#### by Shou-Wu Zhang

For the last 27 years, I have been very fortunate to have Benedict Gross as a mentor. In this essay, I will describe how his papers, our letters, and our discussions have influenced my own development and career.

I first heard the name Benedict Gross in a lecture by Dorian Goldfeld at the Chinese Academy of Sciences in the summer of 1985, when I was a master’s degree student. The following year, Dorian suggested proving extensions of the Gross–Zagier formula [2] for my Ph.D. thesis problem at Columbia. I looked at the Gross–Zagier paper briefly, but I could not really understand much of its proof. Initially, I was unconvinced that their approach was the right direction because the main result only applied to modular elliptic curves; I had studied arithmetic geometry and heights under Szpiro and Faltings until Andrew Wiles unveiled his strategy for proving the modularity conjecture in 1993. Of course, I was convinced then that this was the right direction and a potential area where the theory of heights could be applied.

Unlike the papers in arithmetic geometry where generality is usually
the first priority, Gross and Zagier reduced their formula to the
miraculous equality __\( F_G=F_A \)__ of two modular forms for __\( \Gamma_0(N) \)__
of weight 2, associated to an ideal class character __\( \chi \)__ for an
imaginary quadratic ~~field ~~ with __\( K \)__,__\( F_G \)__ constructed from the
“geometry” of modular curves, and __\( F_A \)__ constructed from the
“analysis” of modular forms. This left a lot of room for
generalization, as one can replace __\( \Gamma_0(N) \)__ by __\( \Gamma_1(N) \)__
and allow __\( \chi \)__ be a general Hecke character. In fact, at the end of
their paper
[2],
there are two suggested generalizations: for
Shimura curves (when a Heegner condition fails) and for higher weight
modular forms. At the end of 1994, I wrote two emails to Dick Gross
asking about new progress towards these generalizations.

On January 3, 1995, he sent me a
letter
[4]
via mail answering all of my questions about
higher-weight
modular
forms. He said that
Brylinski’s
paper in *DMJ*
[e2],
about local heights of
Heegner
cycles
defined by
Deligne,
was the only work on this topic. So
what remained was to develop a theory of global heights. On
January 4, 1995, he sent me another
letter via email
[5]
answering all of my questions about Shimura curves. He made
a point that a Shimura curve over a totally real field is canonically
associated to an odd set of places where the curve has analytic
uniformizations by the work of
Shimura,
Doi–Naganuma,
and
Cerednik–Drinfeld.
These two letters were the beginning of his mentorship of my academic
life, in which he has given me important advice and warm encouragement
at various turning points of my research, always sharing with me his
new ideas and insights.

With so much encouragement from Dick’s letters, I immediately started to work on these two suggested generalizations for higher-weight modular forms and Shimura curves. At the end of 1995, I worked out the suggested Gross–Zagier formula for higher-weight modular forms [e3]. The modification that was needed there was a global definition of heights for Heegner cycles using Gillet–Soulé arithmetic intersection theory on Kuga–Sato varieties instead of local systems. In the summer of 1997, I also proved the suggested Gross–Zagier formula over Shimura curves [e4]. The modification that was needed there was to replace cusps by Hodge classes, which required the development of a multiplicity-one argument for derivations of Hecke operators. In these two papers, the new ideas were height computations that I had learned earlier in Arakelov theory. The analytic parts were completely copied from Gross and Zagier’s paper.

Following these two nice “warm-up exercises”, I thought about a
further extension of the Gross–Zagier formula formulated in Dick’s
paper
[1]
for general anticylotomic characters __\( \chi \)__. A new
modification is that one has to work on level
__\( N\cdot\operatorname{cond}(\operatorname{Ind}^{\mathbb{Q}}_K(\chi)) \)__ directly, where one can only
prove an “almost-equality” __\( F_G\approx F_A \)__ of two modular forms, or
some kind of “arithmetic fundamental lemma” in today’s language. To
deduce the precise identity from this almost-equality of modular
forms, one needs new
multiplicity-one arguments. After he heard about
my plan during his visit to Columbia, Dick sent me a
letter
[7]
(May 14, 1998)
with some papers, including his work with
Prasad
about test vectors.
With these test vectors, I could prove a Gross–Zagier formula in this
setting by developing
the so-called toric newform theory in the summer of
2001
[e5].
With the completion of these three papers with long and tedious
computations, the next step was to consider two natural questions for
a conceptional understanding of the Gross–Zagier formula: *Why
should the Gross–Zagier formula be true?* *How does one
extend the Gross–Zagier formula to higher-dimensional Shimura
varieties?*

For the first question, in December 2001 at an MSRI workshop, Dick
[8]
answered that the Gross–Zagier formula should be an
equality __\( P_G=P_A \)__ of two linear functionals in the
one-dimensional
vector space __\( \operatorname{Hom}_{\mathbb{A}_K^\times} (\pi\otimes \chi, \mathbb{C}) \)__, where __\( \pi \)__
is the automorphic representation generated by modular form __\( f \)__. In
fact, this point of view was already used by
Waldspurger
[e1]
in his toric period integral formula for the central value
__\( L(\pi, \chi, 1/2) \)__ of the Rankin–Selberg __\( L \)__-function.
Unfortunately, only special cases of Waldspurger’s formula had been
cited by others, probably because the general formula is stated in
terms of linear functionals as Proposition 7 on page 50 of a 70-page
paper. Following Dick’s framework
[8],
Xinyi Yuan,
Wei Zhang,
and I were able to give a proof of a Gross–Zagier formula
[e8]
in full generality, combining the strategy of Waldspurger
[e1]
with some “arithmetic” modification such as the *incoherent quaternion algebras* (a reflection of Dick’s point of
view that projective systems of Shimura curves are determined by odd
sets of places).

For the second question, Dick explained his joint work with
S. Kudla
[3]
and
C. Schoen
[6]
to me. As another “warm-up
exercise”, I started to do some computations for the heights of
Gross–Schoen cycles for the triple product of a general curve over a
number field. In
[e6],
I was able to prove an expression of
this height in terms of the self-intersection of dualizing sheaves.
This identity was then used to prove the uniform Bogomolov conjecture
and the uniform Mordell–Lang conjecture by
Z. Cinkir,
R. de Jong,
X. Yuan,
et al. Then for the triple product of Shimura curves, Xinyi Yuan,
Wei Zhang and I could formulate a Gross–Kudla conjecture in full
generality and prove it in certain cases
[e7].
What about the Gross–Zagier formula for other Shimura varieties? In a
hotel lobby in Beijing in December of 2007, Dick explained in a
note
[9]
to me his new conjectures
[10]
with
Wee Teck Gan and
Dipendra Prasad
about the arithmetic diagonal cycles for Shimura varieties
attached to __\( U(n-1,1)\times U(n,1) \)__ or __\( O(n-1, 2)\times O(n, 2) \)__. I
then organized a workshop at the Chinese Academy of Sciences with my
students to study these conjectures and the relative trace formula
approach by
Jacquet–Rallis.
So far, the younger generation, including
Wei Zhang,
Yifeng Liu,
Hang Xue,
and
Chao Li,
has made a lot of
important decisive results for these conjectures.
I myself have focused on related geometric questions in Arakelov
theory: the constructions and computations of Beilinson–Bloch
heights. For example, just recently, I was able to extend
Gross–Schoen’s work to the product of a curve and a surface
[e9].
Even for this small paper, I have already received
warm
encouragement from Dick. It is just another example among many of
his generous support and mentorship, for which I am very grateful.

*Shou-Wu Zhang is Eugene Higgins Professor of Mathematics at Princeton University.
He is known for his work on the Bogomolov conjecture and the Gross–Zagier formula.*