Celebratio Mathematica

Benedict H. Gross

Dick Gross’s marvelous mentoring

by Henri Darmon

I was for­tu­nate to be­come a PhD stu­dent of Dick Gross in 1987. This was a time of great in­tel­lec­tu­al fer­ment at Har­vard, when Dick’s work with Don Za­gi­er [2] was be­gin­ning to as­sert its pro­found im­pact on the the­ory of el­lipt­ic curves. One of the most mys­ter­i­ous in­vari­ants in the sub­ject is the con­jec­tur­ally fi­nite Sha­far­ev­ich–Tate group of an el­lipt­ic curve \( E \) over \( \mathbb{Q} \), which meas­ures the dif­fi­culty of com­put­ing the Mor­dell–Weil group \( E(\mathbb{Q}) \) by Fer­mat’s meth­od of in­fin­ite des­cent. Just as fun­da­ment­al is the (weak) con­jec­ture of Birch and Swin­ner­ton–Dyer, which equates the rank of this Mor­dell–Weil group with the or­der of van­ish­ing of the Hasse–Weil \( L \)-func­tion \( L_E(s) \) at \( s=1 \). The Gross–Za­gi­er for­mula as­sumes that \( E \) is mod­u­lar1 and relates the first de­riv­at­ive \( L_E^{\prime}(1) \) to the height of a Hee­gn­er point arising from the the­ory of com­plex mul­ti­plic­a­tion. If \( L_E(s) \) has a simple zero at \( s=1 \), it fol­lows that the Mor­dell–Weil group \( E(\mathbb{Q}) \) has rank at least 1 be­cause it con­tains a Hee­gn­er point of in­fin­ite or­der. When com­bined with the work of Vic­tor Kolyva­gin [e1], whose an­nounce­ment was one of the high­lights of my first year of gradu­ate stud­ies, the Gross–Za­gi­er for­mula im­plies that both the Sha­far­ev­ich–Tate con­jec­ture and the Birch and Swin­ner­ton–Dyer con­jec­ture are true for \( E \) whenev­er \( L_E(s) \) has a zero of or­der at most 1 at \( s=1 \). This re­mark­able break­through re­mains very close to the state of the art more than 30 years later, es­pe­cially for el­lipt­ic curves of rank 1 over \( \mathbb{Q} \).

I asked Dick to be my su­per­visor right after passing the qual­i­fy­ing ex­ams. In the be­gin­ning, I na­ively thought I should ap­prise him weekly of my rather plod­ding pro­gress. My typ­ic­al ques­tions were quite mundane, and Dick en­cour­aged me to dis­cuss them in­stead with Massimo Ber­to­lini, who at the time was a year ahead of me. This turned out to be ex­cel­lent ad­vice: Massimo (who later moved to Columbia to work with Karl Ru­bin) soon be­came a close friend and col­lab­or­at­or, and we wrote more than 30 pa­pers to­geth­er over the years. The ad­vice was also lib­er­at­ing: not hav­ing to meet with my su­per­visor reg­u­larly gave me the free­dom to ab­sorb the ma­ter­i­al at my own pace and pur­sue the dir­ec­tions that temp­ted me most. Since I was some­what im­ma­ture math­em­at­ic­ally, these were mostly wild goose chases that pro­duced little con­crete pro­gress over long stretches, but en­hanced my ex­per­i­ence of math­em­at­ics as a great in­tel­lec­tu­al ad­ven­ture. Dick’s policy of be­nign neg­lect, which suited me per­fectly, did not ex­tend to all his stu­dents. Some pre­ferred more dis­cip­line and the re­as­sur­ance of reg­u­lar meet­ings, and Dick was al­ways avail­able for them. This il­lus­trates one of the qual­it­ies I most ad­mire in Dick as a su­per­visor: his knack for bring­ing out the best in his dis­ciples by ad­apt­ing to their in­di­vidu­al needs and work­ing styles. I have tried to rep­lic­ate Dick’s ap­proach in my own gradu­ate ment­or­ing, but it is a hard act to fol­low!

The first thes­is prob­lem that Dick pro­posed was to ex­tend Kolyva­gin’s res­ult to the Hasse–Weil \( L \)-func­tions of el­lipt­ic curves twis­ted by an un­rami­fied char­ac­ter, or a more gen­er­al ring class char­ac­ter \( \chi \) of an ima­gin­ary quad­rat­ic field. The goal was to par­lay the non­tri­vi­al­ity of the “\( \chi \)-com­pon­ent” of the Hee­gn­er point in­to the fi­nite­ness of the in­dex of this point in the \( \chi \)-com­pon­ent of the Mor­dell–Weil group, and of the as­so­ci­ated \( \chi \)-com­pon­ent of the Sha­far­ev­ich–Tate group.

Massimo and I solved the prob­lem to­geth­er in the sum­mer of 1989 while at­tend­ing two mem­or­able con­fer­ences back-to-back. The first was a his­tor­ic joint US–USSR meet­ing in Chica­go from mid-June to mid-Ju­ly, where, in the early years of peres­troika, West­ern par­ti­cipants got to meet, in per­son for the first time, many sci­entif­ic lu­minar­ies from what was still called the So­viet Uni­on. One of the high­lights for me was shak­ing the hand of Kolyva­gin, whom Dick in­tro­duced to his star-struck gradu­ate stu­dents. In his ad­dress at the con­fer­ence, Kolyva­gin de­scribed how the full col­lec­tion of Hee­gn­er points over ring class fields of ima­gin­ary quad­rat­ic fields largely de­term­ines the struc­ture of the Selmer group of an el­lipt­ic curve. The Chica­go meet­ing was fol­lowed by a two-week in­struc­tion­al con­fer­ence at the Uni­versity of Durham in Eng­land, where Dick gave a beau­ti­ful sur­vey [3] of Kolyva­gin’s meth­od, which still serves as the stand­ard ini­ti­ation to the sub­ject. Massimo and I wrote up our pa­per [e2] dur­ing a last stop in the Parisi­an sub­urb of Jouy en Jo­sas, where I vis­ited my par­ents be­fore re­turn­ing to Har­vard.

Dick’s ques­tion was per­fect for a be­gin­ning gradu­ate stu­dent be­cause, al­though it did not fol­low im­me­di­ately from Kolyva­gin’s proof, it was very much amen­able to the meth­ods that had been in­tro­duced, and solv­ing it did not pose in­su­per­able bar­ri­ers. Yet it also ad­mits nat­ur­al vari­ants that are sig­ni­fic­antly more dif­fi­cult and in­ter­est­ing. My fa­vor­ite one is what if the ima­gin­ary quad­rat­ic field is re­placed by a real quad­rat­ic field? Dick’s prob­lem re­mains open in this set­ting. My fre­quent ob­sess­ing about it is un­doubtedly what led me, a dec­ade later, to the no­tion of Stark–Hee­gn­er points over ring class fields of real quad­rat­ic fields [e3]. And in 2010, Vic­tor Rot­ger and I answered the real quad­rat­ic ana­logue of Dick’s ques­tion in the more tract­able set­ting of ana­lyt­ic rank 0 [e6]. Namely, we showed that the \( \chi \)-part of the Mor­dell–Weil group of an el­lipt­ic curve is fi­nite when the as­so­ci­ated \( L \)-series does not van­ish, for \( \chi \) a ring class char­ac­ter of a real quad­rat­ic field. The meth­od we fol­lowed dif­fers sub­stan­tially from Kolyva­gin’s and is closer in spir­it to ap­proaches of Coates–Wiles and of Kato, with an im­port­ant fur­ther in­put from Dick’s own ideas on di­ag­on­al cycles and triple product \( L \)-func­tions. Ul­ti­mately, sev­er­al of my most sig­ni­fic­ant math­em­at­ic­al con­tri­bu­tions have their gen­es­is in the mod­est “warm-up ex­er­cise” that Dick pro­posed to Massimo and me in the spring of 1989.

As a more sub­stan­tial thes­is prob­lem, Dick then asked me to re­flect on Kolyva­gin’s meth­od in light of the “tame re­fine­ments” of the Birch and Swin­ner­ton–Dyer con­jec­ture that had been pro­posed by Mazur and Tate around 1986. Dick’s idea was that these tame re­fine­ments would provide the ideal frame­work for un­der­stand­ing and or­gan­iz­ing Kolyva­gin’s meth­od of Euler sys­tems. Like much of what I gleaned in my con­ver­sa­tions with Dick, this in­sight was spot-on and very fruit­ful. Thanks to it, the main res­ults in my thes­is were already in place by early 1990, with a year and a half to spare be­fore gradu­ation, which made for an un­usu­ally pleas­ant and re­laxed fi­nal year of gradu­ate stud­ies.

Over the last thirty years, my math­em­at­ic­al in­terests have nev­er strayed far from the dir­ec­tions that Dick opened up to me. Among the pro­jects that have giv­en me spe­cial pleas­ure over the last two dec­ades, three stand out the most. The first is a col­lab­or­a­tion with Sam­it Dasgupta [e4] and Rob Pol­lack [e5] re­volving around the Gross–Stark con­jec­ture on de­riv­at­ives of De­ligne–Ribet \( p \)-ad­ic \( L \)-func­tions at \( s=0 \), a dir­ec­tion which Sam­it has taken much fur­ther in his more re­cent work with Ma­hesh Kak­de. The second is a pro­ject with Vic­tor Rot­ger [e6] on \( p \)-ad­ic \( L \)-func­tions at­tached to triple products of mod­u­lar forms and as­so­ci­ated di­ag­on­al cycles in triple products of mod­u­lar curves, in­spired by Dick’s ex­ten­sion with Steve Kudla of the Gross–Za­gi­er for­mula to \( L \)-func­tions of auto­morph­ic forms on the product of two or­tho­gon­al groups, at­tached to quad­rat­ic spaces of di­men­sions 3 and 4. The third is an on­go­ing study with Alice Pozzi and Jan Vonk of sin­gu­lar mod­uli for real quad­rat­ic fields via the RM val­ues of ri­gid mero­morph­ic cocycles [e8], [e7]. What de­lights me the most in the lat­ter is the op­por­tun­ity it has giv­en me to re­vis­it the re­mark­ably rich and sem­in­al work of Gross and Za­gi­er on the fac­tor­iz­a­tion of dif­fer­ences of sin­gu­lar mod­uli [1], which lays the found­a­tions for their ground­break­ing work on de­riv­at­ives of \( L \)-series [2].

Like all Har­vard gradu­ate stu­dents, and per­haps even more than most, I was in awe of my su­per­visor, and some­what in­tim­id­ated by him. Be­cause I saw him as an in­tel­lec­tu­al fath­er fig­ure more than as a friend, we nev­er be­came very close. Yet the im­pact he has had on me is tre­mend­ous. In the words of Ral­ph Waldo Emer­son, “Our chief want in life is some­body who will make us do what we can.” As a ment­or, Dick ac­com­plished this su­perbly. His ex­ample has guided me and his ideas have in­spired me throughout my math­em­at­ic­al life, and for this I am im­mensely grate­ful to him.

Henri Dar­mon wrote his PhD from 1987 to 1991 at Har­vard Uni­versity un­der the su­per­vi­sion of Dick Gross. His most not­able con­tri­bu­tions aim to ex­tend the the­ory of com­plex mul­ti­plic­a­tion to real quad­rat­ic and oth­er non-CM fields. He is cur­rently a Dis­tin­guished James Mc­Gill Pro­fess­or at Mc­Gill Uni­versity.


[1] B. H. Gross and D. B. Za­gi­er: “On sin­gu­lar mod­uli,” J. Reine An­gew. Math. 355 (1985), pp. 191–​220. Ded­ic­ated to J.-P. Serre. MR 772491 Zbl 0545.​10015 article

[2] B. H. Gross and D. B. Za­gi­er: “Hee­gn­er points and de­riv­at­ives of \( L \)-series,” In­vent. Math. 84 (1986), pp. 225–​320. To John Tate. This work ex­pands on a short note pub­lished in C. R. Acad. Sci., Par­is 297 (1983). Part II was pub­lished in Math. Ann. 278 (1987). MR 833192 Zbl 0608.​14019 article

[3] B. H. Gross: “Kolyva­gin’s work on mod­u­lar el­lipt­ic curves,” pp. 235–​256 in \( L \)-func­tions and arith­met­ic (Durham, UK, 30 June–11 Ju­ly 1989). Edi­ted by J. Coates and M. J. Taylor. Lon­don Math­em­at­ic­al So­ci­ety Lec­ture Note Series 153. Cam­bridge Uni­versity Press, 1991. MR 1110395 Zbl 0743.​14021 incollection

[4] B. H. Gross and S. S. Kudla: “Heights and the cent­ral crit­ic­al val­ues of triple product \( L \)-func­tions,” Com­pos. Math. 81 : 2 (1992), pp. 143–​209. MR 1145805 Zbl 0807.​11027 article