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
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
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
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
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.