Celebratio Mathematica

That first meet­ing las­ted sev­er­al hours (in the even­ing Hirzebruch had to go home, but Brieskorn in­vited me to a Chinese res­taur­ant to con­tin­ue the dis­cus­sion) and res­ul­ted in new re­search pro­jects for me and in­vit­a­tions to come to Bonn a month later for my first Arbeit­sta­gung (also mem­or­able!) and again in the fall as Hirzebruch’s doc­tor­al stu­dent. (I re­mained im­ma­tric­u­lated in Ox­ford, and Hirzebruch re­ceived a salary of £5 a year for his work.) As my ad­visor, he met me fre­quently, listened to my re­ports with great at­ten­tion, and made such min­im­al­ist­ic com­ments that I al­ways felt the new ideas that emerged were my own, al­though I did some­times won­der why everything was work­ing out so much bet­ter than it ever had be­fore.

My ac­tu­al thes­is was on a some­what dif­fer­ent sub­ject from the co­tan­gent sums that had provided the ini­tial con­tact with Hirzebruch, but dur­ing the two years that I spent in Bonn as his stu­dent and Stu­dentische Hil­f­skraft, we also had many more dis­cus­sions about those things, and he gave a course on the sub­ject which turned in­to our joint book [3] on re­la­tions between in­dex the­or­ems and ele­ment­ary num­ber the­ory. One of the top­ics treated in that book, the cal­cu­la­tion of in­vari­ants of tor­us bundles over the circle, was to lead him later to his beau­ti­ful dis­cov­er­ies, dis­cussed be­low, on the geo­metry of Hil­bert mod­u­lar sur­faces. Some of this work and of Hirzebruch’s own work in this area is beau­ti­fully told in his art­icle [1], whose in­tro­duc­tion ends with the words

In the second half of this lec­ture we shall point out some rather ele­ment­ary con­nec­tions to num­ber the­ory ob­tained by study­ing the equivari­ant sig­na­ture the­or­em for four-di­men­sion­al man­i­folds. Per­haps these con­nec­tions still be­long to re­cre­ation­al math­em­at­ics be­cause no deep­er ex­plan­a­tion, for ex­ample of the oc­cur­rence of Dede­kind sums both in the the­ory of mod­u­lar forms and in the study of four-di­men­sion­al man­i­folds, is known. As a theme (fa­mil­i­ar to most to­po­lo­gists) un­der the gen­er­al title “Pro­spects of math­em­at­ics” we pro­pose “More and more num­ber the­ory in to­po­logy.”

As we will see, these last words were to be proph­et­ic in his own case.

I had in­ten­ded to come to Bonn only for the time needed to com­plete my thes­is, but ended up stay­ing there for my whole life. This de­vel­op­ment, which I could nev­er have ima­gined (not only be­cause I knew no Ger­man when I came and had no re­la­tion­ship with the coun­try, but also be­cause I am half Jew­ish and much of my fath­er’s fam­ily had per­ished in the con­cen­tra­tion camps), was due ex­clus­ively to Hirzebruch’s tre­mend­ous per­son­al­ity and to the at­mo­sphere that he cre­ated. In the first peri­od after my thes­is, I began work­ing more and more in­tens­ively with him, first on co­tan­gent sum-re­lated top­ics and then on Hil­bert mod­u­lar sur­faces. Part of this col­lab­or­a­tion took place on long train trips to Zürich, where he was giv­ing a course on the lat­ter sub­ject and where I reg­u­larly ac­com­pan­ied him be­cause it was the only chance to get him to my­self for long peri­ods at a time. In the even­ings we of­ten ate to­geth­er at the el­eg­ant Zun­fthäuser (guild halls turned in­to res­taur­ants) of Zürich, gradu­ally be­com­ing bet­ter friends and in­creas­ing our al­co­hol con­sump­tion from a mod­est single glass each at the be­gin­ning to a full bottle. On one oc­ca­sion this was in­creased to one and a half bottles, and Pro­fess­or Hirzebruch form­ally pro­posed the use of “Du” and first names. Hence­forth he was al­ways “Fritz” to me, and so he shall re­main for the rest of this art­icle. Dur­ing these years I also got to know his fam­ily well, and this too made Bonn be­come a true home. His daugh­ters, Bar­bara and Ul­rike, also at­ten­ded my course on ele­ment­ary num­ber the­ory. Both had real math­em­at­ic­al tal­ent, but in the end neither one op­ted for a re­search ca­reer, though Ul­rike wrote a mas­ter’s thes­is on el­lipt­ic sur­faces with three ex­cep­tion­al fibers that is still quoted reg­u­larly today.

Fritz had already done earli­er work that is im­port­ant in the the­ory of al­geb­ra­ic and arith­met­ic groups, most not­ably his fun­da­ment­al pa­pers with Ar­mand Borel about ho­mo­gen­eous spaces (in par­tic­u­lar, the de­term­in­a­tion of their char­ac­ter­ist­ic classes) and his pro­por­tion­al­ity prin­ciple, which has proved enorm­ously im­port­ant in the the­ory of auto­morph­ic forms. But start­ing around 1970 his in­terest in the re­la­tions between to­po­logy and num­ber the­ory be­came much more in­tense and led to what one might call a second spring in his math­em­at­ic­al re­search ca­reer. The high point of this was his work on Hil­bert mod­u­lar sur­faces, which I now briefly de­scribe.

In the clas­sic­al the­ory of mod­u­lar forms a cru­cial role is played by the mod­u­lar curve \[ \mathfrak{H}/\operatorname{SL}(2, \mathbb{Z}) \quad (\mathfrak{H} = \text{complex upper half-plane}) \] and its cous­ins. The high­er-di­men­sion­al gen­er­al­iz­a­tion of this curve is the Hil­bert mod­u­lar vari­ety \( \mathfrak{H}^n /\operatorname{SL}(2, \mathcal{O}_k ) \) as­so­ci­ated to a totally real num­ber field \( K \). Here \( \mathcal{O}_K \) is the ring of in­tegers of \( K \) and \( \operatorname{SL}(2, \mathcal{O}_K ) \) is the Hil­bert mod­u­lar group, em­bed­ded in­to \( \operatorname{SL}(2, \mathbb{R})^n \) by the n dif­fer­ent real em­bed­dings of \( K \) and hence act­ing nat­ur­ally (and dis­cretely) on \( \mathfrak{H}^n \). This vari­ety can be com­pac­ti­fied by adding “cusps” to give a pro­ject­ive al­geb­ra­ic vari­ety \( X_K \), but these cusps are highly sin­gu­lar points, with the bound­ary of a small neigh­bor­hood of each cusp be­ing a \( T^n \)-bundle over \( T^{n-1} \) rather than a \( (2n - 1) \)-di­men­sion­al sphere. In par­tic­u­lar, for \( n = 2 \) these neigh­bor­hood bound­ar­ies are pre­cisely the tor­us bundles over a circle that Fritz had already been study­ing in con­nec­tion with the equivari­ant in­dex the­or­em, and it was this that led him to the study of Hil­bert mod­u­lar sur­faces.

He set him­self three main goals:

1. to de­scribe the geo­metry of \( X_K \) and cal­cu­late its nu­mer­ic­al in­vari­ants,
2. to give for \( n = 2 \) the res­ol­u­tion of the sin­gu­lar­it­ies at the cusps, and
3. to ap­ply this to the clas­si­fic­a­tion of \( X_K \) in the sense of the the­ory of al­geb­ra­ic sur­faces.

He achieved these goals in a series of pa­pers pub­lished between 1970 and 1980, par­tially in col­lab­or­a­tion with A. van de Ven and me in the case of part (iii). Each part was math­em­at­ics of the highest or­der. The cal­cu­la­tions of the nu­mer­ic­al in­vari­ants in­volved deep res­ults from both dif­fer­en­tial geo­metry and num­ber the­ory, in­clud­ing Günter Harder’s ex­ten­sion of the clas­sic­al Gauss-Bon­net the­or­em to non­com­pact man­i­folds like Hil­bert vari­et­ies and clas­sic­al res­ults of Hecke, Siegel, and Curt Mey­er about Dede­kind zeta func­tions and class num­bers of num­ber fields and their re­la­tion­ship to co­tan­gent sums. The res­ol­u­tion of the sin­gu­lar­it­ies in terms of peri­od­ic con­tin­ued frac­tions was an amaz­ingly beau­ti­ful res­ult in it­self and also spawned many gen­er­al­iz­a­tions, in­clud­ing the the­ory of tor­oid­al com­pac­ti­fic­a­tions (work of Mum­ford, Falt­ings, and many oth­ers) that now plays a cent­ral role in the the­ory of mir­ror sym­metry. The res­ults in part (iii), which cul­min­ated in the com­plete de­term­in­a­tion of the po­s­i­tion of the Hil­bert mod­u­lar sur­faces with­in the Kodaira clas­si­fic­a­tion, provided a beau­ti­ful col­lec­tion of al­geb­ra­ic sur­faces hav­ing par­tic­u­larly in­ter­est­ing prop­er­ties be­cause of the in­ter­play between their tran­scend­ent­al as­pects (de­scrip­tion as quo­tients of \( \mathfrak{H}^2 \)) and their al­geb­ra­ic as­pects (de­scrip­tion as pro­ject­ive vari­et­ies). This in­ter­play leads to many in­sights that are not avail­able for vari­et­ies pos­sess­ing “merely” an al­geb­ra­ic de­scrip­tion. All as­pects of the the­ory are de­scribed in the mas­ter­ful ex­pos­i­tion [2].

Fritz’s in­vest­ig­a­tion of the geo­metry of the Hil­bert mod­u­lar sur­faces led him to an in­tens­ive study of the mod­u­lar curves \( T_N (N \in \mathbb{N}) \) that are nat­ur­ally em­bed­ded in these sur­faces. This led to a joint pa­per with me [4] show­ing that the gen­er­at­ing func­tion \( \sum_N[T_n]q^N \) of the classes of these curves in the second ho­mo­logy group of the sur­face is it­self a mod­u­lar form in one vari­able, a res­ult that in turn has giv­en rise to many later ap­plic­a­tions and gen­er­al­iz­a­tions (work of Kudla–Mill­son and many oth­ers). There is an­oth­er amus­ing an­ec­dote con­nec­ted with this. Serre, who had stud­ied Fritz’s work on the to­po­lo­gic­al in­vari­ants of Hil­bert mod­u­lar sur­faces, wrote him a let­ter point­ing out a co­in­cid­ence between the num­bers oc­cur­ring here and the for­mu­las for the di­men­sions of cer­tain spaces of mod­u­lar forms. His let­ter and Fritz’s giv­ing the ex­plan­a­tion in terms of our mod­u­lar­ity res­ult crossed in the mail, a nice ex­ample of a ques­tion be­ing answered be­fore it is re­ceived. I should per­haps also men­tion that this col­lab­or­a­tion was one of the most ex­cit­ing math­em­at­ic­al events of my own life and, I think, meant a lot to Fritz too. On the day when we sent off the fi­nal manuscript, we cel­eb­rated with a din­ner to­geth­er with our fam­il­ies in a fancy res­taur­ant at which Bar­bara fam­ously re­acted to the bill by com­put­ing how many por­tions of French fries she could buy with the same money.

In later years Fritz worked on many oth­er top­ics at the in­ter­face between num­ber the­ory and to­po­logy that for lack of space I will not de­scribe in de­tail. A prime ex­ample arose in the late 1980s when Och­an­ine and Wit­ten in­tro­duced el­lipt­ic gen­era, which at­tach mod­u­lar forms to man­i­folds. Not sur­pris­ingly, Fritz was very in­ter­ested in this de­vel­op­ment and wrote some beau­ti­ful pa­pers and a book [7] (joint with his stu­dents Th. Ber­ger and R. Jung) on this top­ic. Oth­er top­ics in­cluded the study of Fuch­sian dif­fer­en­tial equa­tions (alone and in col­lab­or­a­tion with Paula Co­hen) al­luded to above and his really beau­ti­ful work ap­ply­ing the Miyaoka–Yau in­equal­ity and oth­er deep res­ults about char­ac­ter­ist­ic classes of sur­faces to clas­sic­al ques­tions go­ing back to Sylvester (1893) about con­fig­ur­a­tions of points and lines in the plane [5].