the trivial character. If f^\star denotes the modular form obtained from f by complex conjugating its fourier coefficients, then the functional equation of L(f,s) relates it to L(f^*,2-s). A similar discussion holds for the base change of A to H. The L-function of A\times H is self-dual, but it factors as the product of L-series of the type L(f/K,\psi,s) where \psi ranges over the characters of \mathrm{Gal}(H/K). Each of the individual L-functions are not always self-dual (regarding \chi and \psi adelically over \mathbb{Q} and K respectively, L(f/K,\psi,s) is self-dual if and only if the restriction of \psi to the ideles of \mathbb{Q} is the inverse of \chi.) Gross-Zagier formulas are proved in the self-dual setting by Zhang and his collaborators, and also by Howard. And Olivier reminded us that such a formula is not to be expected if we insist to use the point P_\psi. So the question is: for arbitrary pairs (\chi,\psi), does there exist a point P\in (E(H)\otimes \mathbb{C})^{\psi} for which L'(f/K,\psi,1)\overset{\cdot}{=} h(P) up to a well-understood non-zero fudge factor? 4 added 68 characters in body; added 13 characters in body P_\chi P_{\chi} \in E(\mathbb{Q}_\chi)\otimes E(\mathbb{Q}{\chi})\otimes \mathbb{C} such that P_\chi^\sigma mathbb{C}, \quad P{\chi}^{\sigma} = \chi(\sigma) P_\chi P_{\chi}$$ for all $\sigma \in \operatorname{Gal}(\mathbb{Q}\chi/\mathbb{Q})$. {\chi}/\mathbb{Q})$. Do we expect the equality$L'(E,\chi,1) \overset{\cdot}{=} h(P_\chi)$h(P\chi)$ 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)\$, and this is not what one would call a well-understood fudge factor!)

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2 I just edited my question, to focus it on the aspects I'd like to learn about.; edited title
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