Celebratio Mathematica

For the last 27 years, I have been very for­tu­nate to have Be­ne­dict Gross as a ment­or. In this es­say, I will de­scribe how his pa­pers, our let­ters, and our dis­cus­sions have in­flu­enced my own de­vel­op­ment and ca­reer.

I first heard the name Be­ne­dict Gross in a lec­ture by Dori­an Gold­feld at the Chinese Academy of Sci­ences in the sum­mer of 1985, when I was a mas­ter’s de­gree stu­dent. The fol­low­ing year, Dori­an sug­ges­ted prov­ing ex­ten­sions of the Gross–Za­gi­er for­mula [2] for my Ph.D. thes­is prob­lem at Columbia. I looked at the Gross–Za­gi­er pa­per briefly, but I could not really un­der­stand much of its proof. Ini­tially, I was un­con­vinced that their ap­proach was the right dir­ec­tion be­cause the main res­ult only ap­plied to mod­u­lar el­lipt­ic curves; I had stud­ied arith­met­ic geo­metry and heights un­der Szpiro and Falt­ings un­til An­drew Wiles un­veiled his strategy for prov­ing the mod­u­lar­ity con­jec­ture in 1993. Of course, I was con­vinced then that this was the right dir­ec­tion and a po­ten­tial area where the the­ory of heights could be ap­plied.

Un­like the pa­pers in arith­met­ic geo­metry where gen­er­al­ity is usu­ally the first pri­or­ity, Gross and Za­gi­er re­duced their for­mula to the mi­ra­cu­lous equal­ity \( F_G=F_A \) of two mod­u­lar forms for \( \Gamma_0(N) \) of weight 2, as­so­ci­ated to an ideal class char­ac­ter \( \chi \) for an ima­gin­ary quad­rat­ic field \( K \), with \( F_G \) con­struc­ted from the “geo­metry” of mod­u­lar curves, and \( F_A \) con­struc­ted from the “ana­lys­is” of mod­u­lar forms. This left a lot of room for gen­er­al­iz­a­tion, as one can re­place \( \Gamma_0(N) \) by \( \Gamma_1(N) \) and al­low \( \chi \) be a gen­er­al Hecke char­ac­ter. In fact, at the end of their pa­per [2], there are two sug­ges­ted gen­er­al­iz­a­tions: for Shimura curves (when a Hee­gn­er con­di­tion fails) and for high­er weight mod­u­lar forms. At the end of 1994, I wrote two emails to Dick Gross ask­ing about new pro­gress to­wards these gen­er­al­iz­a­tions.

On Janu­ary 3, 1995, he sent me a let­ter [4] via mail an­swer­ing all of my ques­tions about high­er-weight mod­u­lar forms. He said that Bryl­in­ski’s pa­per in DMJ [e2], about loc­al heights of Hee­gn­er cycles defined by De­ligne, was the only work on this top­ic. So what re­mained was to de­vel­op a the­ory of glob­al heights. On Janu­ary 4, 1995, he sent me an­oth­er let­ter via email [5] an­swer­ing all of my ques­tions about Shimura curves. He made a point that a Shimura curve over a totally real field is ca­non­ic­ally as­so­ci­ated to an odd set of places where the curve has ana­lyt­ic uni­form­iz­a­tions by the work of Shimura, Doi–Naganuma, and Cered­nik–Drin­feld. These two let­ters were the be­gin­ning of his ment­or­ship of my aca­dem­ic life, in which he has giv­en me im­port­ant ad­vice and warm en­cour­age­ment at vari­ous turn­ing points of my re­search, al­ways shar­ing with me his new ideas and in­sights.

With so much en­cour­age­ment from Dick’s let­ters, I im­me­di­ately star­ted to work on these two sug­ges­ted gen­er­al­iz­a­tions for high­er-weight mod­u­lar forms and Shimura curves. At the end of 1995, I worked out the sug­ges­ted Gross–Za­gi­er for­mula for high­er-weight mod­u­lar forms [e3]. The modi­fic­a­tion that was needed there was a glob­al defin­i­tion of heights for Hee­gn­er cycles us­ing Gil­let–Soulé arith­met­ic in­ter­sec­tion the­ory on Kuga–Sato vari­et­ies in­stead of loc­al sys­tems. In the sum­mer of 1997, I also proved the sug­ges­ted Gross–Za­gi­er for­mula over Shimura curves [e4]. The modi­fic­a­tion that was needed there was to re­place cusps by Hodge classes, which re­quired the de­vel­op­ment of a mul­ti­pli­city-one ar­gu­ment for de­riv­a­tions of Hecke op­er­at­ors. In these two pa­pers, the new ideas were height com­pu­ta­tions that I had learned earli­er in Arakelov the­ory. The ana­lyt­ic parts were com­pletely copied from Gross and Za­gi­er’s pa­per.

Fol­low­ing these two nice “warm-up ex­er­cises”, I thought about a fur­ther ex­ten­sion of the Gross–Za­gi­er for­mula for­mu­lated in Dick’s pa­per [1] for gen­er­al an­ti­cyl­o­tom­ic char­ac­ters \( \chi \). A new modi­fic­a­tion is that one has to work on level \( N\cdot\operatorname{cond}(\operatorname{Ind}^{\mathbb{Q}}_K(\chi)) \) dir­ectly, where one can only prove an “al­most-equal­ity” \( F_G\approx F_A \) of two mod­u­lar forms, or some kind of “arith­met­ic fun­da­ment­al lemma” in today’s lan­guage. To de­duce the pre­cise iden­tity from this al­most-equal­ity of mod­u­lar forms, one needs new mul­ti­pli­city-one ar­gu­ments. After he heard about my plan dur­ing his vis­it to Columbia, Dick sent me a let­ter [7] (May 14, 1998) with some pa­pers, in­clud­ing his work with Prasad about test vec­tors. With these test vec­tors, I could prove a Gross–Za­gi­er for­mula in this set­ting by de­vel­op­ing the so-called tor­ic new­form the­ory in the sum­mer of 2001 [e5]. With the com­ple­tion of these three pa­pers with long and te­di­ous com­pu­ta­tions, the next step was to con­sider two nat­ur­al ques­tions for a con­cep­tion­al un­der­stand­ing of the Gross–Za­gi­er for­mula: Why should the Gross–Za­gi­er for­mula be true? How does one ex­tend the Gross–Za­gi­er for­mula to high­er-di­men­sion­al Shimura vari­et­ies?

For the first ques­tion, in Decem­ber 2001 at an MSRI work­shop, Dick [8] answered that the Gross–Za­gi­er for­mula should be an equal­ity \( P_G=P_A \) of two lin­ear func­tion­als in the one-di­men­sion­al vec­tor space \( \operatorname{Hom}_{\mathbb{A}_K^\times} (\pi\otimes \chi, \mathbb{C}) \), where \( \pi \) is the auto­morph­ic rep­res­ent­a­tion gen­er­ated by mod­u­lar form \( f \). In fact, this point of view was already used by Wald­spur­ger [e1] in his tor­ic peri­od in­teg­ral for­mula for the cent­ral value \( L(\pi, \chi, 1/2) \) of the Rankin–Sel­berg \( L \)-func­tion. Un­for­tu­nately, only spe­cial cases of Wald­spur­ger’s for­mula had been cited by oth­ers, prob­ably be­cause the gen­er­al for­mula is stated in terms of lin­ear func­tion­als as Pro­pos­i­tion 7 on page 50 of a 70-page pa­per. Fol­low­ing Dick’s frame­work [8], Xinyi Yuan, Wei Zhang, and I were able to give a proof of a Gross–Za­gi­er for­mula [e8] in full gen­er­al­ity, com­bin­ing the strategy of Wald­spur­ger [e1] with some “arith­met­ic” modi­fic­a­tion such as the in­co­her­ent qua­ternion al­geb­ras (a re­flec­tion of Dick’s point of view that pro­ject­ive sys­tems of Shimura curves are de­term­ined by odd sets of places).

For the second ques­tion, Dick ex­plained his joint work with S. Kudla [3] and C. Schoen [6] to me. As an­oth­er “warm-up ex­er­cise”, I star­ted to do some com­pu­ta­tions for the heights of Gross–Schoen cycles for the triple product of a gen­er­al curve over a num­ber field. In [e6], I was able to prove an ex­pres­sion of this height in terms of the self-in­ter­sec­tion of du­al­iz­ing sheaves. This iden­tity was then used to prove the uni­form Bogo­mo­lov con­jec­ture and the uni­form Mor­dell–Lang con­jec­ture by Z. Cinkir, R. de Jong, X. Yuan, et al. Then for the triple product of Shimura curves, Xinyi Yuan, Wei Zhang and I could for­mu­late a Gross–Kudla con­jec­ture in full gen­er­al­ity and prove it in cer­tain cases [e7]. What about the Gross–Za­gi­er for­mula for oth­er Shimura vari­et­ies? In a hotel lobby in Beijing in Decem­ber of 2007, Dick ex­plained in a note [9] to me his new con­jec­tures [10] with Wee Teck Gan and Di­pen­dra Prasad about the arith­met­ic di­ag­on­al cycles for Shimura vari­et­ies at­tached to \( U(n-1,1)\times U(n,1) \) or \( O(n-1, 2)\times O(n, 2) \). I then or­gan­ized a work­shop at the Chinese Academy of Sci­ences with my stu­dents to study these con­jec­tures and the re­l­at­ive trace for­mula ap­proach by Jac­quet–Ral­lis. So far, the young­er gen­er­a­tion, in­clud­ing Wei Zhang, Yifeng Liu, Hang Xue, and Chao Li, has made a lot of im­port­ant de­cis­ive res­ults for these con­jec­tures. I my­self have fo­cused on re­lated geo­met­ric ques­tions in Arakelov the­ory: the con­struc­tions and com­pu­ta­tions of Beil­in­son–Bloch heights. For ex­ample, just re­cently, I was able to ex­tend Gross–Schoen’s work to the product of a curve and a sur­face [e9]. Even for this small pa­per, I have already re­ceived warm en­cour­age­ment from Dick. It is just an­oth­er ex­ample among many of his gen­er­ous sup­port and ment­or­ship, for which I am very grate­ful.

Shou-Wu Zhang is Eu­gene Hig­gins Pro­fess­or of Math­em­at­ics at Prin­ceton Uni­versity. He is known for his work on the Bogo­mo­lov con­jec­ture and the Gross–Za­gi­er for­mula.