Here are some memories about Dick Gross, who sadly just passed away very recently.
Nothing was quite as reassuring as having Dick Gross in your audience. Inevitably, when your talk was done, he would both compliment you on it and have something very interesting mathematical to say. The last time this happened to me was at the Tate 100 conference in March. Dick was a student of Tate, and although he wasn’t able to come in person, he gave some prerecorded reminiscences. (Many of the very nice things Dick says about Tate can also be said about Dick.) But after my own talk (broadcast on Zoom), I still got an email from Dick titled “great talk,” which led to an interesting conversation between the relationship between Serre’s Conjecture and Artin’s Conjecture, as well as some analogs of these questions for \(\mathrm{GSp}_4\). I’m guessing I am not the only speaker at that conference to get such an email!
It’s hard to know where to start. Dick was a great mathematician — his collaboration with Don produced surely one of the greatest theorems in modern number theory (and many more great theorems besides; the paper on difference of singular moduli, for example). He was a great expositor — his Duke paper is a masterly exposition of quite a lot of the arithmetic theory of modular forms (and also a wonderful theorem). He was a great mathematician to chat with at afternoon tea or in the corridor, when you could learn all sorts of clever ideas that weren’t written down anywhere else. He was a pioneer in the arithmetic theory of automorphic forms on higher ranked groups. As Dick himself used to say, you start with \(\mathrm{GL}_2\), then remove the \(\mathrm{L}\), and then the \(2\).
In contrast, the first time I interacted with Dick, a little shy of 25 years ago, he was at the beginning of a second career beyond mathematics in administration, having become math department chair before his tenure as Dean. It was his job to let me know that the department was offering me a BP position. One line from that email was as follows:
For now, let me say how delighted I would be if you would join us next year, as a colleague.
This certainly made me feel pretty good at the time, and it is a line I have come back to and reused myself as a junior hiring chair. (Another line in that email, “I hear you are now an uncle. Behave accordingly”, is less versatile.) More generally, Dick was charming in the best possible way — combining not only the polish that this word suggests, but with an underlying spirit of someone who was attentive, personable, and conscientiously kind.
My interactions with Dick at Harvard mostly continued through his capacity as chair. At one point, I realized that I had been slightly overpaid (I was getting a mix of money both from Harvard and from AIM). His remark at the time, which I can only paraphrase due to the passage of time — said entirely deadpan — was something like, “I have two pieces of advice in life: avoid paying your taxes as much as possible, and don’t tell anyone if you are overpaid.”
There was only one moment where I saw him anywhere approaching being exasperated (though presumably that must have happened quite often as Dean). DeBacker and I had been put in charge of the colloquium committee. The speakers had already been invited by the time it started, so the main task was simply organizing the dinners for the speakers. This was all done by DeBacker, who paid for everything himself and was later reimbursed. On one occasion, Richard Borcherds gave the colloquium, and we ended up going to a quite fancy restaurant (Harvest) in Harvard Square. I don’t think a single senior faculty member came, but lots of graduate students did. We were, I think, quite liberal with the purchase of some nice bottles of Chablis. As you can imagine, the price of the dinner (fully paid by the department) was on the higher side, and at some point it must have gone to Dick’s desk to be approved. I believe Dick’s remark to Stephen was along the lines of, “I don’t want to hear the words ‘colloquium dinner,’ ‘graduate students,’ and ‘$2000’ in the same sentence ever again.”
My best mathematical interactions with Dick mostly came through casual conversations and emails (or even comments on this blog!). I did once answer an actual mathematical question raised by Dick in a joint Inventiones paper with Lubin from 1986. They asked whether a certain Hecke algebra of level \(\Gamma_0(p^2)\) localized at an Eisenstein ideal above \(p\) was always a discrete valuation ring, and I found this could be answered in the positive using ideas of Chenevier and Bellaiche.
Of course, that particular question and answer are no more than mathematical ephemera. But Dick’s legacy — as a mathematician and as a person — will live on.
More from other sources:
An article on Dick in Celebratio Mathematica
Faculty Spotlight Harvard Interview
I hope it’s OK, seeing your post made me want to record the numerous ways Dick influenced my own work. The list is very long! He will be missed dearly.
The first time I met Dick was when I was a senior undergraduate in 2009, and was visiting Harvard to decide if I wanted to go there for graduate school. Dick suggested I go for a walk to help me make up my mind 🙂 I think I somehow decided that I was too intimidated by Dick and Richard Taylor to go to Harvard, and went to Princeton instead. (Not sure why I was less intimidated by Wiles and Skinner and Sarnak and Bhargava etc)
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The second time I met Dick was at Harvard in Spring of 2014. I was finishing my PhD that semester. If I recall correctly, Stefan Patrikis was a BP at Harvard at the time, and had invited me to speak in the number theory seminar. I was speaking on some work with Shrenik Shah on the Spin L-function for cuspidal representations $\pi$ on GSp(6). A pole of this L-function at s=1 detects when $\pi$ is a functorial lift from G_2, and, relying on some work of Wee Teck Gan and Nadya Gurevich, Shrenik and I proved some result in this direction. Dick was in the audience, and I was nervous! I can’t remember what he said, but somehow, before the talk started, Dick found a way to calm my nerves a bit.
Of course, Dick had thought about the lifting from G_2 to PGSp_6, in a Compositio paper with Gordan Savin, and in other ways. This was the first of what would become many times where Dick’s work in some number theoretic area would significantly influence my own work.
I remember another time, when I was a postdoc, when I sent an email to Dick with a preprint I was planning to post to arXiv. It was another paper on the Spin L-function on GSp(6), this time applicable to Siegel modular forms. I didn’t think Dick knew who I was, and I didn’t know if I would get a response. But he replied something like 15 minutes later with positive feedback and specific questions! Probably many people have had this experience with Dick: he immediately understood what it is you are doing (probably because he had thought about it a decade ago), and is quick with encouraging and insightful remarks.
Some of the many ways in which Dick’s work significantly impacted my own:
1.) Orbit parametrization problems: In a short paper with Mark Lucianovic, Dick and Mark understand the orbits of GL_3(Z) on ternary quadratic forms in terms of orders in quaternion algebras, and the orbits of GL_2(Z) on binary cubic forms in terms of cubic rings.
These particular orbit parametrization problems are inspired by automorphic considerations. The orbits of GL_3(Z) on ternary quadratic forms also parametrize Fourier coefficients of Siegel modular forms on GSp(6). So, one could understand Fourier coefficients on GSp(6) in terms of quaternion orders. Thinking about this relationship was crucial to my work on L-functions of Siegel modular forms. Likewise, the orbits of GL_2(Z) on binary cubics parametrize Fourier coefficients of modular forms on G_2, and this orbit problem helped me understand L-functions on G_2.
2. Quaternionic modular forms: Dick and Nolan Wallach pioneered a detailed study of the so-called quaternionic discrete series representations. Dick had the ingenious idea that the automorphic representations with quaternionic components at infinity should be arithmetic in way similar to the holomorphic modular forms that exist on Shimura varieties. This point of view has dominated by research for the past 8 years. Trying to understand their work, and the paper of Gan-Gross-Savin on quaternionic modular forms on G_2, led me to thinking about the explicit Fourier expansion of these automorphic forms.
3.) The automorphic minimal representation: After working out the shape of the Fourier expansion of quaternionic modular forms, Dick encouraged me to compute the Fourier coefficients in an explicit example: The quaternionic automorphic form attached to the minimal representation on a group of type E_8. Dick and Nolan had shown that this minimal representation was quaternionic, so this would be a great place to start computing in an explicit example. (It is a very helpful suggestion because the minimal representation on E_8 is spherical at every finite place, arises as a residue of an Eisenstein series, and most Fourier coefficients are provably 0. So, it is the most accessible example of a quaternionic modular form.)
4.) Constructing cusp forms on G_2: In their beautiful Duke paper, Gan-Gross-Savin asked the question of finding the smallest weight k so that the space of level one cuspidal quaternionic modular forms on G_2 of weight k is nonzero. (This was later resolved by Rahul Dalal: the answer is k=6). Their question inspired me. I made a small amount of progress by constructing a nonzero example in weight k = 20, by lifting from Sp(4) to SO(8) and then restricting to G_2.
5.) Algebraic modular forms: Dick wrote a paper on algebraic modular forms, i.e., automorphic forms on reductive groups G for which G(R) is compact. These objects are great for explicit computation. They later appear in the paper of Gross-Savin, who consider the exceptional theta correspondence between algebraic modular forms on compact G_2 and Siegel modular forms on PGSp_6. Dick and Gordan produce a cuspidal nonzero lift with some ramification, but leave open the problem of producing a nonzero level 1 cuspidal lift to PGSp(6). Again inspired by some of Dick’s work, I worked out how to check the non-vanishing of these lifts (with the help of a computer) in a recent preprint.
6.) Constructing cusp forms on G_2, take 2: In the same preprint, I do the same for algebraic modular forms on compact F_4, now lifting them to quaternionic modular forms on split G_2. This is inspired by many of Dick’s papers, with Wallach, with Gordan Savin, with Wee Teck Gan: One starts with algebraic modular forms, the theta lift uses the quaternionic minimal representation on E_8, one ends with quaternionic cusp forms on G_2, and via the explicit nature of the lift, one can understand the Fourier coefficients on G_2 (which are paramtrized by cubic rings)!
7.) A Waldspurger-like conjecture: At some point several years ago, Chao Li explained to me that Dick had a conjecture about the Fourier coefficients of certain Arthur lifts from PGL_2 to split G_2. Let f be a weight 2k, level one cusp form on PGL_2. Arthur predicts a lift of f to G_2, call it F, which is a weight k cuspidal quaternionic modular form. Dick conjectures that the square of the Fourier coefficients of F are related to the central L-values of f, twisted by Artin motives of cubic fields. The conjecture is something like the famous result of Waldspurger (made more explicit by Kohnen-Zagier) that relates the square of the Fourier coefficients of the Shimura lift of f (which exists on the metaplectic group) to the central L-values of f twisted by quadratic characters. I have had many plane rides where I think about Dick’s conjecture and get nowhere!
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That was a nice read, thanks.
I remember when I first met Dick Gross. He was a friend of the family, and when I came to Harvard as an undergraduate he asked to meet with me in his office. At the time he was no ordinary math professor: he was the Dean of Harvard College. Accordingly, his office wasn’t some dingy affair in the Science Center. At least in my memory, it was a cavernous realm in which an ordinary house could comfortably fit, located in that suitably noble building right behind the statue of John Harvard, in the very center of Harvard Yard.
Anyway, at the other end of this great hall was a mahogany desk with an affable man behind it. The meeting was quick: he just wanted to know what classes I was planning to take. I told him about the scheme theory class which, as I recall, was being taught that year by a colourful tandem of postdocs: the ever-personable Samit Dasgupta, and some sardonic Aussie whose name escapes me. I remember that Dick advised me against taking this class. He said that before I try to learn about things like etale cohomology I should learn the usual topological story. Wise advice, but of course I ignored it.
Actually, at the time I was filled with a youthful fervor which made me very suspicious of Dick. What math professor would voluntary forestall their study of mathematics in order to serve as a bureaucrat? But every interaction I’ve had with Dick over the years has laid bare his warmth and benevolence, and now that I’m older myself, I perfectly understand his motivations: he wanted to do right by the next generation of students. As far as I can attest, he has certainly succeeded, through both his personal interactions and his mathematics.
Dustin Clausen
P.S.: I’m getting a lot out of Dick’s Eilenberg lectures, and I highly recommend them to fellow students of number theory: https://www.youtube.com/playlist?list=PL5E0D6DC4BCD8309D