Celebratio Mathematica

Benedict H. Gross

A second graduate school education

by David E. Rohrlich

I first met Dick Gross in the late sum­mer or early fall of 1976. I had just ar­rived at Har­vard, and Neal Kob­litz had ini­ti­ated some con­ver­sa­tions re­lated to CM types of Jac­obi­ans of Fer­mat curves. At an early stage of our col­lab­or­a­tion Neal sug­ges­ted that it would be worth­while to in­volve a gradu­ate stu­dent named Dick Gross in our dis­cus­sions. At first I did not know what to make of Dick. I don’t think I had ever met any­body whose ac­tu­al giv­en name in Eng­lish was Be­ne­dict, and I cer­tainly had not en­countered any gradu­ate stu­dent whose math­em­at­ic­al in­tens­ity seemed quite as fren­zied. However an early in­cid­ent won my grat­it­ude. In a con­ver­sa­tion among the three of us I had quoted a fact from the Shimura–Tan­iyama volume [e1] that seemed im­port­ant for the mat­ter at hand, but at our next meet­ing we looked at the quoted pas­sage to­geth­er, and I was mor­ti­fied to dis­cov­er, prob­ably thanks to Dick’s guid­ance, that I had mis­quoted it. Worse, I had the im­pres­sion that in the in­ter­ven­ing days Dick had already in­ves­ted some think­ing about the Fer­mat curves based on the false in­form­a­tion I had provided. But Dick seemed em­bar­rassed by my ab­ject apo­logy, and if he felt any an­noy­ance, none was ap­par­ent. He swiftly steered the con­ver­sa­tion in­to a more pro­duct­ive dir­ec­tion.

Per­haps this little in­cid­ent was a har­binger of things to come, be­cause over the next two years, while he was still a gradu­ate stu­dent and I a postdoc, a role re­versal oc­curred in which he be­came something like a second thes­is ad­visor to me — not in the sense of su­per­vising my thes­is but more in the sense of edu­cat­ing me about the tricks of the trade and the niceties of math­em­at­ic­al ex­pos­i­tion. The things I am re­fer­ring to here in­clude not only im­port­ant in­sights but also ba­sic habits of mind, even things as simple as an un­re­mit­ting aware­ness that num­ber fields need not be re­garded as sub­fields of \( \mathbb{C} \). In prin­ciple of course I already had such an aware­ness, but it was in­struct­ive to ob­serve how in Dick’s case, this bed­rock con­vic­tion — not mere aware­ness — in­formed his every thought. In the end it was my good for­tune to wit­ness the gen­es­is of the De­ligne–Gross con­jec­ture: The or­der of van­ish­ing of a mo­tivic \( L \)-func­tion at a cent­ral crit­ic­al point is in­de­pend­ent of the com­plex em­bed­ding of the coef­fi­cient field of the motive (see [e2], Con­jec­ture 2.7(ii), p. 323). It does not di­min­ish my enorm­ous debt to my thes­is ad­visor Serge Lang to re­cog­nize that after re­ceiv­ing my de­gree I was priv­ileged to con­tin­ue my gradu­ate school edu­ca­tion un­der Dick’s tu­tel­age.

After Dick got his de­gree and I moved on from my postdoc, we no longer saw each oth­er with the same reg­u­lar­ity, but he con­tin­ued to be a source of in­sights and in­spir­a­tion at con­fer­ences or sem­inars or wherever I en­countered him. Some­times I did not ap­pre­ci­ate the sig­ni­fic­ance of his re­marks un­til many years after he made them. For ex­ample, as I write these re­min­is­cences, I hap­pen to be work­ing on a little pa­per in­volving idele class char­ac­ters of a CM field which are trivi­al on the ideles of the max­im­al totally real sub­field, and I am re­minded of a re­mark that Dick made to me in the early 1980s, re­fer­ring at that time to an ima­gin­ary quad­rat­ic field \( K \): When a char­ac­ter of the type just men­tioned factors through the Galois group of a \( \mathbb{Z}_p \)-ex­ten­sion of \( K \) then the name an­ti­cyc­lo­tom­ic char­ac­ter is ap­pro­pri­ate, but in the gen­er­al case, the clas­sic­al term ring class char­ac­ter is the cor­rect des­ig­na­tion. I did not at­tach much im­port­ance to Dick’s com­ment at the time, but I have since come to real­ize that this is a case where the his­tory of math­em­at­ics, of which Dick is very con­scious, can use­fully in­form the prac­tice of math­em­at­ics, and what was seem­ingly just a re­mark about ter­min­o­logy has turned out to be in­tel­lec­tu­ally pro­voc­at­ive.

I have nev­er ad­equately ac­know­ledged Dick’s im­mense in­flu­ence on me, even in cases like [e4] where a sug­ges­tion of his was the cata­lyst for un­der­tak­ing a pro­ject. But oth­ers much wiser than I and a good bit less stingy have man­aged to find just the right words to ex­press their ap­pre­ci­ation. I am think­ing, for ex­ample, of the ac­know­ledg­ment at the end of the in­tro­duc­tion to the pa­per [e3] of Ru­bin and Wiles, who write: “Fi­nally, we thank Gross and Mazur for their help­ful sug­ges­tions, con­scious and oth­er­wise.” Sug­ges­tions con­scious and oth­er­wise — that says it all. By the way, De­ligne’s ac­know­ledg­ment in loc. cit. is not dis­sim­il­ar: “Que (ii) soit rais­on­nable m’a été suggéré par B. Gross.” De­ligne’s re­mark, to­geth­er with my own firsthand glimpse of the events which pre­ceded it, is the reas­on that I speak of the “De­ligne–Gross con­jec­ture,” even though I have the im­pres­sion that the epi­thet has not yet gained the wide­spread cur­rency that it de­serves.

So far I have em­phas­ized Dick’s in­flu­ence, but there are two oth­er as­pects of his math­em­at­ic­al per­sona that I would also like to men­tion. One is the breadth of his ap­pre­ci­ation of math­em­at­ics, which en­com­passes both the large and the small, both the soph­ist­ic­ated and the naïve, both the cut­ting-edge and the old and fa­mil­i­ar. He calls to mind a lin­guist who has mastered dozens of lan­guages but is non­ethe­less full of ad­mir­a­tion when he en­coun­ters some­body who speaks only one for­eign lan­guage but speaks it with ex­cep­tion­al gusto. Or per­haps one could com­pare Dick to a pi­an­ist who tours the con­cert halls of the world per­form­ing the Em­per­or Con­certo and then comes home and plays Für Elise for an in­tim­ate circle. About four years ago Dick sent me the pre­print [1], and I was de­lighted to see the re­appear­ance of top­ics that had once en­chanted us both: the Chow­la–Sel­berg for­mula, val­ues of the gamma func­tion, peri­ods, and so on. Ap­par­ently they still en­chant us. The com­bin­a­tion of breadth, depth, and fi­del­ity to one’s math­em­at­ic­al youth is one of the traits that I most ad­mire in a math­em­atician; it is all too rare, but Dick has it in abund­ance.

Fi­nally, let me touch on Dick’s ca­pa­city for con­cen­tra­tion. As es­sen­tial as it has been to his math­em­at­ic­al ca­reer, it is not so much a part of his math­em­at­ic­al per­sona as it is simply a part of his per­son­al­ity, and it is re­flec­ted in all of his en­deavors, not just the math­em­at­ic­al ones. In fact it is crys­tal­lized for me in a men­tal snap­shot from many years ago of one of Dick’s non­mathem­at­ic­al activ­it­ies. The Cor­val­lis con­fer­ence on auto­morph­ic forms was held at Ore­gon State Uni­versity in Cor­val­lis in the sum­mer of 1977, and at the time there was some grumbling about the lack of re­cre­ation­al amen­it­ies avail­able either on cam­pus or in town. But for bet­ter or worse, the res­ult was that the par­ti­cipants in the con­fer­ence had to cre­ate their own forms of en­ter­tain­ment. One af­ter­noon I came upon Joe Buhler and Dick jug­gling bowl­ing pins, toss­ing them back and forth on the lawn in front of the dorm­it­ory which served as our ac­com­mod­a­tions. I no longer re­call how many bowl­ing pins Dick and Joe were able to keep in the air, but the sight was mes­mer­iz­ing. Even so, the im­age I re­tain best is the look of in­tense con­cen­tra­tion on Dick’s face. It was as if jug­gling were the most im­port­ant thing in the world to him. The ser­i­ous­ness of his ex­pres­sion al­lowed for only one in­ter­pret­a­tion: at that mo­ment, it was.

Dav­id Rohr­lich re­ceived his PhD from Yale in 1976 and has taught at Har­vard, Rut­gers, and the Uni­versity of Mary­land. Since 1990 he has held the po­s­i­tion of pro­fess­or of math­em­at­ics at Bo­ston Uni­versity.


[1] B. H. Gross: “On the peri­ods of Abeli­an vari­et­ies,” IC­CM Not. 8 : 2 (2020), pp. 10–​18. MR 4245911 Zbl 1475.​11121 ArXiv 2005.​04194 article