The Gross—Zagier Formula, 40 Years Later

This lecture traces two mathematical developments that evolved largely independently over nearly two thousand years. One begins with Euclid's parametrization of right triangles and culminates in the arithmetic of elliptic curves and Heegner points. The other starts with Euler's and Dirichlet's work on prime numbers and leads to the theory of L-functions and their special values. The Gross–Zagier formula reveals a remarkable connection between these two worlds by expressing the arithmetic height of a Heegner point as the central derivative of an L-function. I will also describe some of its arithmetic applications and its far-reaching generalizations to Shimura curves and Shimura varieties. The story illustrates a recurring theme in mathematics: ideas developed in distant areas, often centuries apart, can ultimately converge to illuminate the same arithmetic phenomena.

In the 1980s, Gross and Zagier studied the norms of the differences between singular moduli. Analogous questions on Shimura curves allow for purely p-adic investigations through their p-adic uniformisations. In earlier work, infinitesimal deformations of Galois representations were used to compute the norms of the differences between these singular moduli without relying on their connection to the geometry of Shimura curves. We explain how more intricate deformations can be used to obtain results beyond the norm, supporting purely p-adic methods to prove the algebraicity of conjecturally algebraic p-adic quantities. This is joint work with Yingkun Li.

A year after the appearance of everyone’s favourite formula, Bernadette Perrin-Riou published an analogous one in p-adic coefficients. I will survey some of the developments of this formula since then, and conclude with some new applications to the p-adic Beilinson-Bloch-Kato conjecture.

The scare quotes around the definite article refer to this talk’s limitation to a certain orthodoxy: further p-adic metamorphoses of the formula will appear in Henri Darmon’s talk.

The protean nature of the celebrated formula whose 40th anniversary serves as the pretext for this conference is nowhere more apparent than in the p-adic variants whose most basic prototypes were described in Daniel Disegni’s lecture. The availability of a plethora of p-adic Gross-Zagier formulae can be attributed in part to the greater richness of the theme of variation in the p-adic theory of modular forms. I will describe a conjecture relating certain first order p-adic deformations of quaternary theta series to the p-adic logarithms of global points in jacobians of modular curves, and discuss the relation of this conjecture to the theory of Heegner and Stark-Heegner points on modular elliptic curves over Q.

Modular forms on a group G with rank at least 2 have Fourier coefficients that satisfy various symmetries, which come from invariance of the modular form with respect to a parabolic subgroup of G. "Automatic convergence" refers to the idea that if one takes a collection of complex numbers that satisfy these symmetries--with no a priori assumption on their growth--then these numbers are the Fourier coefficients of a modular form on G. I will explain some automatic convergence theorems and applications.

The arithmetic Siegel-Weil formula is an identity between arithmetic intersection numbers on Shimura varieties and derivatives of Siegel Eisenstein series. It plays a central role in Kudla's program on establishing a higher dimensional Gross-Zagier type formula. We will discuss its origin and highlight recent progress.

A celebrated result of Gross and Zagier gives an explicit formula for the prime factorization of the norm of the difference of two singular moduli. We use a generalization of this result to CM values of Borcherds products to study genus two curves whose Jacobian has complex multiplication by a biquadratic CM field. In this case the Jacobian is the product of two elliptic curves with complex multiplication by orders in the same imaginary quadratic field. We derive explicit bounds for the primes of stable bad reduction of the curve in terms of the discriminants of the CM orders. Moreover, we show that there always exist primes of stable bad reduction. To establish these results, we provide an explicit description of principally polarized abelian surfaces with CM by a biquadratic field in terms of small CM points on an orthogonal Shimura variety. We use it to prove an arithmetic intersection formula between such CM points with Humbert surfaces. In addition, we make use of the arithmetic properties of higher Green functions. This is joint work with Tonghai Yang and Peng Yu.

In a celebrated paper of 1985, Waldspurger proved a formula relating the squares of toric periods for GL(2) to central values of base-change L-functions. Gross soon recognized this identity to be a rank 0 analogue of his joint work with Zagier on heights of Heegner points. This talk will discuss higher rank generalizations of the Waldspurger's formula, namely the Gan-Gross-Prasad conjectures and their refinements by Ichino-Ikeda, N. Harris and Liu. It will provide a survey of the progress made on these conjectures through the lens of relative trace formulas from the past 15 years as well as some prospects regarding the remaining open cases. Time permitting, I might also sketch an application of the GGP conjecture to the construction of p-adic L-functions, which is joint work with X. Dimitrakopoulou.

Rigid meromorphic cocycles are cocycles for certain p-arithmetic groups acting on p-adic symmetric spaces. Their values at “special points” are conjectured to belong to class fields of some suitable global fields. In this talk, we discuss certain modular generating series for values of rigid cocycles on the Drinfeld p-adic upper half-plane in a framework involving biquadratic extensions inspired by Gross and Zagier. This is joint work with Judith Ludwig, Isabella Negrini, Sandra Rozensztajn and Hanneke Wiersema.

In this talk, I will present joint work with A. Burungale constructing new Euler systems for elliptic modular forms over an imaginary quadratic field, with many arithmetic applications. In a similar fashion to Kato’s celebrated construction, our method imposes no local condition on the modular form at p. The key new ingredient is a change in geometric perspective: we construct p-adic étale cohomology classes on modular curves arising from theta series which serve as a crucial input to construct the Euler system classes.

We prove a rationality theorem for certain linear combinations of traces of cycle integrals of meromorphic Hilbert modular forms. These forms are meromorphic analogues of the Hilbert cusp forms $\omega_m(z_1,z_2)$ introduced by Zagier in his study of the Doi--Naganuma lift.

Our main result gives an explicit formula for these cycle integrals in terms of the Fourier coefficients of harmonic Maass forms. The proof combines the construction of locally harmonic Hilbert--Maass forms with a new regularized theta lift related to the Doi--Naganuma lift and the development of a $\xi$-operator for Hilbert modular surfaces. We conclude by discussing possible extensions of these ideas to orthogonal modular forms.

In the past 40 years, the Gross-Zagier formula has been vastly generalized. However, (essentially) except the work of S. Zhang on elliptic modular forms of even weights under the Heegner condition, all restrict to minimal weights. In this talk, we will explain our recent discovery on the arithmetic meaning of central L-derivatives of cohomological automorphic forms of general weights over CM fields.

I will talk about how my own journey as a mathematician has been shaped by the Gross-Zagier formula, which I encountered at the 2017 Arbeitsgemeinschaft organized by Zhiwei Yun and Wei Zhang. This led to an inspiring collaboration which had a profound impact on my career. I will survey some of the ensuing story, including also joint works with Ben Howard and Mikayel Mkrtchyan. Finally, I’ll try to get some research done in the next 3 weeks so that I can mention some more recent developments.

In this talk, I will discuss Iwasawa theory and its applications to the Birch and Swinnerton-Dyer and Bloch–Kato conjectures. In particular, I will explain how one can deduce from Iwasawa main conjecture(s) for elliptic curves a p-converse to the Gross–Zagier–Kolyvagin theorem, as well as results on Heegner points in situations where the analytic rank is greater than one. I will present some earlier results in the case of Eisenstein primes, together with partial generalisations to the setting of Rankin–Selberg convolutions of cusp forms.

Abstract coming soon.

We consider complex multiplication cycles on integral models of unitary Shimura varieties. These are arithmetic 1-cycles that come with natural Green currents at the archimedean place. Using the trace pairing, they assemble into a generating series which we expect to be modular in the sense of the Kudla program. As a first step in this direction, we consider the case of Shimura curves and intersection against special divisors. The idea is to use the arithmetic relative trace formula approach of W. Zhang to compare the resulting intersection numbers with derivatives of orbital integrals. The main new difficulties arise at the archimedean place; here, one needs to construct Schwartz functions whose orbital integral derivatives capture the star products of Green currents. I will present our solution to this problem in the curve case, which is joint work with S. Sankaran and T. Yang.

In this talk, I will discuss two conjectural families of relative trace formula comparisons motivated by relative Langlands duality, and present some partial results toward these comparisons. I will also explain how these two families of comparisons reduce the study of period integrals on general hyperspherical Hamiltonian spaces to the strongly tempered case. Part of this work is joint with Zhengyu Mao and Lei Zhang.

Colmez's conjecture predicts that Faltings heights of CM abelian varieties appear in subleading terms of certain Artin L-functions. An averaged version is known by the work of Andreatta--Goren--Howard--Madapusi and Yuan--Zhang.

We propose a generalized problem relating diagonal cycles on $(n - 1)$-dimensional unitary Shimura varieties and adjoint L-functions of cohomological tempered cuspidal automorphic representations of $U(n)$. The case $n = 1$ is (a variant of) the averaged Colmez conjecture. I will explain some origins of our conjecture, discuss our predictions in the general case, and report on our results in the case of $n = 2$. Our strategy is a comparison of relative trace formulas (RTF): one side is an arithmetic analogue of an Eichler--Selberg trace formula, and the other is the derivative of a twisted Jacquet--Zagier RTF.

This is joint work in progress with Weixiao Lu and Wei Zhang.

The higher Gross–Zagier formula, introduced by Yun and Zhang, is the function field analogue of the classical Gross–Zagier formula. It relates intersection numbers of special cycles on PGL₂ Shtukas to higher central derivatives of base change L-functions. It may be viewed as a cycle-theoretic enhancement of the classical paradigm that periods of automorphic forms are related to special values of L-functions. The relative Langlands duality, proposed by Ben-Zvi, Sakellaridis, and Venkatesh, provides a systematic and conceptual framework for understanding the relationship between periods and L-functions. In this talk, I will explain how this framework can be extended to relate special cycles to higher derivatives of L-functions. This perspective not only provides conceptual explanations and new proofs of some function field analogues of Gross–Zagier type formulas, but also suggests new phenomena and conjectures concerning derivatives of L-functions away from the central point.

The Ceresa class is a canonical, cohomologically trivial algebraic cycle on the Jacobian of a curve. Modified diagonal classes are closely related, and almost equivalent, algebraic cycles defined on the triple product of the curve with itself. These canonical cycles are known to be non-torsion in the Chow group for sufficiently general curves, but the proof is not explicit. It is therefore interesting to ask for which curves these cycles are non-torsion.

This talk will report on joint work in progress with Ari Shnidman, in which we calculate the ramification of an l-adic Abel-Jacobi image of a modified diagonal class for any semistable curve over a finite extension of Z_p. The calculation is in terms of purely combinatorial data on the special fiber. As an application, we are able to give new and explicit conditions under which a curve over a number field has nonzero or nontorsion Ceresa class.