Axiomatizing Gross-Zagier formulae

Question

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 $P_{\chi} \in E(\mathbb{Q}_{\chi})\otimes \mathbb{C}$ lying in the $\chi$-eigenpart of the Modell-Weil group under the Galois action.

Do we expect the equality $L'(E,\chi,1) \overset{\cdot}{=} h(P_\chi)$ 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?

Answer 1

UPDATE: I have updated this answer slightly to take into account Victor's remark.

I think that the precise questions being asked admit a straightforward answer. At the moment, no such formula is known and the proofs of Gross-Zagier, Waldspurger, Zhang et al. and Howard all absolutely and crucially require the hypothesis of self-duality. The reason for this is that the representation-theoretic part of the proof requires an understanding of test-vectors, as in the works of Tunell and Saito or as in the conjecture of Gross-Prasad. This is explained for instance in the article of Gross entitled Heegner points and representation theory as well as in Non-triviality of CM points in ring class field towers. Aflalo, Esther and Nekovář, Jan. Israel J. Math. 175 (2010), 225--284 (in which the formal setting is explored).

As for whether similar formula could hold, I am not too optimistic. A Gross-Zagier formula should involve the $\psi$-eigenpart of the action of $\textrm{Gal}(H/\mathbb Q)$ on the projection of a CM point on the $\pi(f)$-component of the Jacobian of a Shimura curve. However, comparing the Galois action on CM points with the adelic action on the Jacobian, we see that this $\chi$-eigenpart can be non-trivial only when the restriction of $\psi$ to $\mathbb A_{\mathbb Q}$ is equal to $\chi$, or equivalently only in the self-dual case. This is proved for instance in Cornut, Christophe; Vatsal, Vinayak Nontriviality of Rankin-Selberg L-functions and CM points. Lemma 4.6.

Note also that this is what we should expect from the conjectures on special values of $L$-functions applied to $L(f/K,\psi,s)$ when $f$ is not self-dual. In that case, the conjecture implies that $L'(f/K,\psi,s)$ should be related to cohomology classes in the dual of the motive of $f$. Only in the self-dual case does these collapse in a formula involving the height of a point. Finally, in the situation you describe, even though $L(f,\psi,s)$ might vanish at 1, it is is expected that it doesn't generically (say in a relevant $\mathbb Z_{p}$-extension), so the conjectural relation between $L(f/K,\psi,s)$ (or its Selmer group) and putative point could hold only "locally at the specialization corresponding to $f$" in a $p$-adic family of automorphic representation containing $\pi(f)\otimes\psi$. All the arguments that I know relating these objects would then simply vanish.

Now an argument from ignorance is not a very good one, and I would very much like to be proven false, if only to learn something. Hidden behind all this is the question of the link between Kato's Euler system and rational points on modular varieties. The link is mysterious to me, but David Loeffler and Sarah Zerbes have some ideas.

Answer 2

Here is a more detailed version of my comment above.

I will consider only the setting of your Example 3, namely $E/\mathbf{Q}$ is an elliptic curve and $\chi$ is a Dirichlet character such that $L(E \otimes \chi,1)=0$ and $L'(E \otimes \chi,1) \neq 0$.

Let $m \geq 1$ be an integer. The base change $L$-function $L(E \otimes \mathbf{Q}(\zeta_m),s)$ is the product of twisted $L$-functions $L(E \otimes \chi,s)$ where $\chi$ runs through the characters of conductor dividing $m$. The BSD conjecture for the base change $E \otimes \mathbf{Q}(\zeta_m)$ can be refined for each factor $L(E \otimes \chi,s)$. Roughly, the idea is that each arithmetical invariant appearing in the conjecture for $E \otimes \mathbf{Q}(\zeta_m)$ should factor in a way that reflects the decomposition of the motive $h^1(E \otimes \mathbf{Q}(\zeta_m)) = \bigoplus_{\chi} h^1(E \otimes \chi)$. The motive $M_\chi = h^1(E \otimes \chi)$ has dual $M_{\overline{\chi}}$, so it is self-dual only when $\chi$ is quadratic.

Assume $L(E \otimes \chi,s)$ vanishes at order one at $s=1$. The main invariant to consider is the discriminant of the Néron-Tate height pairing $\langle,\rangle$ on $E(\mathbf{Q}(\zeta_m)) \otimes \mathbf{R}$. We can extend this pairing to a positive definite hermitian form on $V=E(\mathbf{Q}(\zeta_m)) \otimes \mathbf{C}$. There is a decomposition of $V$ into isotypical components $V_\chi$ which are pairwise orthogonal with respect to the pairing. In this particular case we indeed expect $L'(E \otimes \chi,1) \sim \langle P_{\chi},P_{\chi} \rangle$ where $P_\chi$ is a generator of $V_\chi$. Usually $P_\chi$ takes the form $\sum_{\sigma} P^{\sigma} \otimes \chi(\sigma)$ for some $P \in E(\mathbf{Q}(\zeta_m))$.

In order to convince you this is reasonable, here is an example I computed using Magma. The rank $0$ elliptic curve $E=X_0(20):y^2 = x^3 + x^2 + 4x + 4$ acquires two independent points of infinite order over the cubic extension $\mathbf{Q}(a)$ with $a=2\operatorname{cos}(2\pi/9))$. Letting $P=(a+1,2a+3)$, we check numerically that $L'(E \otimes \chi,1) \sim \Omega_E \cdot \langle P_\chi,P_\chi \rangle$. The fudge factor appears to be an algebraic number of degree 6.

E:=EllipticCurve("20a1");
G:=FullDirichletGroup(9);
chi:=(G.1)^2; // Character of conductor 9 and order 6
LEchi:=TensorProduct(LSeries(E),LSeries(chi)); // L-Series L(E \otimes \chi,s)
Evaluate(LEchi,1);
//2.22329881577004394873515961159E-30
LEchi1:=Evaluate(LEchi,1:Derivative:=1);
LEchi1;
//2.78851510267155729197040153856 + 0.491690448714030907428058875920*$.1

Q<x>:=PolynomialRing(Rationals());
K<a>:=NumberField(x^3-3*x+1); // Cubic sufield of Q(zeta_9)
EK:=BaseChange(E,K);

G,_,map:=AutomorphismGroup(K); // Galois group of K
P:=[EK![(map(g))(a)+1,2*(map(g))(a)+3] : g in G]; // Set of conjugates of P=[a+1,2a+3] with a=2cos(2pi/9)
M:=HeightPairingMatrix(P);

C:=ComplexField();
PchiPchi:=C!M[1,1] + C!chi(2)*C!M[1,2] + C!chi(4)*C!M[1,3];

ratio:=LEchi1/(Periods(E)[1]*PchiPchi);
PowerRelation(ratio,6);
//27*x^6 - 9*x^3 + 1

I'm not sure about the situation for general abelian varieties $A_f/\mathbf{Q}$, but I guess some of the above might extend.

Of course, the real question seems to be whether or not these points $P_\chi$ are related in some way to Heegner points!