by Chao Li
I first met Dick as an arriving graduate student at Harvard in Fall 2010. But even before that, my link to Dick had already begun: my final project in the undergraduate algebraic number theory course at Tsinghua was to survey the Gauss class number problem, which gave me the chance to learn about the celebrated work of Gross–Zagier and Goldfeld. At the beginning of June 2010, I was told by the math office that certain documents for me had to be processed the following week because they were busy running a large math conference that week, which I later realized, happened to be Dick’s 60th birthday conference “Number Theory and Representation Theory”! I was unlucky to miss this unique spectacular event, but lucky enough to learn more about the Gross–Zagier formula (and even make my own contribution years later).
When I was a first-year student, I didn’t know much about elliptic curves and Dick kindly agreed to supervise a minor thesis on the Gross–Zagier paper on singular moduli. It was a fascinating three-week learning experience, and no doubt motivated me to study more. In Fall 2011, while Dick was on leave to be the Eilenberg Lecturer at Columbia, he sent me his notes on a problem of constructing rational points on certain elliptic curves with 2-Selmer rank 1. I was immediately attracted by this concrete (and, as it turns out, rather deep) problem, which eventually became my thesis topic. Even though Dick was away, I still learned a huge amount from him by watching (and transcribing) the recording of his Eilenberg Lectures. (Recording was an uncommon resource in the pre-Zoom era!) As always, his lectures were strikingly beautiful, with the magic power of energizing the entire audience with deep mathematical ideas.
Dick was popular among students by any standard. Every year there is a special dinner for which every Harvard undergraduate student invites an instructor of choice to join. One year I was humbled to be invited by a few students from my calculus class. When I entered Memorial Hall with my hosts to find a table, I noticed that one long dinner table had been completely occupied. I got closer to see what was going on. Sitting in the middle was none other than Dick, surrounded by perhaps more than 20 inspired undergraduate students eager to talk to him!
Dick was an amazing advisor. During my stay at Harvard, he also had many graduate students. Every Wednesday I tried to find him before the number theory seminar to report my progress and ask for guidance, then go to the seminar talk. Dick listened with his characteristic enthusiasm, and for innocent graduate students his questions for speakers were sometimes more illuminating than the talks. Being a graduate student is fun but also there are many frustrating days with little progress, and in retrospect I certainly asked Dick more than my share of dumb questions, but Dick was always patient and encouraging. Dick often says, “Graduate students are smarter than we are, but we know more. When the students know enough? We graduate them.”
To be honest I didn’t really solve the problem originally suggested by Dick and it still lies in the back of my mind. Even today the so-called \( p \)-converse theorem of Gross–Zagier and Kolyvagin remains mysterious for \( p=2 \) in general. But perhaps Dick decided that I knew enough to be able to graduate. Dick’s advice is to come back to one’s thesis many times, just as he did. When Dick told John Tate that the elliptic curves in his thesis (now known as Gross curves) were already known (over \( \mathbb{C} \)) by Hecke, Tate said not to worry, Hecke had anticipated many of the results in his thesis too!
When I asked for Dick’s advice on the first talk, first publication, first job, etc., he always responded with wisdom and often an amusing anecdote. For example, when I served as a journal referee for the first time as a graduate student, Dick told me that when he was a graduate student, he was asked to referee a paper on Fermat curves and he wrote a report with the assessment between acceptance and declination. The author then responded to the editor with a 10-page rebuttal arguing that the referee had not understood the paper at all and insisting, “There is a young man who knows all about Fermat curves — Dick Gross — why don’t you ask him to referee?”
After graduation I had the opportunity to chat with Dick at various occasions, and his great works continue to be a major inspiration for me. I also had a memorable visit to Dick in San Diego in February 2020, right before the pandemic hit. Many of us became better at our jobs because we had the chance to watch Dick do his, and for that privilege I am forever grateful!
Chao Li is Associate Professor of Mathematics at Columbia University. He earned his Harvard Ph.D. degree under the supervision of Benedict Gross in 2015.
