Celebratio Mathematica

Benedict H. Gross

A tribute to Dick Gross

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.