#### 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.*