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 field with
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
For the first question, in December 2001 at an MSRI workshop, Dick
[8]
answered that the Gross–Zagier formula should be an
equality
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
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.