Celebratio Mathematica

Barry Mazur has had a long­stand­ing in­terest in ques­tions of de­cid­ab­il­ity in num­ber the­ory, con­tinu­ing a long tra­di­tion go­ing back at least to Hil­bert. This has guided him to in­ter­est­ing new ques­tions and the­or­ems in arith­met­ic geo­metry.

Hil­bert’s tenth prob­lem, from the list of 23 prob­lems he pub­lished after a fam­ous lec­ture in 1900, asked for an al­gorithm to de­cide, giv­en a mul­tivari­able poly­no­mi­al equa­tion with in­teger coef­fi­cients, wheth­er it has a solu­tion in in­tegers. In 1970, Matiy­a­sevich [e2], build­ing on work of Dav­is, Put­nam, and Robin­son [e1], proved that no such al­gorithm ex­ists; Mazur him­self wrote a beau­ti­ful sur­vey art­icle on this [2].

But if the ques­tion is changed by re­pla­cing \( \mathbb{Z} \) by \( \mathbb{Q} \), it is not known wheth­er an al­gorithm ex­ists. One pos­sible ap­proach: If \( \mathbb{Z} \) is di­o­phant­ine over \( \mathbb{Q} \) (that is, there is a \( \mathbb{Q} \)-vari­ety \( X \) with a morph­ism to \( \mathbb{A}^1 \) such that the im­age of \( X(\mathbb{Q}) \to \mathbb{A}^1(\mathbb{Q}) = \mathbb{Q} \) is \( \mathbb{Z} \)), then the neg­at­ive an­swer for \( \mathbb{Z} \) im­plies a neg­at­ive an­swer for \( \mathbb{Q} \).

The sug­ges­tion that \( \mathbb{Z} \) might be di­o­phant­ine over \( \mathbb{Q} \) led Mazur, in a con­trari­an mood, to pose a new type of ques­tion, about “to­po­logy of ra­tion­al points”: he asked wheth­er, for every vari­ety \( X \) over \( \mathbb{Q} \), the clos­ure of \( X(\mathbb{Q}) \) in \( X(\mathbb{R}) \) (with re­spect to the Eu­c­lidean to­po­logy) has at most fi­nitely many con­nec­ted com­pon­ents ([1], Con­jec­ture 3). Mazur ob­served that a pos­it­ive an­swer to this ques­tion would im­ply that \( \mathbb{Z} \) is not di­o­phant­ine over \( \mathbb{Q} \) and hence would rule out one ap­proach to a neg­at­ive an­swer to Hil­bert’s tenth prob­lem over \( \mathbb{Q} \). There has been little pro­gress on Mazur’s ques­tion bey­ond the spe­cial cases that Mazur him­self proved in [1]. But this is not for a lack of in­terest! Rather, it is a test­a­ment to the dif­fi­culty of un­der­stand­ing ra­tion­al points on high­er-di­men­sion­al vari­et­ies. Mazur also asked about stronger ver­sions of his ques­tion: for ex­ample, he asked wheth­er, for a smooth vari­ety \( X \) in which \( X(\mathbb{Q}) \) is Za­r­iski dense, the clos­ure of \( X(\mathbb{Q}) \) must be a uni­on of con­nec­ted com­pon­ents of \( X(\mathbb{R}) \) ([1], Con­jec­ture 1). The an­swer to this turned out to be no ([e9], Sec­tion 5).

In a dif­fer­ent dir­ec­tion, Denef and Lip­shitz con­jec­tured in 1978 that for every num­ber field \( K \), Hil­bert’s tenth prob­lem over the ring of in­tegers \( \mathcal{O}_K \) has a neg­at­ive an­swer [e4]. This was proved in the 1970s and 1980s for sev­er­al classes of num­ber fields, by us­ing unit groups [e3], [e5], [e4], [e6], [e7]; build­ing on these meth­ods, Mazur, Ru­bin, and Sh­lapen­tokh re­cently con­struc­ted a uni­form defin­i­tion of \( \mathbb{Z} \) in the rings of in­tegers of many in­fin­ite al­geb­ra­ic ex­ten­sions of \( \mathbb{Q} \), and answered a 1948 ques­tion of Tarski by prov­ing that the first-or­der the­ory of the field of con­struct­ible num­bers is un­de­cid­able [10]. On the oth­er hand, vari­ous au­thors ob­served that el­lipt­ic curves could be used in place of unit groups [e11], [e10], [e12]. In par­tic­u­lar, for an ex­ten­sion of num­ber fields \( L/K \), if there ex­ists an el­lipt­ic curve \( E \) over \( K \) with \( 0 < \operatorname{rank} E(K) = \operatorname{rank} E(L) \), then \( \mathcal{O}_K \) is di­o­phant­ine over \( \mathcal{O}_L \), so a neg­at­ive an­swer to Hil­bert’s tenth prob­lem for \( \mathcal{O}_K \) im­plies a neg­at­ive an­swer for \( \mathcal{O}_L \); one might hope to reach every num­ber field in­side a tower of such ex­ten­sions.

Mazur and Ru­bin then began to in­vest­ig­ate this phe­nomen­on of “di­o­phant­ine sta­bil­ity”. In par­tic­u­lar, us­ing an ap­proach in­spired by [e8] and de­veloped fur­ther in [3], Mazur and Ru­bin gave a con­di­tion­al proof that for every prime-de­gree cyc­lic ex­ten­sion of num­ber fields \( L/K \), such an el­lipt­ic curve ex­ists ([4], The­or­em 1.13). This was enough to yield a con­di­tion­al proof that for every num­ber field \( K \), Hil­bert’s tenth prob­lem over \( \mathcal{O}_K \) has a neg­at­ive an­swer [4]. The res­ults were con­di­tion­al on the fi­nite­ness of Tate–Sha­far­ev­ich groups, but this fi­nite­ness was widely be­lieved to hold, so the Mazur–Ru­bin res­ult was strong evid­ence for the Denef–Lip­shitz con­jec­ture. The strategy of Mazur and Ru­bin in­volved a del­ic­ate ana­lys­is of how Selmer groups change in a fam­ily of quad­rat­ic twists: for many el­lipt­ic curves over num­ber fields, they showed that by twist­ing care­fully, they could make the Selmer rank go up or down. By it­er­at­ing this, they proved that over every num­ber field \( K \), there is an el­lipt­ic curve of rank 0, and they could also force rank 1 over \( K \) and \( L \) sim­ul­tan­eously, as­sum­ing fi­nite­ness of Tate–Sha­far­ev­ich groups (used to en­sure par­ity, to pre­vent the rank over \( K \) from drop­ping to 0). In later joint work with Klags­brun, they also used these ideas to­wards res­ults in arith­met­ic stat­ist­ics, con­cern­ing the dis­tri­bu­tion of 2-Selmer ranks in a fam­ily of quad­rat­ic twists of an el­lipt­ic curve [5], [6]. Mazur and Ru­bin re­turned to ques­tions of di­o­phant­ine sta­bil­ity in [7], [8]; the former art­icle pro­duced new fam­il­ies of field ex­ten­sions \( L/K \) and \( K \)-vari­et­ies \( V \) such that \( V(K)=V(L) \), with ap­plic­a­tions to new un­con­di­tion­al cases of Hil­bert’s tenth prob­lem over rings of in­tegers, and ap­plic­a­tions to de­term­in­ing the pos­sible field ex­ten­sions gen­er­ated by a vary­ing al­geb­ra­ic point on a giv­en vari­ety.

Koy­mans and Pa­gano found a way to use a the­or­em of ad­dit­ive com­bin­at­or­ics to con­struct rank-stable el­lipt­ic curves for enough ex­ten­sions to prove a neg­at­ive an­swer to Hil­bert’s tenth prob­lem over \( \mathcal{O}_K \) for every num­ber field \( K \) [e13]. About two months later, Alpöge, Bhar­gava, Ho, and Shnid­man, pos­ted a second ar­gu­ment, us­ing a less soph­ist­ic­ated in­put from ad­dit­ive com­bin­at­or­ics but con­struct­ing only an abeli­an vari­ety ex­hib­it­ing rank sta­bil­ity, not an el­lipt­ic curve [e14]. For­tu­nately for them, Mazur, Ru­bin, and Sh­lapen­tokh had just proved what they needed, that a rank-stable abeli­an vari­ety would do just as well as a rank-stable el­lipt­ic curve for prov­ing the neg­at­ive an­swer to Hil­bert’s tenth prob­lem for rings of in­tegers [9]!

Bjorn Poon­en, Dis­tin­guished Pro­fess­or in Sci­ence at MIT, is a mem­ber of the Amer­ic­an Academy of Arts of Sci­ences. His book “Ra­tion­al Points on Vari­et­ies” was awar­ded the 2023 Joseph L. Doob Prize. Twenty-sev­en math­em­aticians have re­ceived a Ph.D. un­der his su­per­vi­sion.