Celebratio Mathematica

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 \( G_2 \), auto­morph­ic rep­res­ent­a­tions over func­tion fields over fi­nite fields, his re­cent proof, joint with Man­jul Bhar­gava, that a “pos­it­ive dens­ity” of hy­per­el­lipt­ic curves have no ra­tion­al points.

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 \( aX + b \) (with \( a \) and \( b \) re­l­at­ively prime) “con­tains” in­fin­itely many prime num­bers. Di­rich­let did this by the in­tro­duc­tion of \( L \)-func­tions as a bridge between the ana­lys­is (that was geared to count­ing dens­it­ies) and the arith­met­ic (that deals with, for ex­ample, prime num­bers). This fits in­to the amaz­ing chapter of ana­lyt­ic num­ber the­ory re­lated to what are called ana­lyt­ic for­mu­las which broadened to in­clude sweep­ing con­jec­tures (e.g., the Birch–Swin­ner­ton–Dyer Con­jec­ture, the Bloch–Kato Con­jec­ture, and the con­jec­tures of Beil­in­son) that con­nect arith­met­ic­ally im­port­ant quant­it­ies to ana­lyt­ic com­pu­ta­tions through the me­di­um of “\( L \)-func­tions.” Now these re­main, largely, con­jec­tures.

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 \( L \)-func­tion, or of its de­riv­at­ive, and the struc­ture of ex­tremely im­port­ant and del­ic­ate arith­met­ic (the di­o­phant­ine be­ha­vi­or of el­lipt­ic curves over the ra­tion­al num­ber field). In fact, this the­or­em is in­stru­ment­al in the ac­tu­al proof of (a piece of) the Birch–Swin­ner­ton–Dyer Con­jec­ture in cer­tain con­texts. But even more broadly, the Gross–Za­gi­er the­or­em holds something of a stra­tegic po­s­i­tion in the sub­ject and there­fore it sets the stage for — and has ap­plic­a­tions in — new dir­ec­tions. For ex­ample:

• the \( p \)-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.