by Barry Mazur
The one word to describe conversations with Dick Gross is joyfulness. Dick’s accomplishments — his extremely important research, his phenomenal teaching, and the range of his engagement with the wider circle of mathematical and intellectual culture — are most definitely a delight to be thinking about.
And his mathematical presentations are celebrations of our subject; they’re very much presents to us. And this began, for me, fairly early in Dick’s career. I remember the first time I met him. He knocked on my office door, and came in launching a conversation, illuminating to me, with all the fervor and richness of curiosity that he always has. It was about — I think — the Eisenstein ideal in the Hecke algebra of modular curves. I wondered: Is this a new colleague? Is he a visiting professor? But this was in September of his first year as an incoming graduate student. So, it began early.
Dick’s many important contributions, crucial to our subject, can
hardly be even touched in this brief Celebratio — think, for example, of
his work on the local Langlands correspondence, automorphic
representations, the exceptional Lie group
So I will focus only on one example in the extraordinary world of Dick’s contributions to mathematics.
There are formulations of mathematical theories or mathematical vocabulary that encompass a particular subject, enliven it, make it newly exciting to work on, etc. And there are also results that radiate outwards: that explain and illuminate similar past results, that inspire new developments.
A striking example of a radiating contribution is the Gross–Zagier Theorem.
In the early 19th century Dirichlet proved that every arithmetic
progression
But the Gross–Zagier Theorem is the one shining modern result that
does radiate backwards to enliven aspects of Dirichlet’s result (and
its subsequent development by
Dedekind,
and
Kronecker)
and radiates
forwards — when paired with the key theory of
Kolyvagin — establishing a
bridge between a special value of a certain type of
- the
-adic counterpart to the Birch–Swinnerton–Dyer Conjecture; - when paired with the work of Goldfeld it establishes some much-sought asymptotic bounds on class numbers of quadratic imaginary fields: this type of question was already an implicit issue in Gauss’s Disquisitiones Arithmeticae.
The Gross–Zagier theorem nowadays continues to play the role of a model and a magnificent guide. On the one hand, it is being specifically generalized to larger contexts, and on the other, it is an inspiration in itself, and for the formation of larger conjectures and research projects.
What a great joy it is to celebrate Dick’s work.
Barry Mazur is the Gerhard Gade University Professor at Harvard University.