# “Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the derivative of the $L$-function of an elliptic curve $E$ to the canonical height of a special rational point on $E$.

Let me describe the Gross-Zagier formula in the simplest case.

Setup. Let $E/\mathbf Q$ be an elliptic curve. According to the modularity theorem, there exists a finite $\mathbf Q$-morphism $p :X_0(N) \to E$ mapping the cusp $\infty$ to the origin of $E$, where $N$ is the conductor of $E$. Let $K\subseteq \mathbf C$ be an imaginary quadratic field, other than $\mathbf Q[i]$ or $\mathbf Q[\sqrt{-3}]$, in which the prime factors of $N$ split. Then we can choose an ideal $\mathcal J$ in $\mathcal O_K$, such that $\mathcal O_K/J \simeq \mathbf Z/N\mathbf Z$. Viewing $\mathcal J$ as a lattice in $\mathbf C$, we can form the elliptic curves $C_\mathcal J = \mathbf C/\mathcal J$ and $C_K = \mathbf C/\mathcal O_K$. While they are a priori elliptic curves over $\mathbf C$, we know from the theory of complex multiplication that they are actually defined over the Hilbert class field $H$ of $K$. Anyways, the inclusion $\mathcal J \subseteq \mathcal O_K$ induces by passage to the quotient an isogeny $C_\mathcal J \to C_K$ of degree $N$, which is precisely the kind of gadget that $X_0(N)$ parametrizes as a moduli space. Thus we get a point in $X_0(N)(H)$. We can take the image of this point via the modular parametrization $p$ to get a point $y_J \in E(H)$. Taking the sum of its Galois conjugates down to $K$ we get a point $y_K \in E(K)$, which, it turns out, doesn't depend on the choice of $\mathcal J$, up to sign and up to torsion. This means that its Néron-Tate height $\hat{h}(y_K)$ is a well-defined non-negative real number, which, by the non-degeneracy of the height pairing, is zero if and only if $y_K$ is a torsion point on $E$.

With this setup, Gross and Zagier proved:

Theorem: $$L'(E/K, 1) = \hat{h}(y_K) \frac{\iint_{E(\mathbf C)} \omega \wedge i\omega}{\sqrt D},$$ where $\omega$ is the invariant differential on $E$ (suitably normalized), and $D$ is the discriminant of $K$.

Since the factor $\frac{\iint_{E(\mathbf C)} \omega \wedge i\omega}{\sqrt D}$ is never zero, we have:

Corollary: Suppose that $L(E/K, s)$ has a simple zero at $s=1$. Then $y_K$ is a point of infinite order in $E(K)$, and in particular, $\text{Rank}_\mathbf{Z}E(K)>0$.

Thus, we have an example of a situation where the vanishing of $L(E/K, s)$ at $s=1$ implies the existence of an (explicit!) point of infinite order in $E(K)$ -- a behavior which is, of course, predicted by the BSD conjecture (minus the "explicit" part, which comes as a surprise).

If we believe BSD, it is natural to make the following conjecture:

Conjecture (Gross-Zagier): Suppose that $L'(E/K, 1)$ is nonzero, or equivalently, that $\hat{h}(y_K)$ is nonzero, or still equivalently, that $y_K$ is non-torsion. Then $E(K)$ has rank exactly one, and $\left<y_K\right>$ has finite index in it.

This conjecture was proven by Kolyvagin.

In any case, since $\hat{h}(y_K) = \left<y_K, y_K\right>$ where $\left<,\right>$ is the Néron-Tate height pairing on $E$, we can rewrite the Gross-Zagier formula as

$$L'(E/K, 1) = \left<y_K, y_K\right> \times C$$ where $C$ is the nonzero constant of the formula.

Let me now tell you about something completely different, namely the Minakshisundaram-Pleijel zeta function. (I don't know enough about this, so please forgive any inaccuracies.) Given a compact Riemannian manifold $M$, one has the Laplace-Beltrami operator $\Delta$ on $M$ which generalizes the familiar Laplacian on $\mathbf R^n$. The spectrum of this operator is an important invariant of $M$, and it is encoded in the M-P zeta function $$\zeta(\Delta, s) = \sum_{n=1}^\infty |\lambda_n|^{-s}.$$ It admits a meromorphic continuation to the whole plane, and is holomorphic at $s=0$. Its Taylor coefficients at $0$ around contain geometric data about $M$. Let us specialize to the case where $M$ is a surface; then $$\zeta'(\Delta, 0) = \frac{1}{12}\int_M K dA$$ where $K$ is the Gaussian curvature. According to the Gauss-Bonnet theorem, this is essentially the Euler characteristic, so: $$\zeta'(\Delta, 0) = \chi(M) \times C, \qquad C\neq 0.$$ But there is a more suggestive way of writing the Euler characteristic. The diagonal $D \subseteq M \times M$ determines a cohomology class on $M \times M$; moreover it lies in the piece of the cohomology of $M\times M$ which is self-dual with respect to Poincaré duality, so the expression $\left<D, D\right>$ makes sense (it is the "self-intersection" number of the diagonal). As is well-known, $\left<D, D\right>$ is precisely the Euler characteristic of $M$. So, we can write $$\zeta'(\Delta , 0) = \left<D, D\right> \times C.$$

Corollary: If $\zeta'(\Delta, 0) \neq 0$, then $D$ is a nontrivial cohomology class on $M\times M$.

Thus, we have a completely different example of a situation where the non-vanishing of the derivative of a zeta function implies the existence of an explicit non-trivial cohomology class which "accounts" for the non-vanishing (in the first case, viewing a rational point on $E$ as a degree $0$ Galois cohomology class).

As René points out, $D$ is always nontrivial, for trivial reasons. What is interesting to me is not so much the nontriviality of $D$, but rather that one can deduce this nontriviality from $\zeta'(\Delta, 0)\neq 0$.

Another important difference that I should point out betweeen the two situations is that $\zeta'(\Delta, 0)\neq 0$ does not imply $\zeta(\Delta, 0)= 0$, because there is no analogue of BSD. In fact, $\zeta(\Delta, 0)$ is essentially $\mathrm{vol}(M)$.

Nevertheless, I am still curious.

Questions:

1. Is the similarity between the two formulas, and between their Corollaries, coincidental? (This question probably doesn't have a precise answer, but I feel that it's worth asking.) And, assuming that the answer to this question is not completely disappointing...

2. Have things like this been pointed out before? Are there more examples of formulas outside of number theory which bear resemblance to the Gross-Zagier formula?

• Are there examples where the class of $D \subset M \times M$ in the cohomology of $M \times M$ is trivial? (Since it has non-zero intersection number with $M \times \{ \operatorname{pt} \}$, I am tempted to think the answer can't be yes.) – RP_ Mar 2 '14 at 16:59
• @René Good point. In any case, it is the resemblance between the two formulas which I am really curious about. – Bruno Joyal Mar 2 '14 at 19:14
• I knew Zagier pretty well, as he was at UMCP when I was a graduate student in number theory. Then I was Kolyvagin's postdoc at Hopkins for a year. So you'd think I'd have a clue about this. Have you tried contacting Zagier? He's pretty approachable and so is Gross for that matter, or at least he used to be. Kolyvagin, probably less so. – Gregory Grant Mar 9 '15 at 2:03

I think you know all this, but nevertheless...

These two formulas are arguably incarnations of the general philosophy of Arakelov geometry according to which derivatives of zeta functions (regularized determinants) compute (or are computed by, depending on your perspective) arithmetic intersection numbers. See for instance the arithmetic Riemann-Roch theorem of Deligne, Gillet-Soulé and Bismut.

Both the Gross-Zagier formula and its momentous generalizations in the Kudla's program (expressing intersection numbers on Shimura varieties in terms of coefficients of automorphic forms) and Arakelov's Riemann Roch results rely crucially in their proofs on dynamical or deformation arguments, meaning that the computation is carried over on a favorable locus of the variety (or for favorable cycles) then moved to the actual locus or cycles of interests. In the context of Arakelov geometry, this is possible thanks to the study of the variation of the determinant of Laplacian operators through deformation by J-M.Bismut and G.Lebeau (see for instance here). It is tempting to speculate that Kudla's program type result (and the Gross-Zagier formula) could likewise follow from a study of the $p$-adic variation under deformation of the determinant of étale cohomology.

At present, this is certainly backwards, in the sense that we usually prove results about the determinant of the cohomology using intersection of cycles on Shimura varieties and not the other way around, but it remains true that $p$-adic variants of the Gross-Zagier formula are usually easier to prove, and I don't think this is at all coincidental.

For a concrete instance, though I don't think this is written anywhere, the proof of the Iwasawa Main Conjecture in dihedral $\mathbb Z^{d}$-extensions of CM extensions of totally real fields for the motive attached to a quaternionic automorphic form of parallel weight $(2,\cdots,2)$ over a Shimura curve (which is accessible if not known by results of B.Howard and X.Wan) combined with a global divisibility in $p$-adic families of such automorphic forms (by someone) and the appropriate control theorem (of T.Ochiai, J.Saha and maybe someone else) is probably enough to prove the non-triviality of many $p$-adic intersection pairings between higher weights cycles (on the same Shimura varieties).

Roughly speaking, you're really looking at quite similar objects if you take the following perspective:

a. The Gross-Zagier formula computes the first derivative of $L(E,s)$. By modularity, one may associate to the elliptic curve a modular form $f$ such that $L(E,s)=L(f,s)$. The resulting formula is due to Waldspurger, was reinterpreted by Jacquet as a `relative trace formula'. In this setting one has the Petersson norm of $f$ instead of the height pairing, and similar the other constants have parallel interpretations.

b. The M-P formula that you mention is Selberg's Trace Formula, with a special choice of test function. When you take the derivative with respect to the $s$ parameter, it is again a trace formula (ignoring possible analytic difficulties). For a compact Riemann surface, indeed the main term is the volume factor, but if you pass to a noncompact setting other terms appear. Jacquet's Relative Trace Formula is so-called in comparison to Selberg's. Indeed, the $L^2$ spectrum of your manifold is built out of modular forms, or more generally, automorphic forms.

• Could you expand on "The $L^2$ spectrum of your manifold is built out of modular forms"? – TreFox Apr 17 '18 at 14:19
• Essentially, the $L^2$ spectrum is described by the action of the Laplace operator, and modular forms turn out to be eigenfunctions, if we are looking at the upper half plane $\mathbb H$ modulo some discrete cocompact lattice $\Gamma$. Generalisations of this sort lead to automorphic forms, which decompose the $L^2$ spectrum of some homogeneous space $G/K$, and instead of the Laplacian one considers the right regular representation. – TA Wong Apr 20 '18 at 9:26