Jean-Marc Fontaine (1944–2019) Pierre Berthelot, Luc Illusie, Nicholas M

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Jean-Marc Fontaine (1944–2019) Pierre Berthelot, Luc Illusie, Nicholas M. Katz, William Messing, and Peter Scholze

Jean-Marc Fontaine passed away on January 29, 2019. For half a century, and up until his very last days, he devoted his immense talent and energy to the development of one of the most powerful tools now available in number theory and arithmetic geometry, 푝-adic Hodge theory. His work has had and will continue to have a tremendous influence. It has many deep applications. With the following texts, we would like to express our admiration, gratitude, and friendship. Jean-Marc was born on March 13, 1944, in Boulogne- Billancourt, a city located in the southwest suburbs of Paris. Boulogne-Billancourt was then mainly known as hosting one of the biggest car plants in France, and as a labor union stronghold. However, Jean-Marc’s family background was different. Close to the Christian Democratic movement, his father, Andr´eFontaine, was a very influential journal- Figure 1. Jean-Marc Fontaine during a visit to China, standing ist and historian, specialist on the Cold War. He worked next to a modern copy of the “Lantingji Xu”. from 1947 to 1991 for the well-known daily newspaper Le Monde, where he served in particular as editor-in-chief In just a few years, Schwartz succeeded in creating an active (1969–85) and director (1985–91). mathematical research center, and became very influential Jean-Marc himself had a natural taste for intellectual de- among the students who wanted to specialize in mathe- bates, and very independent judgement. In most cases, his matics. It is no surprise that many chose a thesis topic point of view was original, sometimes almost provocative. in analysis. However, Jean-Marc became more interested This is already apparent in the choices he made when he in number theory, and it is by attending the lectures of was a student at Ecole´ Polytechnique, from 1962 to 1965. Charles Pisot that he found his path towards mathemati- Indeed, high-level teaching and research in mathematics cal research. were still somewhat undeveloped at Ecole´ Polytechnique Pisot was Jean-Marc’s advisor for two years. In Decem- in the late fifties. However this began to change with the ber 1967, Jean-Marc gave a lecture at the Delange–Pisot– appointment of Laurent Schwartz as a professor in 1958. Poitou seminar on ramification for local field extensions [1]. Jean-Pierre Serre was very positively impressed. He Communicated by Notices Associate Editor William McCallum. got in touch with Jean-Marc, and, as a thesis topic, he pro- For permission to reprint this article, please contact: posed his conjecture on the rationality of Artin represen- [email protected]. tations of local fields. Jean-Marc proved the conjecture DOI: https://doi.org/10.1090/noti2118 pretty rapidly and defended his thesis in 1972 [2]. Then,

1010 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 with the encouragement of Serre, he began to develop a how to think about such problems. Further development program for a systematic study of 푝-adic representations of his ideas led to more general rings of this type, which of Galois groups of local fields, and this is where the story are some of the fundamentals of 푝-adic Hodge theory, but of Fontaine rings begins. this was the starting point. I refer to Illusie’s piece for his keen insight into Jean- Marc as a colleague and as a courageous human being. It Jean-Marc Fontaine was an honor to have known Jean-Marc. He will be sorely missed. Nicholas M. Katz I first met Jean-Marc Fontaine and his family in the early 1970s. Our first children were born the same year (1974), and they remain friends to this day. In the following years we made several family visits to them in Grenoble, and had many dinners together when we were at IHES and they were in Paris. Let me say a few words about some of the mathematics we owe to Jean-Marc. His work is described in detail in the articles of Berthelot and Scholze, but I want to give Nicholas M. Katz a “baby” version of one part, and a reminiscence of the impact it had. From the early 1970s, Jean-Marc was fasci- Remembering Jean-Marc nated by Grothendieck’s “mysterious functor.” Recall the background of this. Tate in 1966 had shown that a 푝-divisible group Γ over Luc Illusie the ring of integers, (e.g., ℤ푝) of a local field (e.g., ℚ푝) I am not sure when I met Jean-Marc for the first time. It was was determined by its Galois representation (the action of probably at his 1972 thesis defense, which was directed by 퐺푎푙(ℚ푝/ℚ푝) on the Tate module formed from the 푝-power Serre, about the rationality of Artin representations. That

torsion points of Γ(ℚ푝)). On the other hand, Messing, car- summer we both attended the Antwerp summer school on rying out an idea of Grothendieck, had shown that for 푝 > modular forms, and two years later we were both at the 2, such a Γ was determined by a filtration of the Dieudonné AMS Summer Institute on Algebraic Geometry, in Arcata. I also attended his Cours Peccot at Collège de France in module of the special fibre Γ픽푝 , a 푝-divisible group over 픽푝. The “mysterious functor” was to pass between the Galois 1975. After his thesis, Jean-Marc took a position at Greno- representation and the filtered Dieudonnémodule. The ble. In 1988, Jean-Marc moved to Orsay, where he was a mystery was how to do this without invoking Γ. When, for professor until his retirement in 2009, when he became example, Γ is the 푝-divisible group of an abelian variety emeritus. 푝 퐴/ℚ푝 with good reduction (i.e., an abelian scheme 픸/ℤ푝), His Cours Peccot was about the classification of - then the Galois representation is the 푝-adic étalecohomol- divisible groups over local fields. He was fascinated by 1 questions of Grothendieck about 푝-divisible groups and ogy group 퐻et́ of 퐴ℚ , while the Dieudonnémodule is the 푝 the “mysterious functor” relating de Rham cohomology de Rham cohomology group 퐻1 of 픸/ℤ . So more gen- dR 푝 and 푝-adic cohomology of varieties over local fields. At erally, the “mysterious functor” was to pass between the that time, no one had the slightest idea of how to proceed, 푝-adic étalecohomology and the filtered de Rham coho- or even how to formulate precise conjectures. Jean-Marc mology of (eventually quite general) varieties over local fields of mixed characteristic (0, 푝). Luc Illusie is a retired professor of mathematics at UniversitéParis-Saclay, Or- At the Journéesde GéométrieAlgébriquede Rennes in say, France. His email address is [email protected]. 1978, Jean-Marc astounded all of us by introducing his This text is the translation by Nicholas M. Katz of a slightly edited version of “Barsotti-Tate rings,” which, he showed, realized the “mys- the eulogy delivered by the author at Jean-Marc Fontaine’s funeral on Febru- terious functor” in the case of 푝-divisible groups. This not ary 5, 2019, which appeared in the October 2019 issue of La Gazette des only solved an existing problem, but was a revolution in Mathématiciens, and is translated and published here by permission. The author is very grateful to Nick Katz for the translation of his piece, and thanks Nicholas M. Katz is a professor of mathematics at Princeton University, Prince- the editors of the Gazette for granting the Notices permission to include this ton, New Jersey. His email address is [email protected]. translation.

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1011 Périodes 푝-adiques, Astérisque223, which grew out of the seminar, remains today a basic reference for 푝-adic Hodge theory. The 1990s were also rich in new developments: work of Jean-Marc and Perrin-Riou on Bloch–Kato, formulation of the Fontaine–Mazur conjecture, a “푝-adic semester” at IHP in spring 1997 co-organized with Berthelot, Kato, Rapoport, and me and devoted in particular to work of Tsuji and Faltings on 푝-adic comparison theorems. Let me now turn to Jean-Marc’s interaction with China. After going to Beijing to give an invited talk at the 2002 ICM, Jean-Marc thought it was the right time to start a cooperative program with China in arithmetic geometry. This was realized by a chaired professorship of arithmetic geometry at Tsinghua University which ran from 2003 to 2006, of which Jean-Marc was the first. He convinced several of his Orsay colleagues (including me) to go teach at Tsinghua, to give a solid foundation to students who were both talented and anxious to learn. Each year, some of the best were chosen to come to Orsay or ENS to write a Figure 2. Jean-Marc Fontaine with Michel Raynaud thesis. Many then returned to China to hold permanent (Conference in honor of L. Illusie, IHES, 2005). academic positions. They became the “dorsal fin” of arithmetic geometry in China. Jean-Marc was the guiding spirit devoted himself to this, and over the next years constructed of this cooperation, to which he devoted all his energy un- an elaborate algebraic formalism in which the conjectures til he fell ill in 2016. In an article published this past Jan- could be formulated, and which eventually led to their uary 30 by the Chinese Mathematical Society, Yi Ouyang complete solution. His formalism, based on the construc- (who gave us invaluable assistance in Beijing, and who was tion of certain rings called “period rings” which now bear a coauthor with Jean-Marc of an introductory book on 푝- his name, was at first received with a combination of skep- adic Hodge theory) wrote a moving tribute4 to Jean-Marc. 푝 ticism and fear. It became the pillar of -adic Hodge the- Miaofen Chen, who had been our student in Tsinghua in ory. His rings play a central role in present-day arithmetic 2004, and is now a professor at ECNU in Shanghai, made geometry. Over the past twenty years, they have had many a video, together with former classmates, in Jean-Marc’s spectacular applications, featuring in the work of Taylor– honor in December of 2018. Wiles on Fermat, and of Khare–Wintenberger on Serre’s Jean-Marc was extremely generous. He loved to convey modularity conjecture. his ideas and his vision of mathematics. This was of great Throughout my career, I have been closely involved benefit to his students, especially Wintenberger, Colmez, with Jean-Marc, not only at Orsay but also as we traveled and Breuil, each of whom went on to contribute enor- together all around the (mathematical) world: Europe, the mously to both his work and its influence. US, India, China, Japan. Let me say a few words about his Very much like Michel Raynaud, whom we lost in March arrival at Orsay in 1988. It brought us a breath of fresh air, of 2018, Jean-Marc did not care for honors, and never a renewed sense of energy. Together with Raynaud, Win- 1 spoke of those he had received (invited speaker at the 1983 tenberger (who had himself just arrived at Orsay), Perrin- ICM in Warsaw and the 2002 ICM in Beijing, Prix Petit Riou, Kato (who was visiting Orsay), and Mazur, who d’Ormoy, Carrière et Thébautin 1984, Gay-Lussac Hum- was visiting IHES, Jean-Marc organized a seminar at IHES boldt Prize in 2002, election to the Académiedes Sciences 푝 on “ -adic periods.” There was great enthusiasm about in 2002). 2 퐶 the “semistable conjecture” 푠푡 of Jean-Marc and Jannsen. Let me end on a personal note. Jean-Marc loved to 퐶 The geometric formulation of 푠푡, following work of Hy- argue about all sorts of things. At the Orsay dining odo and Kato, led to new subjects of research, in particu- hall, or around the coffee machine, we would have long lar what is now called logarithmic geometry.3 The book in July 1988, on the train taking us to Oberwolfach. 1 Sadly deceased on January 23, 2019. 4This tribute is essentially reproduced in Ouyang’s piece “The China legacy 2It was fully proven some twelve years later by Tsuji. of Jean-Marc Fontaine” in the October 2019 issue of La Gazette des 3In fact, logarithmic geometry was born in a conversation Jean-Marc and I had Mathématiciens.

1012 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 discussions. I could barely state an opinion before Jean- Marc objected; often he was right. But in November 2017, after his major surgery, I noticed a change. We were both in Padova, each giving a course. He knew his way around the back streets; I did not. On rainy nights we would go by means of narrow, winding, poorly lit streets to a good restaurant he knew. When we got there he wouldn’t eat much but we talked for a long time. And we agreed on everything. It was a bad omen. Right up to the end, Jean-Marc fought valiantly against the disease which was destroying his body. In March of 2016 I was in Taipei, and suggested Jean-Marc to Chia Fu Yu, who then invited him to come give a course. He accepted, but had to postpone coming several times, because of his operation and its ensuing complications. He finally went in October of 2018, despite his cancer having Figure 3. Jean-Marc Fontaine in Cambridge (UK), with the returned with a vengeance. With incredible stoicism, he Messing and Berthelot families, after Wiles’s lectures on Fermat (June 28, 1993). gave three two-hour lectures, before having to be rushed home. In our last conversations, we talked as though every- Souvenirs de Jean-Marc thing were normal—about math, about current events, about music, about Schoenberg’s La nuit transfigurée, a fa- William Messing vorite of his. Some mathematical things he cared deeply about were the re-edition, with errata and addenda, of the I first encountered Jean-Marc on Monday, June 7,1971 book Périodes 푝-adiques, Astérisque223; the publication of when he was sitting with Serre during the pot before Serre’s 5 his last article, on almost 퐶푝-representations; the publica- SéminaireBourbaki talk. I next encountered him on Mon- tion of his book with Yi Ouyang; and the publication of day, June 5, 1972, again sitting with Serre, before Serre’s his work with Fargues on the “fundamental curve” in 푝- SéminaireBourbaki talk. I do not recall speaking to him adic Hodge theory. Of these, only the last, Astérisque406, on either of these occasions. That summer, we both at- was done in time for Jean-Marc to hold in his hands. tended the Antwerp Modular Forms of One Variable for three weeks and we did discuss 푝-divisible groups a cou- ple of times. It was during the AMS Summer Institute in Algebraic Geometry, held in Arcata in August 1974, that I really got to know Jean-Marc. He was there with his first wife, Lau- rence, and their infant daughter, Isabelle, and I discussed with him our common interests in 푝-divisible groups, 푝- adic Galois representations, and Grothendieck’s problem of the “mysterious functor.” He attended my two lectures and I was very impressed by his August 15 lecture, given Luc Illusie in the Serre–Tate seminar, Arithmetic, Formal Groups and their Division Points. I visited Berthelot and Breen in Rennes during the period September–December 1976. In November 1976 I flew from Rennes to Lyon, where Jean-Marc picked meup at the airport and drove to his Grenoble apartment where I spent a week staying with him, Laurence, and their three daughters. The twins, Camille and Caroline, were infants

William Messing is a professor of mathematics at the School of Mathemat- 5 This article has now appeared: Jean-Marc Fontaine, Almost 퐂푝 Galois repre- ics, University of Minnesota, Minneapolis, Minnesota. His email address is sentations and vector bundles, Tunis. J. Math. 2 (2020), no. 3, 667–732. [email protected]. MR4041286.

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1013 at the time. It was a terrific visit, both because of our children, Katell and Erwan, until I picked him up and we mathematical interaction and my getting to know both drove to our small apartment in Zup-Sud Rennes. Jean- him and Laurence as wonderful hosts and friends. Jean- Marc visited Rennes and again stayed with the Berthelots. Marc was then a believing Catholic. I was surprised, as We continued to work together. the French mathematicians whom I had known previously During the Fall Semester 1983, I visited Jean-Marc in were strongly anticlerical. He told me a wonderful story Grenoble and stayed at the Institut Fourier tower where about how in the past he had traveled with four priests there were a small number of apartments. Jean-Marc and to Normandy, where they enjoyed a large sumptuous lun- I discussed Dieudonnégauges. Jean-Marc was an Ord- cheon on a Friday featuring Marmite Dieppoise, agneau de way Visitor at the Minnesota Mathematics Department and pré-salé, camembert, and several trous normands. I asked stayed at the Messing house November 12–December 11, Jean-Marc about the priests eating meat on a Friday, and 1984. We continued to work on 푝-adic comparison theo- he told me that he had asked them the same question and rems. He gave a series of lectures on Wintenberger’s the- their response was “The peasants refrain from eating meat; orem showing when 퐾 is a characteristic zero, absolutely the priests do not.” I was both amused and amazed with unramified local field with perfect residue field of charac- this story, as he was in recounting it. teristic 푝 > 0 and 푋/퐾 is proper and smooth and has good ∗ On Thursday and Friday, July 6 and 7, 1978, Jean-Marc reduction, the Hodge filtration on 퐻푑푅(푋/퐾) is canonically spoke at the Rennes conference, Journéesde Géométrie split. Algébriquede Rennes. Jean-Marc announced and outlined I saw Jean-Marc next at the conference for Tate’s sixti- his solution to the mysterious functor problem for abelian eth birthday at Harvard, in April 1985. Jean-Marc and I varieties and 푝-divisible groups. To put it mildly, this was worked intently at a blackboard for three days and then a striking result and was, hands down, the highlight of the Jean-Marc lectured on the comparison theorem we had conference. I spent the summer and autumn first at the proven. The AMS conference Current Trends in Arith- IHES and then in Rennes. Jean-Marc visited Rennes and metic Algebraic Geometry was held in Arcata, August 18– stayed at chez Berthelot for a week. The three of us talked 24, 1985. I gave four lectures on our joint work and our at length. I again visited Jean-Marc in Grenoble, staying paper was published in the proceedings of that conference. again with him and Laurence. Immediately after the conference, Jean-Marc came to our The Fontaine family spent the month of January 1980 house in Saint Paul, where he stayed a week and gave a visiting my wife, Rita, and me at UC Irvine. We had rented beautiful lecture on the Frey curve and Serre’s strategy for a small bungalow for them on Balboa Island. Jean-Marc proving Fermat’s Last Theorem. Shortly after, I returned to and I began seriously working together during that month. Grenoble, where I again spent the Fall Semester. In addi- Let 푘 be a perfect field of characteristic 푝 > 0, let 푊푛 be the tion to Jean-Marc’s presence, Kato was also a visitor and Witt vectors of length 푛 of 푘, and let 푆푛 = 푆푝푒푐(푊푛). He this was when I first met him. Kato learned and immedi- told me that, with covering morphisms being flat, locally ately understood what Jean-Marc and I had done and saw complete intersections the functor assigning to a scheme how to generalize some of our results. It was thrilling for 0 푋/푘, 퐻푐푟푦푠(푋/푆푛) is a sheaf. I told him that this topology both Jean-Marc and me to see and learn from a young mas- had previously been considered by Mazur, who named ter. it the syntomic topology, and I gave Jean-Marc a copy of Jean-Marc and I exchanged emails in November 2018. I Mazur’s unpublished notes on the syntomic topology. We wrote to him on November 11 and he responded on No- 푐푟푦푠 named this sheaf 풪푛 . The next day, I gave Jean-Marc a vember 22. Here are those letters: proof that if 푋푆푌푁 denotes the big syntomic site of the 푘 Cher Jean-Marc, scheme 푋, then we have a canonical isomorphism, com- Please permit me to write in English, as I want to be as pre- ∗ patible with the action of Frobenius, between 퐻푐푟푦푠(푋/푆푛) cise as possible. Luc wrote and told me of your relapse and the ∗ 푐푟푦푠 and 퐻 (푋푆푌푁 , 풪푛 ). He was delighted and our work took lack of hope for your recovery. I have recalled and reflected on off. the more than forty-four years we have known each other, my The Messing family spent the period June–December visits to you in Grenoble, your visits to me in both Irvine and 1980 abroad. Jean-Marc and I corresponded that summer Minnesota. When I last saw you at the Gabber conference in and, in particular, on August 19, I wrote the letter per- June, you seemed to be in good spirits and I was very happy mitting the formulation of the 퐶푐푟푦푠 conjecture in Jean- that you were a cancer survivor. Thus, I was shocked to learn Marc’s Annals of Mathematics paper. In September, my of the recurrence. I remember vividly the hospitality you and nine-year-old son, Charles, and I went to Rennes, where he Laurence extended to me in Grenoble and getting to know both entered the écoleprimaire near the Berthelot house. Each of you and your daughters. Later getting to know Hélène. I re- afternoon, after school, he would play with the Berthelot member how much I learned from you, your incredible drive and

1014 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 work ethic and mathematical vision which reshaped the 푝-adic landscape. I recall my interest in your Catholic faith, as most A Quest for Rings mathematicians I knew were noncroyant. I remember too being quite surprised when you told me, that you had stopped going Pierre Berthelot to Mass each week. Please let me know if there is anything that At the heart of Fontaine’s program for the study of 푝-adic Rita and I might do to aid and bring you solace. representations of Galois groups of local fields, Fontaine’s Amiti´es, rings have often be viewed as mysterious and highly im- Bill pressive objects. Yet, their construction has been the result of a very coherent strategy, initiated by Fontaine in Dear Bill, the early seventies, and regularly enriched with new view- Sorry for being so slow to answer but I was too tired to write points and key results. In this survey, I would like to anything reasonable. Thank you heartily for your message which put into perspective the main steps of these developments, touched me deeply. Needless to say that, despite my bad temper, from the early contributions of Tate and Serre to the com- it has been a pleasure to work with you. I know very well that our plete solution of Fontaine’s conjectures some thirty years joint work on 푝-adic comparison theorems has been the hardest later. thing I ever contributed to and that, without you, I would have In the whole story, 푝 denotes a fixed prime number, and had no chance to get into this kind of thing. That we did not 퐾 a local field6 of characteristic 0 and residue characteris- succeed to continue to work together is entirely my fault and I tic 푝 (퐾 is also called a 푝-adic field). Let 퐾 be an algebraic regret it deeply. I also have a lot of nice souvenirs of when we closure of 퐾, 퐶 its completion, 푂 , 푂 , 푂 the rings of inte- were together, in particular of your warm hospitality with Rita 퐾 퐾 퐶 in Minneapolis. Everything concerning my catholic faith seems gers in 퐾, 퐾, 퐶, and 푘 the residue field of 푂퐾 . When needed, to me now as if it never existed. I am ready to die (even if I the result of a base change will be denoted by adding the would love to wait 20 more years!). I am neither interested or new base as a subscript. frightened by what is going to happen after. I can not say the The objects of interest here are the 푝-adic representa- 7 same about what is going to happen before. I wish you, Rita tions of the absolute Galois group 퐺 = Gal(퐾/퐾). Let and your family a long, nice and peaceful life. me first recall a few important results already known in With all my affection, the late sixties. Jean-Marc 1. Hodge–Tate Representations Adieu Jean-Marc. A simple example of a 푝-adic representation of 퐺 ∶= Gal(퐾/퐾) is given by the natural action of 퐺 on the roots

of unity in 퐾: use the transition maps 휇푝푛+1 (퐾) → 휇푝푛 (퐾) given by elevation to the 푝th power to define

푇 (휇) ∶= lim 휇 푛 (퐾), 푉 (휇) ∶= ℚ ⊗ 푇 (휇). 푝 ←−− 푝 푝 푝 ℤ푝 푝 푛

Then 푉푝(휇) is a 1-dimensional ℚ푝-representation of 퐺, defined by the (푝-adic) cyclotomic character 휒 of 퐺. For any 푖 ∈ ℤ, we denote by ℚ푝(푖) the ℚ푝-representation of dimension 1 given by multiplication by 휒푖. If 퐸 is a ℚ -representation William Messing 푝 of 퐺, we denote by 퐸(푖) the ℚ푝-representation 퐸 ⊗ℚ푝 ℚ푝(푖), and by 퐸{푖} the subspace {푥 ∈ 퐸 | ∀푔 ∈ 퐺, 푔푥 = 휒(푔)푖푥}.

Pierre Berthelot is a retired professor of mathematics at Universit´ede Rennes 1, Rennes, France. His email address is pierre.berthelot@univ-rennes1 .fr. This text is partly based on an unpublished report due to Luc Illusie. The author thanks him heartily for letting him use that document.

6i.e., a complete discrete valuation field with perfect residue field. 7 i.e., the finite-dimensional ℚ푝-vector spaces endowed with a continuous linear action of 퐺. We will often write simply “ℚ푝-representation.”

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1015 Let 푉 be a ℚ푝-representation of 퐺. Then the 퐶-vector Conjecture/Theorem 1 (퐂퐇퐓). Let 푋 be a proper and smooth space 퐶 ⊗ℚ푝 푉 is endowed with a semilinear action of 퐺, scheme over 퐾. For any 푛 ∈ 퐍, there exists a canonical 퐺- equivariant isomorphism and (퐶⊗ℚ푝 푉){푖} is a 퐾-subspace of 퐶⊗ℚ푝 푉. A fundamental result of Tate on the Galois cohomology of 퐶 implies that ∼ 퐻푗(푋, Ω푖 ) ⊗ 퐶(−푖) → 퐶 ⊗ 퐻푛 (푋 ,ℚ ). the 퐺-equivariant map ⨁ 푋/퐾 퐾 ℚ푝 et́ 퐾 푝 푖+푗=푛 휀 ∶ 퐶 ⊗ ((퐶 ⊗ 푉){푖}) → 퐶 ⊗ 푉 ⨁ 퐾 ℚ푝 ℚ푝 푖 We’ll see later how this conjecture has been included is injective (see [Serre, MR0242839] and [Tate, in the broader context of Fontaine’s conjectures. By anal- MR0231827]). One says that 푉 is a Hodge–Tate represen- ogy with the Hodge decomposition for the cohomology tation if 휀 is an isomorphism. The integers 푖 for which of complex varieties, the subject became known as 푝-adic (퐶 ⊗ 푉){푖} ≠ 0 are then called the Hodge–Tate weights of Hodge theory. 푉. 8 2. The “Mysterious Functor” Let 퐻 = ⋃푛 퐻[푛] be a 푝-divisible group of height ℎ over 퐾, 퐻[푛] denoting the kernel of multiplication by 푝푛 in Another major input for the development of 푝-adic Hodge 퐻. The above construction can be repeated using the sub- theory came from Grothendieck’s work on Dieudonné theory for 푝-divisible groups (cf. his lecture at ICM70 groups 퐻[푛] in place of the 휇푝푛 ’s. This gives the Tate mod- in Nice [MR0578496] and his 1970 Montreal course ule 푇푝(퐻) and the associated 푝-adic representation 푉푝(퐻) of 퐺, of dimension ℎ. [MR0417192]). A natural question is then: is 푉(퐻) a Hodge–Tate rep- Dieudonnétheory was initiated by a series of articles written by Dieudonnéin the fifties. Their goal was to clas- resentation? When 퐻 has good reduction over 푂퐾 , i.e., is sify formal Lie groups over a perfect field 푘 of characteristic the generic fiber of a 푝-divisible group ℋ on 푂퐾 , Tate gave a positive answer by constructing a 퐺-equivariant isomor- 푝 by some kind of (semi)linear data analogous to the Lie phism algebra in characteristic 0. It has been developed by many authors, in several flavors, covariant or contravariant, each ∼ ∗ ∨ ∨ 푡ℋ′ (퐶) ⊕ [푡ℋ(퐶) ⊗ 푉푝(휇) ] −→푉(퐻) , with its own hypotheses. Over 푘, one of the most commonly used constructions where ∨ denotes the 퐶-valued dual, ℋ′ is the Cartier dual is due to Manin (see Manin [MR0157972] or Demazure ∗ of ℋ, and 푡 (resp., 푡 ) denotes the tangent (resp., cotan- [MR0344261]). Let 푊 = 푊(푘) = lim 푊푛(푘) be the ring of ←−−푛 gent) space [MR0231827]. Using the semistable reduction Witt vectors with coefficients in 푘 and let 휎 be its Frobenius theorem [SGA7], Raynaud extended Tate’s proof to the case automorphism. For a finite unipotent 푝-group scheme Γ of the 푝-divisible group 퐻 = 퐴[∞] defined by an arbitrary over Spec(푘), Manin defines the Dieudonnémodule 푀(Γ) abelian variety 퐴. In 1982, Fontaine gave another proof for by 푀(Γ) ∶= 햧허헆(Γ, 핎), where 핎 is the ind-group scheme 푝-divisible groups, using a completely different method representing the “ring of Witt vectors functor.” Then 푀(Γ) based on the study of the modules of absolute 1-forms of is a finitely generated torsion 푊-module, endowed with 푂 푂 −1 퐾 and 퐾 [6]. 퐹, 푉 휎 휎 ∨ a pair of endomorphisms , respectively, and - When 퐻 = 퐴[∞], 푉(퐻) can classically be identified semilinear, such that 퐹푉 = 푉퐹 = 푝 and 푉 is nilpotent. The 퐻1 (퐴 ,ℚ ) with the first étalecohomology group et́ 퐾 푝 . Tate’s functor Γ ↦ (푀(Γ), 퐹, 푉) is an equivalence of categories. isomorphism can then be written By an inverse limit argument, it yields an equivalence between unipotent 푝-divisible groups and free finitely gen- (퐻1(퐴, 풪 ) ⊗ 퐶) ⊕ (퐻0(퐴, Ω1 ) ⊗ 퐶(−1)) 퐴 퐾 퐴 퐾 erated 푊-modules endowed with 퐹, 푉 as before, 푉 being ∼ 1 now topologically nilpotent. Finally, the theory can be ex- −→퐶 ⊗ℚ푝 퐻et́ (퐴퐾 ,ℚ푝). tended to all finite or 푝-divisible 푘-groups using the canon- Tate asked whether a similar decomposition might actu- ical decomposition of a 푝-group scheme on a perfect field ally exist for any proper and smooth 퐾-scheme, thus giv- and Cartier duality. ing the first formulation of the Hodge–Tate decomposition Grothendieck’s interest in Dieudonnétheory was mo- conjecture : tivated by the development of crystalline cohomology in a relative situation, and by the possibility of using deformation theory to go from characteristic 푝 to characteristic 8 퐻[푛] i.e., a direct system of finite locally free commutative group schemes over 0 푆 푆, of rank 푝푛ℎ, such that the transition maps identify 퐻[푛] with the kernel of . His idea was that over a nonperfect base , one should multiplication by 푝푛 on 퐻[푛 + 1]. Following Grothendieck, 푝-divisible groups replace the Dieudonnémodule of a 푝-divisible group 퐻 are also called Barsotti–Tate groups. by a Dieudonnécrystal 픻(퐻). This would provide a locally

1016 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 free finitely generated 풪푆′ -module 픻(퐻)푆′ for any nilim- In 1970, these questions seemed to be completely inac- mersion 푆 ↪ 푆′ defined by an ideal endowed with a suit- cessible. Following Grothendieck, they were referred to as able divided power structure,9 together with base change iso- the problem of the mysterious functor. morphisms when 푆′ varies. For 푆 = Spec 푘, the datum of a crystal on 푆 is equivalent to that of a 푝-adically complete 3. Dieudonn éTheory Revisited and separated 푊-module, and 픻(퐻) should give back the During a lecture at IHES10 in 2009, Fontaine pointed out usual Dieudonnémodule. that he met Grothendieck only once: in 1972, during the Grothendieck proposed two constructions of the crystal Antwerp Summer School on Modular Functions. As is well 픻(퐻), which yield in addition a locally split monomor- known,11 there could have been better circumstances for phism 휔퐻 ↪ 픻(퐻)푆, where 휔퐻 is the module of an introduction to the problem of the mysterious functor. invariant differential forms on 퐻. His program was Fortunately, Serre took over: he shared with Fontaine his completed by Messing [MR0347836] (see also [Mazur– interest in these questions, and he convinced him to work Messing, MR0374150]), who proved in particular that if on their solution. the divided powers satisfy some nilpotency condition, the Fontaine’s strategy unfolded progressively. Its first step datum of a 푝-divisible group 퐻′ on 푆′ lifting 퐻 is equiva- was to revisit completely the previous results about the lent to the datum of a locally split submodule ℒ ⊂ 픻(퐻)푆′ classification of finite 푝-groups and 푝-divisible groups over lifting 휔퐻 ↪ 픻(퐻)푆. 푘 and 푂퐾 . More specifically, Fontaine’s goal was to make Assume again that 퐾 is a 푝-adic field with residue field as explicit as possible the construction of a quasi-inverse 푘. Let 퐾0 be the fraction field of 푊, which is in a nat- functor retrieving the initial 푝-divisible group from the ural way a subfield of 퐾. Using Messing’s theorem, one classifying data. gets that, up to isogeny, a 푝-divisible group 퐻 over 푂퐾 is Over 푘, Manin’s construction of the Dieudonnémod- determined by the data of a 퐾0-vector space 퐸 endowed ule has an unpleasant feature: in order to define the with a 휎-semilinear automorphism 퐹, and of a 퐾-subspace Dieudonnémodule of an arbitrary 푝-divisible group 퐻, one must decompose 퐻 as the product of its unipotent 퐿 ⊂ 퐾 ⊗퐾0 퐸. It follows that the 푝-adic representation 푉(퐻퐾 ) is determined uniquely by the triple (퐸, 퐹, 퐿). But and multiplicative components and give an ad hoc defi- the converse is also true, as Tate proved in [MR0231827] nition for each of the two components. Fontaine looked that the functor 퐻 ↦ 퐻퐾 is fully faithful. for an alternate definition that would unify the treatment Then Grothendieck emphasized the following striking of the two cases. His solution was to modify a construc- fact. Using the theory of 푝-divisible groups on 푂퐾 as an in- tion of Barsotti and to use the sheaf of Witt covectors. If 푅 termediate, we obtain a correspondence between two very is a 푘-algebra, define different kinds of data. However, both are described in (… , 푎−푖, … , 푎−1, 푎0) | 푎−푖 ∈ 푅, purely algebraic terms, without ever using the notion of 퐶푊(푅) = { ∃ 푟 > 0 such that } . a 푝-divisible group, nor any other geometric construction. the ideal (푎−푖)푖≥푟 is nilpotent This led him to ask in his ICM70 talk [MR0578496] the following questions: Let 퐷푘 be the Dieudonnéring, generated by 푊 and by (1) Can one give a direct construction of the 푝-adic elements 퐹, 푉, with relations 퐹푉 = 푉퐹 = 푝, 퐹푎 = 휎(푎)퐹, 푉휎(푎) = 푎푉 for all 푎 ∈ 푊. Then 퐶푊(푅) has a func- representation 푉(퐻퐾 ) in terms of the data (퐸, 퐹, 퐿), without using 푝-divisible groups? Conversely, is there a direct torial structure of left-퐷푘-module. Let 풜 be the category of finite 푘-algebras. Fontaine de- construction recovering the triple (퐸, 퐹, 퐿) from 푉(퐻퐾 )? (2) Is there a higher dimension analogue of this corre- fines the Dieudonnémodule of a finite 푝-group scheme 퐻 as being the left 퐷푘-module 풟(퐻) = 햧허헆(퐻, 퐶푊), where spondence relating, for a proper and smooth 푂퐾 -scheme ∗ 햧허헆 is the set of morphisms of group functors on 풜. If 푋, the 푝-adic representations 퐻et́ (푋퐾 ,ℚ푝) with the crys- ∗ 푀 is a 퐷푘-module of finite length as a 푊-module, he talline cohomology spaces 퐻cris(푋푘/푊) ⊗푊 퐾0 endowed with their Frobenius action and a Hodge-type filtration? also defines a functor 풢(푀) on 풜 by setting 풢(푀)(푅) = 햧허헆퐷푘 (푀, 퐶푊(푅)). Then he proves that the functors 퐻 ↦ 풟(퐻) and 푀 ↦ 풢(푀) are left adjoint functors, providing 푝 9 an antiequivalence between the category of finite -groups A divided power structure, or PD-structure, on an ideal 퐽 is a family of oper- over 푘 and the category of 퐷 -modules of finite length over ators on 퐽, often denoted by 푥 ↦ 푥[푛], which behave “as the operators 푥 ↦ 푘 푥푛/푛! in characteristic 0” (see [MR0384804] or [MR0491705]). Endowed 푊. Taking into account appropriate topologies allows him with this extra structure, 퐽 is called a PD-ideal. For example, for any ℤ푝-algebra to obtain similar results for 푝-divisible groups. 푅, the ideal 푝푅 has a canonical PD-structure given by (푝푥)[푛] = (푝푛/푛! )푥푛 for all 푛 ≥ 1 and all 푥 ∈ 푅. We always require that a PD-structure on 퐽 induce the 10Available at https://www.dailymotion.com/video/x8jeru. canonical PD-structure on 퐽 ∩ 푝푅. 11See for example [Jackson, MR2104915, p. 1201].

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1017 At this point, Fontaine has recovered the main results of classical Dieudonnétheory over 푘, with an explicit de- scription of a quasi-inverse functor 풢. Then he proceeds with the classification of 푝-divisible groups over 푂퐾 , first up to isomorphism in the unramified case (for 푝 ≠ 2), next up to isogeny in the general case. We cannot give the details of his constructions here, but we would like to emphasize a feature that will become the core of his method for classifying 푝-adic representations: in order to define a pair of functors relating two different categories of algebraic structures, he constructs an object endowed with both structures in a compatible way, and uses bifunc- Figure 4. Jean-Marc Fontaine giving a lecture. tors such as 햧허헆 or ⊗ to get functors in both directions. In the relation between finite 푝-group schemes over 푘 and share some features, such as the Künneth formula or Dieudonnémodules, this role is played by the family of Poincaréduality, and one could hope that the “mysterious (ℤ푝, 퐷푘)-bimodules 퐶푊(푅). To give another example, we functor” would respect these structures. More precisely, now sketch briefly the correspondence between isogeny he wanted the functor to be an equivalence between two classes of 푝-divisible groups over 푂퐾 and triples (퐸, 퐹, 퐿) Tannakian categories ,13 and to preserve the multiplicative as considered by Grothendieck. structures on both. He concluded that the “bistructured For any discrete ring 퐴, let 퐵푊(퐴) be the additive group objects” such as 퐶푊(푅) or 퐵푊(Res(푂 )) used in the con- 12 퐶 of Witt bivectors with coefficients in 퐴; the functor 퐵푊 struction of the “mysterious functor” for 푝-divisible groups can be extended in a natural way to complete linearly should be replaced by similar objects having a ring struc- topologized rings. When 퐴 is a 푘-algebra, 퐵푊(퐴) has ture. a natural structure of a 퐷푘-module. Denote by Res(푂퐶) With his lecture at the 1978 Rennes conference [5], (푛) the set of families 푥 = (푥 )푛∈ℤ of elements of 푂퐶 such Fontaine began to develop a very impressive program, (푛+1) 푝 (푛) that (푥 ) = 푥 for all 푛. One can endow Res(푂퐶) which is still at the heart of 푝-adic Hodge theory nowadays. with a structure of 푘-algebra, complete for a linear topol- Let Rep(퐺) be the category of finite-dimensional continu- 퐾 휃 ∶ ogy. Fontaine defines a 0-linear homomorphism 0 ous ℚ푝-representations of 퐺. Fontaine’s strategy consists 푛 (푛) 퐵푊(Res(푂퐶)) → 퐶 by 휃((푥푛)푛∈ℤ) = ∑푛∈ℤ 푝 푥푛 , and ex- of describing various Tannakian subcategories of Rep(퐺) tends it by linearity as 휃 ∶ 퐾 ⊗퐾0 퐵푊(Res(푂퐶)) → 퐶. Then by means of correspondences with other tensor categories he endows 퐾 ⊗퐾0 퐵푊(Res(푂퐶)) with the length 1 filtration having as objects 퐾-vector spaces endowed with some extra defined by Ker(휃), and constructs a 퐺-equivariant isomor- structures (such as a graduation, a filtration, etc.). Let 풞 be phism [3, V, Theorem 1] such a tensor category. In each case, the correspondence is to be defined by an object of 풞 endowed with a commu- ∼ 푢 ∈ 햧허헆 (퐸, 퐵푊(Res(푂 ))) 푉(퐻) −→ { 퐾0[퐹,푉] 퐶 } . tative 퐾-algebra structure and a compatible 퐺-action. For such that 푢퐾 (퐿) ⊂ Ker(휃) such algebras, which he calls (ℚ푝, 퐺)-rings, Fontaine uses With these results, Fontaine answers Grothendieck’s the generic notation 퐵 (as a tribute to Barsotti), generally questions about the “mysterious functor” for 푝-divisible enriched by some relevant abbreviation. groups. They were the topic of his 1975 Cours Peccot at The algebra 퐵 defines a functor 풟퐵 ∶ Rep(퐺) → 풞 by the Collège de France, and were published in 1977 as an setting, for any representation 푉 ∈ Rep(퐺), Astérisquevolume [3]. However, the method used for 푝- 퐺 풟퐵(푉) = (퐵 ⊗ℚ 푉) . divisible groups could not suffice to construct the “mysteri- 푝 ous functor” for varieties of any dimension, because it only Let 퐹 denote the 퐾-algebra 퐵퐺. By adjunction, one gets a gives representations with Hodge–Tate weights in {0, 1}. So canonical 퐵-linear map new ideas were again needed. .. . 훼퐵(푉) ∶ 퐵 ⊗퐹 풟퐵(푉) → 퐵 ⊗ℚ푝 푉. 4. Fontaine Rings I : 퐁 퐇퐓 Fontaine defines 퐺-regular (ℚ푝, 퐺)-rings by certain condi- In his quest for a correspondence relating the 푝-adic étale tions which insure in particular that 퐹 is a field, and that and de Rham/crystalline cohomologies of algebraic vari- 훼퐵(푉) is injective for all 푉. Assuming that 퐵 is 퐺-regular, eties over 퐾, Fontaine had a guideline: these cohomologies a representation 푉 is said to be 퐵-admissible if 훼퐵(푉) is an

12 13 i.e., families (푥푛)푛∈ℤ satisfying the condition used above to define 퐶푊(퐴). See [Deligne, MR0654325].

1018 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 isomorphism, or, equivalently, if dim 풟 (푉) = dim 푉. ♭ 퐹 퐵 ℚ푝 Let 휃 ∶ 푂퐾 ⊗푊 푊(푂퐶) → 푂퐶 be the 푂퐾 -linear extension When 푉 is 퐵-admissible, one says that 푉 and the object of 휃0. The 푂퐾 -algebra 퐴inf is then defined as the comple- 풟 (푉) 풞 ♭ 퐵 of are associated (under the correspondence de- tion of 푂퐾 ⊗푊 푊(푂퐶) for the adic topology defined by the fined by 퐵). ideal (푝) + Ker(휃), and one sets 퐴inf,퐾 = 퐾 ⊗푂퐾 퐴inf. As a first example, let us consider the Hodge–Tate ring Keeping the notation 휃 for the natural extensions of 휃 −1 퐵HT = 퐶[푇, 푇 ]. On the one hand, it has an obvious to 퐴inf and 퐴inf,퐾 , one denotes by 퐽퐾 ⊂ 퐴inf,퐾 the kernel graded 퐾-algebra structure. On the other hand, it can be of 휃. It is a maximal principal ideal in 퐴inf,퐾 . One de- endowed with an action of 퐺 by setting, for all 푔 ∈ 퐺, + notes by 퐵dR the completion of 퐴inf,퐾 with respect to the 푖 푖 푖 퐽 -adic topology. It is a complete discrete valuation ring 푔 ⋅ (∑ 휆푖푇 ) = ∑ 푔(휆푖)휒 (푔)푇 . 퐾 푖∈ℤ 푖∈ℤ with residue field 퐶. Finally, one defines the ring 퐵dR as + being the fraction field of 퐵dR. Then 퐵HT is (ℚ푝, 퐺)-regular, and a ℚ푝-representation is 퐵HT- admissible if and only if it is a Hodge–Tate representation As all the steps leading to 퐵dR are compatible with the as defined in our first section. action of 퐺, the algebras 퐴inf, 퐴inf,퐾 , 퐵dR are endowed with 퐺 퐵 an action of 퐺. We have (퐵dR) = 퐾, and there is a natural The notion of -admissibility can be extended so as to × equivariant injection 휈 ∶ ℤ푝(1) ↪ 퐴 , defined as follows. take into account finite extensions of the field 퐾: a ℚ푝- inf An element 휀 ∈ ℤ푝(1) is a sequence (휀푛)푛∈ℕ such that 휀0 = 1 representation 푉 of 퐺 = Gal(퐾/퐾) is said to be potentially 퐵- 푝 and 휀 = 휀 for all 푛. It can be viewed as an element admissible if there exists a finite extension 퐾′ of 퐾 in 퐾 such 푛+1 푛 푂♭ 휀(푛) = 휀 푛 휈 that 푉 is 퐵-admissible as a representation of Gal(퐾/퐾′). of 퐶 by setting 푛 for all , and one defines by 휈(휀) = 1 ⊗ [휀], where [휀] is the Teichmüller representative 5. Fontaine Rings II : 퐁퐝퐑 of 휀. Then 휈(휀) ∈ Ker(휃), and one can define an additive + In [7], [8], and [14], Fontaine constructs three new rings map ℤ푝(1) → 퐵dR by which play a central role in his classification of 푝-adic rep- 휀 ↦ log(휈(휀)) = ∑(−1)푛+1(휈(휀) − 1)푛/푛. resentations: 퐵dR, 퐵cris, and 퐵st. We now sketch the con- 푛≥1 struction of the biggest one, 퐵dR, which will control the 푝- Fontaine proved that the image in 퐵dR of any nonzero ele- adic ´etalecohomology of general algebraic varieties over ment of ℤ푝(1) is a uniformizing parameter. 퐾. The ring 퐵dR has another fundamental structure: it is The first step is to construct a “universal thickening” 퐴inf endowed with the decreasing filtration defined by its dis- of 푂퐶 [14]. Note that the Frobenius endomorphism 퐹 of crete valuation. The Galois action is compatible with this ♭ 푂퐶/푝 is surjective. We denote by 푂퐶 the ring defined by filtration. Thanks to the previous results, one sees that ♭ 푂 ∶= lim 푂 /푝. the choice of a nonzero element 푡 ∈ ℤ푝(1) provides a 퐺- 퐶 ←−− 퐶 퐹 equivariant graded isomorphism ∼ If 푥 = (푥 ) is an element of 푂♭ , and if, for all 푛 ≥ 0, •퐵 −→ 퐶(푖) = 퐶[푡, 푡−1] = 퐵 , 푛 푛≥0 퐶 gr dR ⨁ HT 푛̂푥 ∈ 푂퐶 is a lifting of 푥푛, then, for all 푚 ≥ 0, the se- 푖∈ℤ 푛 푝 (푚) quence 푛+푚̂푥 converges towards an element 푥 ∈ 푂퐶 where 퐵HT is the previous Hodge–Tate ring. (푚) when 푛 → ∞. The elements 푥 do not depend upon Following the general pattern, one says that a ℚ푝- the choice of the liftings 푛̂푥 , and the mapping (푥푛)푛≥0 ↦ representation 푉 is a de Rham representation if it is 퐵dR- (푚) ♭ (푥 )푚≥0 allows one to identify 푂퐶 with the set of families admissible, i.e., if the canonical map (푚) (푚+1) 푝 (푚) (푥 )푚≥0 of elements of 푂퐶 such that (푥 ) = 푥 for 퐺 훼퐵dR (푉) ∶ 퐵dR ⊗퐾 (퐵dR ⊗ℚ푝 푉) → 퐵dR ⊗ℚ푝 푉 all 푚 ≥ 0. Using Teichmüller representatives, this provides is an isomorphism. Using the previous results, one gets a 푘-algebra structure on 푂♭ . 퐶 that de Rham representations are Hodge–Tate. Fontaine defines a surjective 푊-algebra homomor- ♭ phism 휃0 ∶ 푊(푂퐶) → 푂퐶 by setting, for a Witt vector Remarks. (a) The ring 퐵dR is functorial with respect to ♭ ′ (푥푛)푛≥0 with coordinates in 푂퐶, (퐾, 퐾). Fontaine shows that if 퐾 ⊂ 퐾 is a finite extension (푛) of 퐾, the functoriality map is a filtered isomorphism. As 휃 ((푥 ) ) = ∑ 푝푛푥 . 0 푛 푛≥0 푛 a consequence, 퐵 does not depend upon 퐾, and has a 푛≥0 dR natural 퐾-algebra structure. He shows that 휃 is an initial object in the category of sur- + 0 (b) As 퐵 is a complete discrete valuation ring of jective 푊-algebra homomorphisms 훽 ∶ 퐷 → 푂 , with ker- dR 퐶 residue characteristic 0, it has a (noncanonical) 퐶-algebra nel 퐼, such that 퐷 and the 퐷/퐼푚+1 are 푝-adically separated ∼ structure. However, one can show that there does not exist 푚+1 and complete, and 퐷 −→lim 퐷/퐼 . 퐺 퐵+ → 퐶 ←−−푚 any -equivariant section of the reduction map dR .

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1019 (c) There exist Hodge–Tate representations which are 푉 can be recovered from the associated filtered module 퐺 not de Rham. 푀 = (퐵cris ⊗ℚ푝 푉) by

∼ 6. Fontaine Rings III : 퐁퐜퐫퐢퐬 푉 → 햧허헆filt.휑-mod.(퐾0, 퐵cris ⊗퐾0 푀). The smallest of the three main Fontaine rings, 퐵cris, was introduced at the very beginning of the theory, in order to Finally, one sees from the above diagram that crystalline control the 푝-adic ´etalecohomology of algebraic varieties representations are de Rham. 14 with good reduction over 푂퐾 . There are actually several 7. Fontaine Rings IV : 퐁퐬퐭 subrings of 퐵dR which can be used for that purpose, and the ring 퐵 initially used by Fontaine in [7] was later replaced by Using de Jong’s theorems [MR1423020], many questions relative to varieties over a discrete valuation field can be the ring now known as 퐵cris, better adapted to comparison with crystalline cohomology (see [8], [14]). reduced to the case of varieties with semistable reduction. It A basic tool in crystalline cohomology is the PD- is therefore essential to understand where the cohomology envelope of an ideal: given a commutative ring 퐴, and an of these varieties stands in Fontaine’s classification. ideal 퐽 ⊂ 퐴, there exist a ring homomorphism 휌 ∶ 퐴 → In analogy with the complex and ℓ-adic cases, Fontaine and Janssen [R1012170] conjectured that the de Rham co- 풫퐴(퐽) and a 푃퐷-ideal 퐽 ⊂ 풫퐴(퐽) such that 휌(퐽) ⊂ 퐽, and which are universal for data of this kind (see footnote 9). homology of these varieties could be endowed with an ad- ditional structure provided by a nilpotent endomorphism, We return to the setup of the construction of 퐵dR, and the monodromy operator. This was proved by Hyodo and we consider again the kernel of the surjective map 휃0 ∶ ♭ Kato [MR1293974], and Fontaine introduced a new ring 푊(푂퐶) → 푂퐶. Let 퐴cris be the 푝-adic completion of the 퐵st in order to control the cohomology of varieties with PD-envelope of Ker(휃0). The group 퐺 acts by functoriality semistable reduction. on 퐴cris. Repeating the construction made for 퐵dR, one gets We will describe 퐵st as a subring of 퐵dR, which requires an equivariant map ℤ푝(1) → 퐴cris. If 푡 ∈ 퐴cris is the image + us to fix a prime element 휋 ∈ 푂퐾 . Let 푠 = (푠푛)푛≥1 be a se- of a generator of ℤ푝(1), one defines 퐵cris = 퐾0 ⊗푊 퐴cris and 푝 + quence of elements of 푂 such that 푠 = 휋 and 푠 = 푠 퐵 = 퐵 [푡−1]. Extending by functoriality the action of 퐺 퐶 0 푛+1 푛 cris cris for all 푛 ≥ 0, and let 푠 ∈ 푂 /푝 be the reduction of to 퐵 , one gets (퐵 )퐺 = 퐾 . 푛 퐶 cris cris 0 푠 푠 = (푠 ) 푂♭ From the constructions of 퐵 and 퐵 , one checks that 푛. The sequence 푛 푛≥0 is an element of 퐶, and dR cris 휀(푠) = [푠] there are natural injective maps we denote by its Teichmüller representative in 푊(푂♭ ) ⊂ 퐴 휀(푠)휋−1 ∈ 1 + 1퐵 / / / 퐶 cris. Then Fil dR, and its loga- 퐴 퐵+ 퐾 ⊗ 퐵+ 퐵+ + cris cris _ 퐾0 _ cris dR _ rithm converges in 퐵dR towards an element 푢푠. Fontaine + + defined 퐵st = 퐵cris[푢푠], 퐵st = 퐵cris[푢푠]. He proved that 푢푠 is + transcendental over 퐵cris, and that the canonical morphism / / 퐾 ⊗퐾 퐵st → 퐵dR is injective. One endows 퐾 ⊗퐾 퐵st with 퐵cris 퐾 ⊗퐾0 퐵cris 퐵dR. 0 0 the filtration induced by the filtration of 퐵dR. This allows one to endow 퐾 ⊗ 퐵 with the filtration 퐾0 cris These subrings of 퐵dR are stable under the action of 퐺, 퐵 퐺 induced by the filtration of dR. and we again have (퐵st) = 퐾0. The action of Frobenius A functoriality argument shows that the Frobenius of on 퐵cris extends to 퐵st by setting 휑(푢푠) = 푝푢푠. In [15], ♭ 푂퐶 defines a Frobenius action 휑 on 퐴cris; as 휑(푡) = 푝푡, this Fontaine defined the monodromy operator 푁 ∶ 퐵st → 퐵st + 15 action extends to 퐵cris and 퐵cris and commutes with the 퐺- as being the unique 퐵cris-derivation such that 푁(푢푠) = 1. action. Fontaine calls filtered 휑-module the datum of a 퐾0- He called filtered (휑, 푁)-module the datum of a filtered 휑- vector space 푀0 endowed with a Frobenius action and a module endowed with a 퐾0-linear endomorphism 푁 such 퐾 ⊗ 푀 퐵 filtration on 퐾0 0. With these definitions, cris has that 푁휑 = 푝휑푁. In this way, 퐵st is endowed with a struc- a structure of a filtered 휑-module, compatible with the 퐺- ture of filtered (휑, 푁)-module, compatible with the action + action. of 퐺. Note that the subrings 퐵st and 퐵st, as well as 휑 and 푁, A 푝-adic representation 푉 is said to be crystalline if it is depend upon the choice of the prime element 휋, but not 퐵cris-admissible, i.e., if the canonical map upon 푠. 퐺 As with the previous rings, we will say that a ℚ푝- 훼퐵cris (푉) ∶ 퐵cris ⊗퐾0 (퐵cris ⊗ℚ푝 푉) → 퐵cris ⊗ℚ푝 푉 representation 푉 is semistable if it is 퐵st-admissible. When is an isomorphism. When this is the case, 훼퐵 (푉) is a cris 푉 is semistable, the isomorphism 훼퐵 (푉) is compatible 퐺-equivariant filtered isomorphism, commuting with the st Frobenius actions. Moreover, a crystalline representation 15To insure a better compatibility with crystalline cohomology, Tsuji uses in 14 i.e., which can be extended as a proper and smooth scheme over 푂퐾 . [MR1705837] the opposite convention 푁(푢푠) = −1.

1020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 with all structures, and the representation 푉 can be recovered from the associated filtered module 푀 by ∼ 푉 → 햧허헆filt.(휑,푁)-mod.(퐾0, 퐵st ⊗퐾0 푀). Semistable representations are de Rham, crystalline representations are semistable, and a semistable representation is crystalline if and only if 푁 = 0. 8. Fontaine Conjectures As expected, a first, and fundamental, application of Fontaine’s theory is a classification of 푝-adic representations. In contrast with ℓ-adic representations of 퐺퐾 (ℓ ≠ 푝), whose structure is rather simple,16 푝-adic representations of 퐺퐾 have a complicated structure: for example, even when a variety 푋 over 퐾 has good reduction, the inertia group may act wildly on 퐻et́ (푋퐾 ,ℚ푝), while it acts trivially in the ℓ-adic case, ℓ ≠ 푝. Until Fontaine set up his formalism of 퐵 rings, the understanding of this wildness and a classification of 푝-adic representations of 퐺퐾 seemed hope- less. Using Fontaine’s rings 퐵cris, 퐵st, 퐵dR, 퐵HT, and the cor- Figure 5. Jean-Marc Fontaine making his point with Takeshi responding 퐵-admissibility and potential 퐵-admissibility Saito. conditions, we now have a hierarchy (with a “?” discussed as Conjecture/Theorem 6): theorem of Berthelot–Ogus [MR0700767], there is a crystalline ⇒ semistable ⇒ de Rham canonical isomorphism ∼ ⇓ ⇓ ? ‖ 푚 푚 ⇐ 퐾 ⊗푊 퐻cris(풳푘/푊) −→퐻dR(푋/퐾) pot. cryst. ⇒ pot. semist. ⇒ de Rham where 퐻푚 (풳 /푊) is the crystalline cohomology of the ⇓ cris 푘 special fiber 풳푘. This endows the 퐾0-vector space 퐾0 ⊗ Hodge–Tate 푚 퐻cris(풳푘/푊) with a structure of filtered 휑-module. Com- Hints for this classification came of course from geome- bining this structure with the analogous one on 퐵cris, try. Fontaine expected that the geometric properties of an Fontaine made the following conjecture [7]: algebraic variety 푋 over 퐾 would be reflected in the position of its 푝-adic ´etalecohomology spaces in this hierarchy, Conjecture/Theorem 3 (퐂퐜퐫퐢퐬). With the above notation, 푚 and he made a series of conjectures to that effect. 퐻et́ (푋퐾 ,ℚ푝) is crystalline, and associated with 퐾0 ⊗푊 푚 Assume only that 푋 is proper and smooth over 퐾. 퐻cris(풳푘/푊); i.e., there exists a canonical isomorphism ∼ Fontaine made the following conjecture [7]: 푚 푚 퐵cris ⊗푊 퐻cris(풳푘/푊) → 퐵cris ⊗ℚ푝 퐻et́ (푋퐾 ,ℚ푝) Conjecture/Theorem 2 (퐂퐝퐑). Under the previous assump- tions, 퐻푚(푋 ,ℚ ) is de Rham, and associated with 퐻푚 (푋/퐾) compatible with Frobenius and Galois actions on both sides, and et́ 퐾 푝 dR 퐵 endowed with its Hodge filtration; i.e., there exists a canonical with the filtrations after extension to dR. isomorphism In the semistable reduction case, where 푋 = 풳퐾 for a ∼ proper semistable 푂 -scheme 풳, Hyodo and Kato used the 퐵 ⊗ 퐻푚 (푋/퐾) → 퐵 ⊗ 퐻푚(푋 ,ℚ ), 퐾 dR 퐾 dR dR ℚ푝 et́ 퐾 푝 theory of logarithmic structures initiated by Fontaine and Il- compatible with filtrations and Galois actions on both sides. lusie, and developed by Kato and many others, to replace the crystalline cohomology of 풳푘 by its log crystalline co- Using the results of section 5, one sees that 퐂퐝퐑 implies homology. Generalizing the aforementioned theorem of 퐂퐇퐓. Berthelot–Ogus, they defined a canonical isomorphism ∼ Assume now that 푋/퐾 has good reduction, and let 풳 be a 푚 푚 퐾 ⊗푊 퐻logcris(풳푘/푊) → 퐻dR(푋/퐾) (see [MR1293974]) proper and smooth 푂퐾 -scheme such that 푋 = 풳퐾 . By a 푚 which endows 퐾0 ⊗ 퐻logcris(풳푘/푊) with a structure of fil- (휑, 푁) 16By Grothendieck’s local monodromy theorem, when no finite extension of 푘 tered -module. Combining this structure with the contains all the roots of unity of order a power of ℓ, the action of the inertia analogous one on 퐵st, Fontaine and Jannsen made the fol- group is quasi-unipotent. lowing conjecture ([11], [MR1293975]):

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1021 Conjecture/Theorem 4 (퐂퐬퐭). With the above notation, semistability condition for vector bundles on curves: these 푚 퐻et́ (푋퐾 ,ℚ푝) is semistable, and associated with 퐾0 ⊗푊 polygons have the same terminal points, and, for any sub- 푚 퐻logcris(풳푘/푊); i.e., there exists a canonical isomorphism module, Newton is above Hodge. He called the filtered (휑, 푁) ∼ -modules satisfying this property weakly admissible, 푚 푚 퐵st ⊗푊 퐻logcris(풳푘/푊) → 퐵st ⊗ℚ푝 퐻et́ (푋퐾 ,ℚ푝) and he made the following conjecture ([5], [15]): compatible with 휑, 푁, and Galois actions on both sides, and Conjecture/Theorem 5. Weakly admissible (휑, 푁)-modules with the filtrations after extension to 퐵dR. are admissible.

By now conjectures 퐂퐝퐑, 퐂퐜퐫퐢퐬, and 퐂퐬퐭 (and in particular This conjecture resisted all attempts for over twenty 퐂퐇퐓) have been fully proved. Actually, 퐂퐜퐫퐢퐬 turns out to be years. Exploiting the analogy between weak admis- a particular case of 퐂퐬퐭 (with 푁 = 0), and it was shown, us- sibility and semistability, Faltings proved a partial re- ing de Jong’s alterations, that 퐂퐬퐭 implies 퐂퐝퐑. Conjecture sult in this direction, namely, the stability (in the crys- 퐂퐬퐭 was first proved by Tsuji [MR1705837]. talline case) of weak admissibility under tensor product The history of the proofs of these conjectures is long [MR1403965]. The semistable case was then treated by and involves many mathematicians. Bloch and Kato Totaro [MR1388844]. Conjecture 5 was eventually proved had proved 퐂퐇퐓 in the case of good ordinary reduction in 2000 by Colmez and Fontaine [MR1779803]. More re- [MR0849653]. However their method did not extend to cently, Fargues and Fontaine gave a beautiful new proof, the general case, and gave no idea on how to attack 퐂퐜퐫퐢퐬. making the analogy become a true correspondence (see The first breakthrough was made by Fontaine and Messing section 12). [10], who proved 퐂퐜퐫퐢퐬 for 퐾 = 퐾0 and dim 푋 < 푝. For 9.2. Potentially semistable representations. As semi- this they introduced a new and fruitful technique, that of stable representations are de Rham, and 퐵dR does not syntomic sheaves and complexes. Kato’s proof of 퐂퐬퐭 (under change if 퐾 is replaced by a finite extension contained in certain restrictions of dimension) and Tsuji’s proof (in the 퐶, potentially semistable representations are also de Rham. general case) basically follow Fontaine–Messing’s method. However, no example was known of a de Rham represen- At about the same time as Fontaine–Messing’s work, Falt- tation that would not be potentially semistable, which is ings developed quite a different approach, based on the symbolized by the “?” in the hierarchy displayed in sec- theory of almost ´etaleextensions, a far-reaching generaliza- tion 8. This led Fontaine to the conjecture [15]: tion of some fundamental results of Tate in [MR0231827]. Conjecture/Theorem 6 (퐂 ). De Rham representations are A flaw was found in his first proofs of 퐂 and 퐂 , which 퐩퐬퐭 퐝퐑 퐜퐫퐢퐬 potentially semistable. he corrected in [MR1922831], giving a new proof of 퐂퐬퐭. Finally, Niziol [MR2372150] proved 퐂퐬퐭 by still another This conjecture was hard, too. Several proofs were (in- method, based on higher 퐾-theory, and showed that the dependently) given around 2002, by Andr´e[MR1906151], comparison isomorphisms established by all these differ- Kedlaya [MR2119719], and Mebkhout [MR1906152], us- ent methods were actually the same. More recently, Beilin- ing new ingredients: the theory of (휑, Γ)-modules (another son gave simple proofs of these isomorphisms, by a totally fundamental contribution of Fontaine [12]; see the end of different approach, using techniques of derived algebraic section 10), and that of 푝-adic differential equations over geometry (see [MR2904571], [MR3272051]). the Robba ring (see [Berger, MR1906150]). A simplified proof was later given by Colmez [MR2493217]. 9. More Conjectures In addition to these conjectures, the development of 10. Impact on Arithmetic Geometry Fontaine’s theory has raised some new questions involving Fontaine’s work has rapidly found deep applications to al- the structures used in his classification of ℚ푝-representions. gebraic geometry. A beautiful example was given by his The solutions of the next two conjectures have been very proof of a conjecture of Shafarevich: a good control of important steps in 푝-adic Hodge theory. the ramification of finite commutative group schemes over 9.1. Weak admissibility. From the very beginning, Fon- ℤ allowed him to show that there is no nonzero abelian taine wanted to give an intrinsic characterization of the scheme over Spec ℤ [9] (the conjecture was also proved, filtered (휑, 푁)-modules that could be associated with a independently, by Abrashkin). In the same vein, the full crystalline (and, later on, semistable) representation, call- force of Fontaine’s theory served as a crucial ingredient in ing them in short “admissible modules.” He investi- many global results and conjectures over number fields. gated the Newton and Hodge polygons defined by the Let us present here very briefly a few examples. Frobenius action and the filtration of such a module, and 10.1. The Bloch-Kato conjecture on special values of he discovered a remarkable property, reminiscent of the 퐿-functions. Let 푘 be a number field, and let 푉 be the

1022 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 푝-adic realization of a Chow motive 푀 over 푘, e.g., 푉 = finite number of places, and potentially semistable at every place 푚 퐻 (푋푘,ℚ푝) for 푋 projective and smooth over 푘. Using the above 푝. Then 휌 is a subquotient of a representation of the form 푚 rings 퐵cris associated with the places of 푘 over 푝, Bloch and 퐻 (푋푘,ℚ푝) for some proper and smooth 푋/푘 and 푚 ∈ ℤ. Kato define in [MR1086888] a subspace The case 푛 = 1 had been known for a long time. For 퐻1 (푘, 푉) ⊂ 퐻1(푘, 푉), 푓 푛 = 2 and 푘 = ℚ, Conjecture 8 takes the more precise from which they in turn define a subgroup form: 1 1 퐻푓(푘, 푀) ⊂ 퐻 (푘, 푀) Conjecture/Theorem 9. Let 휌 ∶ 퐺ℚ → GL2(ℚ푝) be a contin- of a motivic cohomology group, closely related to a Tate– uous representation. Assume that 휌 is irreducible, odd, unrami- Shafarevich group when 푀 is the motive of an abelian va- fied outside a finite number of primes, and potentially semistable riety over 푘. The Bloch–Kato conjecture predicts the or- at 푝. Then there exists 푖 ∈ ℤ and a newform 푓 as above such 1 ∨ 푖 ∗ der of vanishing of 퐿(푀, 푠) at 푠 = 0 as dim 퐻푓(푘, 푀 (1)) − that 휌 is isomorphic to 휌푓 ⊗ 휒푝, where 휒푝 ∶ 퐺ℚ → ℤ푝 is the dim 퐻0(푘, 푀∨(1)) and determines the leading coefficient cyclotomic character. of the Taylor series expansion of 퐿(푀, 푠) at zero, by a formula generalizing the Birch and Swinnerton-Dyer conjec- Conjecture 9 has been proved independently by Emer- ture. An equivariant version, due to Kato, plays an impor- ton [MR2251474] and by Kisin (with some mild technical tant role in Iwasawa theory. See [MR1086888], [13]. restrictions [MR2505297]). In addition to Kisin’s theory of 10.2. Serre’s modularity conjecture. The following con- potentially semistable deformation rings ([MR2373358], jecture is due to Serre [MR0885783]: [MR2827797]), the proofs use a deep, new ingredient, a 푝-adic local Langlands correspondence between absolutely ir-

Conjecture/Theorem 7. Let 푟 ∶ 퐺ℚ = Gal(ℚ/ℚ) → GL2(퐹) reducible 2-dimensional 푝-adic representations of 퐺ℚ푝 = be a continuous, irreducible, and odd representation, where 퐹 is Gal(ℚ푝/ℚ푝) and certain absolutely irreducible unitary Ba- a finite field of characteristic 푝. Then 푟 is modular, i.e., is the re- nach representations of GL2(ℚ푝), constructed by Breuil duction modulo the maximal ideal of the ring of integers 풪 of a in some cases, and by Breuil and Colmez in general (see finite extension of ℚ푝 of the representation 휌푓 ∶ 퐺ℚ → GL2(풪) [MR2906353] and Ast´erisque319, 330, 331). Their con- associated with a newform 푓 ∈ 푆푘(Γ1(푁)), for a suitable weight struction heavily relies on Fontaine’s theory of (휑, Γ)-modules. 푘 and level 푁 coprime to 푝. This theory, introduced by Fontaine [12], and developed The precise conjecture prescribes 푘 and 푁 in terms of by Berger, Cherbonnier, Colmez, and Kedlaya, describes 푝 퐺 푟. It is known that this conjecture implies the Shimura– -adic representations of ℚ푝 in terms of modules over Taniyama–Weil modularity conjecture for elliptic curves power series rings in one variable over ℚ푝 (or a finite ex- over ℚ, proved by Wiles, Taylor–Wiles et al. Con- tension) equipped with certain operators (a “Frobenius” ∗ jecture 7 has been proved by Khare and Wintenberger 휑 and an action of a group Γ isomorphic to ℤ푝). [MR2551764], modulo some hypothesis, later proved by Kisin [MR2551765]. 11. Norm Fields and Tilts The proof heavily relies on modularity lifting theorems Spectacular generalizations and applications of Fontaine’s due to Kisin [MR2600871] et al. for 푝-adic representations theory have been developed by Peter Scholze. His theory of perfectoid spaces has its roots in: of 퐺ℚ with prescribed ramification at 푝, expressed in terms of Fontaine’s classification (crystalline, semistable, poten- (a) Fontaine’s construction of the 퐵 rings, as seen above; tially semistable). (b) Fontaine–Wintenberger’s theory of norm fields. These constructions make sense for complete, valuative 10.3. The Fontaine–Mazur conjecture. Let 푘 be a num- extensions 퐿 of 퐾 of height one such that the Frobenius en- ber field, and let 푝 be a prime number. Let 퐺 = Gal(푘/푘) 푘 domorphism of 푂 /푝 is surjective. In ([4], [MR0719763]), be the absolute Galois group of 푘. Let 푋/푘 be proper 퐿 Fontaine and Wintenberger constructed, for a large class of and smooth and let 푚 ∈ ℤ. The 푝-adic representation infinite extensions 퐸 of 퐾 (called arithmetically profinite), a 퐻푚(푋 ,ℚ ) of 퐺 is unramified oustside a finite setof et́ 푘 푝 푘 field 퐸˜ of characteristic 푝, called the norm field of 퐸, with places of 푘, and, by the 푝-adic comparison Theorem 3 the remarkable property that the absolute Galois groups of (together with 6), is potentially semistable at any prime 퐸 and 퐸˜ are canonically isomorphic. Moreover, if 퐿 is the above 푝. The same is true of any subquotient. Fontaine 푝-adic completion of 퐸, the fraction field 퐿♭ of 푂♭ is the and Mazur conjectured the following converse [16]: 퐿 completion of the radicial closure of 퐸˜ (hence the abso- ♭ Conjecture/Theorem 8. Let 휌 ∶ 퐺푘 → GL푛(ℚ푝) be a contin- lute Galois groups of 퐿 and 퐿 are also canonically isomor- 1/푝∞ uous, irreducible representation which is unramified outside a phic). As an example, for 퐾 = ℚ푝, 퐸 = ℚ푝(푝 ), we have

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1023 1/푝∞ 퐸˜ = 픽푝((푡)), and the absolute Galois groups of ℚ푝(푝 ) contributors to the development of 푝-adic Hodge theory. and 픽푝((푡)) are canonically isomorphic. Jean-Marc Fontaine has left us now, but he will still be Scholze called fields such as 퐿 perfectoid, and 퐿♭ the tilt present among us for a very long time, thanks to the rich- of 퐿. It was the starting point of his theory of perfectoid ness of his mathematical heritage. spaces, which has had several striking applications outside of the 푝-adic world, e.g., the proof of the weight monodromy conjecture in ℓ-adic cohomology in mixed characteristic for complete intersections in the projective space [MR3090258]. 12. The Fargues–Fontaine Curve In the first decade of this century, most of the conjectures that fueled the development of 푝-adic Hodge theory had been proved, and the landscape of Fontaine’s rings was looking almost familiar. One could then have thought that the theory had essentially reached its final shape. But that would have been ignoring how much Fontaine liked to change perspective and develop new points of view on everything, including his own work. So, in a joint work with Laurent Fargues, he surprised everyone again. In [18], [19] (considerably amplified in Figure 6. Jean-Marc Fontaine attending a lecture with Pierre Colmez and Jean-Pierre Serre. [20]), they constructed and studied a Noetherian scheme 푋 over ℚ푝, regular of dimension 1, now called the Fargues– Fontaine curve. This geometric object, obtained by gluing pieces of Fontaine’s ring 퐵cris, sheds a totally new light on 푝-adic Hodge theory. Though not being of finite type over ℚ푝, it resembles 0 the projective line over ℚ푝: 퐻 (푋, 풪) = ℚ푝, and every vector bundle over 푋 is a sum of line bundles. If 퐾 is a finite extension of ℚ푝, the absolute Galois group 퐺퐾 = Gal(퐾/퐾) acts on 푋, and Fargues and Fontaine establish an equivalence of categories between ℚ푝-representations of 퐺퐾 and 퐺퐾 -equivariant vector bundles on 푋 which are Pierre Berthelot semistable of slope zero. This leads to simpler, more con- ceptual proofs of Conjectures 5 and 6. But there is a much deeper potential application, namely, the Fargues Fontaine’s Rings and Geometry geometrization conjecture for the local Langlands correspondence [arXiv:1602.00999], intensely studied today with Scholze’s theory of diamonds [arXiv:1709.07343]. Peter Scholze One of the most striking results of Fontaine is his theory To Conclude. . . of the field of norms, developed jointly with his student Solving the problem of the “mysterious functor” was one Jean-Pierre Wintenberger. While the field of the biggest challenges left by Grothendieck in the early 푛 seventies. To that effect, Jean-Marc Fontaine patiently ℚ푝 = { ∑ 푎푛푝 ∣ 푎푛 ∈ {0, 1, … , 푝 − 1}} 푛≫−∞ and systematically developed a program which unlocked a whole new area of number theory and algebraic geome- of 푝-adic numbers and the field 푛 try, apparently out of reach before his work. He has been 픽푝((푡)) = { ∑ 푎푛푡 ∣ 푎푛 ∈ {0, 1, … , 푝 − 1} = 픽푝} a leading force in the subject during his whole life, con- 푛≫−∞ stantly enriching it with new methods, results, and con- Peter Scholze is a professor of mathematics at the Mathematisches Institut, Uni- jectures. He was always ready to share his ideas, and sev- versität Bonn, and Max-Planck-Institut für Mathematik, Bonn, Germany. His eral of his former students have been themselves major email address is [email protected].

1024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 18 share many formal similarities, there is no direct algebraic Similarly, a conjecture of Fontaine, known as (퐶dR), as- relationship. For example the absolute Galois group of ℚ푝 serts that for a smooth variety 푋 over ℚ푝, there is a natural is topologically finitely generated, while that of 픽푝((푡)) is isomorphism not. However: 푖 푖 퐻 (푋ℂ푝 ,ℤ푝) ⊗ℤ푝 퐵dR ≅ 퐻dR(푋) ⊗ℚ푝 퐵dR Theorem (Fontaine–Wintenberger, [4]). Consider the fields 퐵 푛 between ´etaleand de Rham cohomology. Here dR is the 퐾푛 = ℚ푝(휁푝푛 ) obtained from ℚ푝 by adjoining a primitive 푝 th + fraction field of 퐵dR. At a concrete level, this means that root of unity 휁푝푛 , and let 퐾∞ = ⋃ 퐾푛 be their compositum. given a differential form and a cycle, one can get periods The absolute Galois groups of 퐾∞ and 픽푝((푡)) are canonically 푑푧 in 퐵dR. This can be applied to analogues of 휔 = and 훾 isomorphic. 푧 + as above, leading in that case to an element 푡 ∈ 퐵dR that is The result holds for many other infinite extensions of + in fact a uniformizer of 퐵dR. This is the so-called “푝-adic ℚ푝, e.g., the one obtained by adjoining all 푝-power roots of analogue of 2휋푖.” In particular, the periods thus obtained 푝. Contemplating this result was a main inspiration for the do not lie in ℂ푝. theory of perfectoid spaces. While I was working on these + Now I claim that by defining 퐵dR, Fontaine was the first results, still a PhD student, I was given the opportunity to person to get a glimpse of a part of the elusive “arithmetic present this work at a course at the IHES in the fall of 2011. surface” It was an enormous honor that Fontaine attended these Spec ℤ × Spec ℤ . lectures, and the support he expressed was most significant Recall that in algebraic geometry, there is a space Spec 퐴 to me. associated with any commutative ring 퐴. For the simplest In this note, I want to focus however on another object (namely, initial) ring 퐴 = ℤ of the integers, the points of introduced by Fontaine: his ring of 푝-adic periods 퐵+ . Let dR Spec ℤ are exactly the prime numbers 2, 3, 5, 7, 11, …, plus me give a brief summary of its character. a generic point 휂. Any other ring 퐴 admits a unique map The ring 퐵+ is a complete discrete valuation ring whose dR ℤ → 퐴, which induces a map Spec 퐴 → Spec ℤ. In other residue field is the field ℂ푝, the completion of ℚ푝. More- words, all of algebraic geometry lives over the base space over, it has a natural topology, inducing the usual 푝-adic Spec ℤ. topology on ℂ푝, and a natural action of the absolute Ga- However, as emphasized for example by Weil, in many lois group 퐺 . ℚ푝 respects Spec ℤ behaves like a curve, in fact very much like Note that by Cohen’s structure theorem, there is an iso- the affine line Spec 픽 [푇] over a finite field 픽 . The the- + 푝 푝 morphism of 퐵dR with a power series ring over ℂ푝, as com- orem of Fontaine–Wintenberger is one of the most pro- plete discrete valuation rings whose residue field ℂ푝 is of found manifestations of this analogy. In the arithmetic of 17 characteristic 0: Spec 픽푝[푇], a critical role is played by the arithmetic surface + 퐵dR ≅ ℂ푝[[푡]]. Spec 픽푝[푇] ×픽푝 Spec 픽푝[푇] = Spec 픽푝[푇1, 푇2]. However, there is no such isomorphism that is continuous, In particular, Weil’s proof of his conjectures for curves is and no such isomorphism that is Galois-equivariant. based on intersecting curves on this surface. The original motivation for introducing the ring 퐵+ dR Unfortunately, no such space is available for Spec ℤ, pre- (and for its name, the “ring of 푝-adic (de Rham) periods”) cisely because all objects in algebraic geometry come with is the role that it plays in 푝-adic Hodge theory. Recall that a unique map to Spec ℤ. It has long been dreamt that there if 푋 is a smooth complex variety, then there is a natural is still some way to make sense of isomorphism Spec ℤ × Spec ℤ , 푖 푖 퐻 (푋, ℤ) ⊗ℤ ℂ ≅ 퐻dR(푋) but it is hard to fill this dream with interesting between singular and de Rham cohomology. On a con- mathematics—taken inside schemes, the best one can do crete level, this means that one has a pairing between dif- is Spec ℤ ×Spec ℤ Spec ℤ, which is just Spec ℤ itself. Dream- × ferential forms and cycles; e.g., if 푋 = ℂ , one can take ing on for a moment, there should be a diagonal embed- 푑푧 휔 = and the circle 훾 = 푆1 ⊂ ℂ× to get the period ding 푧 Δ ∶ Spec ℤ ↪ Spec ℤ × Spec ℤ , 1 1 푑푧 푑(푒2휋푖푡) ∫ = ∫ = ∫ 2휋푖푑푡 = 2휋푖 ∈ ℂ . and the completion of Spec ℤ × Spec ℤ along Δ should be 2휋푖푡 푆1 푧 0 푒 0 some 1-parameter deformation of Spec ℤ. More generally, for any ring 퐴, there should be an object Spec 퐴 × Spec ℤ, 17From what I hear, it took several decades for the mathematical community to + 18 understand the significance of 퐵dR, not least because it’s just ℂ푝[[푡]] after all! First proved by Tsuji, with now many different proofs known.

AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1025 and the natural map Spec 퐴 → Spec ℤ should induce a graph Γ퐴 ∶ Spec 퐴 ↪ Spec 퐴 × Spec ℤ . The completion of Spec 퐴 × Spec ℤ should be a natural 1- parameter deformation of 퐴. This is precisely what Fontaine has defined in the case + of 퐴 = ℂ푝! Indeed Spec 퐵dR ≃ Spec ℂ푝[[푡]] is a certain canonical 1-parameter deformation of Spec ℂ푝. In joint work with Fargues on the Fargues–Fontaine curve, [20], Fontaine has actually gone further and has ar- Peter Scholze guably described all of Spec ℂ푝 × Spec ℤ푝, making it into References an actual mathematical object. By leaving the diagonal [1] Jean-Marc Fontaine, Extensions finies galoisiennes des corps Δ(Spec ℤ) ⊂ Spec ℤ × Spec ℤ near a prime 푝, Fontaine has valués complets à valuation discrète (French), Séminaire led us into a whole new and mysterious world. Delange-Pisot-Poitou: 1967/68, Théorie des Nombres, This was actually the motivation for my work on “di- Secrétariatmathématique, Paris, 1969, Fasc. 1, Exposé6, amonds,” about which I have also given a course at the 21 pp. MR0246857 IHES in March/April 2017. At the time, Fontaine had al- [2] Jean-Marc Fontaine, Groupes de ramification et ready been on cancer treatment. It was a very pleasant sur- représentations d’Artin (French), Ann. Sci. Ecole´ Norm. prise, and again a deep honor, that Fontaine was able to Sup. (4) 4 (1971), 337–392. MR289458 attend the talks. In fact, I was very happy to see him in [3] Jean-Marc Fontaine, Groupes 푝-divisibles sur les corps locaux (French), Astérisque, No. 47-48, SociétéMathématiquede good spirits, and as energetic as ever in his mathematics. France, Paris, 1977. MR0498610 His passion for mathematics was truly inspiring. [4] Jean-Marc Fontaine and Jean-Pierre Wintenberger, Le Where will his mathematics lead us? In joint work with “corps des normes” de certaines extensions algébriquesde corps Clausen, on January 24, 2019, we found an analogue of locaux (French, with English summary), C. R. Acad. Sci. + the ring 퐵dR for the real numbers ℝ. Again, it appears to Paris Sér. A-B 288 (1979), no. 6, A367–A370. MR526137 be just ℝ[[푡]] at first sight. However, we are working with [5] Jean-Marc Fontaine, Modules galoisiens, modules filtrés et the notion of what we termed an analytic ring, which is anneaux de Barsotti-Tate (French), Journéesde Géométrie roughly a topological ring 푅 equipped with a notion of 푅- Algébriquede Rennes (Rennes, 1978), Astérisque, vol. 65, valued measures on any profinite set 푆. It turns out that for Soc. Math. France, Paris, 1979, pp. 3–80. MR563472 0 < 푝 < 1, the space of “푝-measures” ℳ (푆, ℝ) with real [6] Jean-Marc Fontaine, Formes différentielles et modules de 푝 Tate des variétés abéliennes sur les corps locaux (French), coefficients has a canonical nonsplit self-extension, which + 2 Invent. Math. 65 (1981/82), no. 3, 379–409, DOI may be called ℳ푝(푆, 퐵dR(ℝ)/푡 ). It comes from the entropy 10.1007/BF01396625. MR643559 function and some avatars of it had been known in the [7] Jean-Marc Fontaine, Sur certains types de représentations functional analysis literature. We realized that this non- 푝-adiques du groupe de Galois d’un corps local; construc- split self-extension could even be put into an infinite tower, tion d’un anneau de Barsotti-Tate (French), Ann. of Math. + leading to spaces of measures ℳ푝(푆, 퐵dR(ℝ)), which have (2) 115 (1982), no. 3, 529–577, DOI 10.2307/2007012. a ring structure in the suitable sense. Moreover, the def- MR657238 inition of 퐵+ (ℝ) is similar to Fontaine’s definition, as a [8] Jean-Marc Fontaine, Cohomologie de de Rham, cohomolo- dR 푝 certain completion of a ring generated by “Teichmüller rep- gie cristalline et représentations -adiques (French), Alge- braic geometry (Tokyo/Kyoto, 1982), Lecture Notes in resentatives” of the reals. Math., vol. 1016, Springer, Berlin, 1983, pp. 86–108, DOI I immediately wondered whether I should inform 10.1007/BFb0099959. MR726422 Fontaine that his ideas had now truly become real; how- [9] Jean-Marc Fontaine, Il n’y a pas de variétéabéliennesur 퐙 ever, I first wanted to be sure that these ideas were correct. (French), Invent. Math. 81 (1985), no. 3, 515–538, DOI Fontaine died five days later, on January 29, 2019. 10.1007/BF01388584. MR807070 [10] Jean-Marc Fontaine and William Messing, 푝-adic periods and 푝-adic étalecohomology, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 179– 207, DOI 10.1090/conm/067/902593. MR902593 [11] Jean-Marc Fontaine, Letter to U. Jannsen, Nov. 26, 1987.

1026 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 [12] Jean-Marc Fontaine, Représentations 푝-adiques des corps locaux. I (French), The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 249–309. MR1106901 [13] Jean-Marc Fontaine, Valeurs spécialesdes fonctions 퐿 des motifs (French, with French summary), SéminaireBour- baki, Vol. 1991/92, Astérisque 206 (1992), Exp. No. 751, 4, 205–249. MR1206069 [14] Jean-Marc Fontaine, Le corps des périodes 푝-adiques (French), Périodes 푝-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 59–111.With an appendix by Pierre Colmez. MR1293971 [15] Jean-Marc Fontaine, Représentations 푝-adiques semi-stables (French), Périodes 푝-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 113–184. With an appendix by Pierre Colmez. MR1293972 [16] Jean-Marc Fontaine and Barry Mazur, Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 41–78. MR1363495 [17] Pierre Colmez and Jean-Marc Fontaine, Construction des représentations 푝-adiques semi-stables (French), Invent. Math. 140 (2000), no. 1, 1–43, DOI 10.1007/s002220000042. MR1779803 [18] Laurent Fargues and Jean-Marc Fontaine, Vector bundles and 푝-adic Galois representations, Fifth International Con- gress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt. 1, vol. 2, Amer. Math. Soc., Providence, RI, 2012, pp. 77–113. MR2908062 [19] Laurent Fargues and Jean-Marc Fontaine, Vector bundles on curves and 푝-adic Hodge theory, Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp. 17–104. MR3444231 [20] Laurent Fargues and Jean-Marc Fontaine, Courbes et fibrés vectoriels en théoriede Hodge 푝-adique (French, with English and French summaries), Astérisque 406 (2018), xiii+382. With a preface by Pierre Colmez. MR3917141

Credits Figures 1 and 4 are courtesy of Isabelle Fontaine. Figure 2 is courtesy of G´erardLaumon. Figure 3 is courtesy of Pierre Berthelot. Figures 5 and 6 are courtesy of the IHES/MCV. Photo of Nicholas M. Katz is courtesy of Charles Mozzochi. Photo of Peter Scholze is courtesy of the Archives of the Math- ematisches Forschungsinstitut Oberwolfach. The other author photos are courtesy of the authors.

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