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Bernard Dwork (1923–1998)

Nicholas M. Katz and

Bernard Morris describe that proof and sketch Bernie’s other main Dwork, “Bernie” to contributions to mathematics. those who had the But let us start at the beginning. Bernie was privilege of knowing born on May 27, 1923, in the Bronx. In 1943 he him, died on May 9, graduated from the City College of New York with 1998, just weeks a bachelor’s degree in electrical engineering. He short of his seventy- served in the United States Army from March 30, fifth birthday. He is 1944, to April 14, 1946. After eight months of survived by his wife training as repeaterman at the Central Signal Corps of fifty years, Shirley; School, he served in the Asiatic Pacific campaign

All photographs courtesy of Shirley Dwork. his three children, with the Headquarters Army Service Command. He Bernard Dwork, May 27, 1996. Andrew, Deborah, was stationed in Seoul, Korea, which, according to and Cynthia; his four reliable sources, he once deprived of electricity granddaughters; his brothers Julius and Leo; and for twenty-four hours by “getting his wires his sister, Elaine Chanley. crossed”. This is among the first of many “Bernie We mention family early in this article, both be- stories”, some of which are so well known that they cause it was such a fundamental anchor of Bernie’s are referred to with warm affection in shorthand life and because so many of his mathematical as- or code. For instance, “Wrong Plane” refers to the sociates found themselves to be part of Bernie’s time Bernie put his ninety-year-old mother on the extended family. The authors of this article, al- wrong airplane. “Wrong Year” refers to the time though not family by blood, felt themselves to be Bernie was prevented from flying to Bombay in Jan- son and older brother to him. uary 1967 by the fact that his last-minute request Bernie was perhaps the world’s greatest p-adic for a visa to attend a conference at the Tata Insti- analyst. His proof of the rationality of the zeta func- tute was denied on the grounds that the confer- tion of varieties over finite fields, for which he ence was to be held the following year, in 1968. was awarded the AMS in Number The- ory, is one of the most unexpected combinations After his discharge from the army, Bernie of ideas we know of. In this article we will try to worked as an electrical engineer by day and went to school by night, getting his master’s degree in electrical engineering from Brooklyn Polytechnic Nicholas M. Katz is professor of mathematics at Prince- Institute in 1948. Bernie and Shirley Kessler were ton University. His e-mail address is nmk@math. married on October 26, 1947. He worked succes- princeton.edu. sively for I.T.T. (1943–48, minus his army years), John Tate is professor of mathematics at the University the Atomic Energy Commission (1948–50), and the of Texas at Austin. His e-mail address is tate@math. Radiological Research Laboratory of Columbia utexas.edu. Medical Center (1950–52). During these years he

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wrote a number of technical reports and published formal thesis adviser (de- a few papers, including the earliest paper of his spite being two years listed by MathSciNet, “Detection of a pulse super- Bernie’s junior) when he imposed on fluctuation noise”, dating from 1950. spent 1953–54 visiting Co- In view of Bernie’s later interest in differential lumbia. equations (albeit from a p-adic point of view), it is Bernie received his interesting to note that this paper was reviewed by Ph.D. from Columbia in Norman Levinson. 1954, with a thesis enti- In the summer of 1947 Bernie, encouraged by tled “On the root number his brother, took an evening math course at New in the functional equation York University (NYU) with . The course of the Artin-Weil L-series”, was Higher Modern Algebra Part I: Galois Theory. about the possibility of The course notes, taken by Albert A. Blank, are still defining “local root num- one of the best “textbooks” on Galois theory avail- bers” for nonabelian Artin able. The following summer Bernie was back, to L-functions, whose prod- take Higher Modern Algebra Part II: Algebraic Num- uct would be the global ber Theory, again with Artin. In the summer of 1949 root number (:= constant In Europe, en route to 1962 ICM in Bernie took a course from Harold Shapiro on Se- in the functional equation Stockholm. Shirley and Bernie Dwork lected Topics in Additive Number Theory, and in connecting L(s,χ) and − with children Cynthia (left) and the summer of 1950 he took a course from Artin L(1 s,χ¯ )). Bernie solved Andrew. which was the precursor of Artin’s 1950–51 Prince- this problem up to sign; ton course Algebraic Numbers and Algebraic Func- Langlands (unpublished) tions.1 solved it definitively in 1968, and soon after, Deligne, inspired by the work of Langlands, found Bernie was hooked.2 He continued taking a more conceptual solution. evening courses, both at NYU and at Columbia. He Before we discuss Bernie’s later work, let us tried to enter NYU as a graduate student in math- record the key dates of his professional career as ematics, but NYU found his educational back- a mathematician. He spent 1954–57 at Harvard as ground wanting. Fortunately for us, Columbia was a Peirce Instructor, then 1957–64 at Johns Hopkins less picky: in February 1952 Columbia admitted (1957–60 as assistant professor, 1960–61 as as- him as a math graduate student, with a scholar- sociate professor, 1961–64 as professor). In 1964 ship for the year 1952–53. He quit his job and he moved to Princeton, where in 1978 he was took up full-time study in September of 1952. This named Eugene Higgins Professor of Mathematics. was hardly a light decision. He was walking away During this time, Bernie spent numerous sabbati- from a secure career as an engineer, and he had a cal years in France and Italy. In 1992 he was named wife and son to support. Years later he wrote, Professore di Chiara Fama by the Italian govern- “Imagine my horror when I found that the schol- ment and was awarded a special chair at the Uni- arship was only for tuition.” The family savings versity of Padua, which he occupied until his death. soon dwindled, and to make ends meet, Bernie We now return to Bernie’s work. In 1959 he taught night courses at Brooklyn Polytechnic from electrified the mathematical community when he February 1953 to June 1954. proved the first part of the Weil conjecture in a Bernie had intended, once at Columbia, to study strong form, namely, that the zeta function of any under Chevalley. But Chevalley returned to France algebraic variety (“separated scheme of finite type” that year, so he turned to Artin for advice.3 Artin in the modern terminology) over a finite field was gave Bernie a thesis problem and introduced him a rational function. What’s more, his proof did not to the second author of this article, then a young at all conform to the then widespread idea that the instructor at Princeton who later became Bernie’s would, and should, be solved by the construction of a suitable cohomology theory 1Artin was in those years professor of mathematics at for varieties over finite fields (a “Weil cohomology” , but he regularly taught evening courses at NYU in algebra and number theory. in later terminology) with a plethora of marvelous properties.4 2Strictly speaking, he was “rehooked”, because he had been Bernie’s proof of rationality was an incredible very drawn to mathematics in high school and college, but his parents convinced him that he would never earn a liv- tour de force, making use of a number of new and ing as a mathematician and that engineering would be an adequate outlet for his mathematical enthusiasm. Par- 4We should point out, however, that by 1964 the `-adic ents are not always right. étale cohomology with compact supports of M. Artin and 3The (alphabetically) first author of this article, who was Grothendieck had caught up with Bernie and provided an Bernie’s thesis student in 1964–66, has fond memories of alternate proof not only of Bernie’s rationality result but Bernie’s describing his anxieties and his car troubles, also of most of the rest of the Weil conjecture for projec- which included once breaking down on the George Wash- tive smooth varieties over finite fields, all but the “Riemann ington Bridge, on his early trips to Princeton to see Artin. Hypothesis”, which Deligne proved in 1973.

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An important fact is that Zeta(X/k, T ) has integer coefficients. To see this, let us define a closed point ℘ of X/k to be an orbit of the Galois group Gal(¯k/k) on the set X(¯k), and let us define the de- gree of ℘ to be the cardinality of the orbit which ℘ “is”. If we denote by Bd the number of closed points of degree d, we have the relation X Nd = rBr , r|d

hence the Euler product expansion Y Y − − Zeta(X/k, T )= (1 − T r ) Br = (1 − T deg(℘)) 1. r≥1 ℘ This last formula makes clear that Zeta(X/k, T ) as power series has integer coefficients. Let us interpret what it means for Zeta(X/k, T ) to be a rational function of T, say Y . Y Zeta(X/k, T )= (1 − αiT ) (1 − βj T ) . i j Shirley and Bernie Dwork with grandchildren at Bernie’s Taking logarithms of both sides and equating co- retirement party, October 1993. efficients of like powers of T, we see that rationality means precisely that all the integers Nd are de- unexpected ideas. We will describe it in some de- termined by the finitely many numbers αi and βj tail, because even after nearly forty years it remains by the rule strikingly fresh and original and gives a good idea X X d − d of the way Bernie thought. But before we say more Nd = βj αi . about Bernie’s proof, we must digress to say a few j i words about zeta functions of varieties over finite This has the striking consequence that all the N fields. d are determined by the first few of them. More pre- Thus, we begin with a finite field k—e.g., k cisely, once we know upper bounds, say A and B, might be Z/pZ for p a prime number—and an al- for the degrees of the numerator and denomina- gebraic variety X/k—e.g., X might be an “affine va- tor of Zeta, then all the integers N are deter- riety”, namely, the common zeroes of a finite col- d mined by the N for d ≤ A + B. lection of polynomials f (X ,... ,X ) in some finite d i 1 n Let us now describe Bernie’s proof of the ra- number n of variables with coefficients in the field tionality of Zeta(X/k, T ). By an elementary inclu- k. In this affine example, by a k-valued point of sion-exclusion argument, he reduces first to the X we mean an n-tuple (a ,... ,a ) in kn such that 1 n case when X/k is affine, then to the case when X/k f (a ,... ,a )=0in k for each defining equation i 1 n is defined by one equation f =0 for some poly- fi. In any case, for any algebraic variety X/k we nomial f in k[X ,... ,X ], and finally to the case have the notion of a k-valued point of X, and an 1 n when X/k is the open subset of f =0where all the easy but essential observation is that the set X(k) coordinates x are nonzero. To count points over of k-valued points of X/k is a finite set. It is known i k, he then uses a nontrivial additive character that inside a given algebraic closure ¯k of k there of k with values in a field K of characteristic zero. is one and only one field extension kd/k of each For a while in the argument, K could be C, but a degree d ≥ 1. Each field kd is itself finite, so the different choice of K will be handy in a moment. sets X(kd) are finite for all d ≥ 1. We denote by Bernie exploits the classical orthogonality rela- Nd ≥ 0 the integer tion: for a ∈ k, we have, in K, N := Card (X(k )). ( d d X 0 , if a =06 The integers N are fundamental diophantine in- (ya)= d ∈ q := Card (k) , if a =0. variants of X/k. The zeta function of X/k, y k

Zeta(X/k, T ), is simply a convenient packaging of n these integers: it is defined as the formal series in Taking for a the value f (x) at a point x in k , we T with coefficients in Q and constant term 1 by get ( X 0 , if f (x) =06 X (yf(x)) = d ∈ q, if f (x)=0. Zeta(X/k, T ) := exp NdT /d . y k d≥1

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Papers and Books by Bernie Ph.D. Students of Bernard Dwork Under Dwork, B., MathSciNet lists over seventy Kenneth Ireland (1964) items, including three books. Bernie also wrote two Alvin Thaler (1966) papers as Maurizio Boyarsky. Rather than recopy his Nicholas Katz (1966) complete list of publications, we give a complete Daniel Reich (1967) alphabetical list of Bernie’s coauthors, from which Stefan Burr (1968) the reader can correctly infer Bernie’s delight and enthusiasm in sharing ideas with colleagues around Philippe Robba* (c. 1972) the world. Steven Sperber* (1975) Edith Stevenson (1975) A. ADOLPHSON F. LOESER Jack Diamond (1975) F. BALDASSARRI A. OGUS S. BOSCH P. ROBBA Alan Adolphson (1976) S. CHOWLA S. SPERBER Mark Heiligman (1982) G. CHRISTOL F. J. SULLIVAN R. EVANS F. TOVENA *Unofficial students of Bernie G. GEROTTO A. J. VAN DER POORTEN

Letting k∗ = k r {0} we get The simplest example of a splitting function Θ(T ) q p−1 X relative to Fq is exp(π(T − T )), where π = −p, (yf(x)) = qN1 . although the fact that this is a splitting function y∈k is nontrivial. ∈ ∗n x k To see what this has bought us, lift the poly- The terms with y =0 in the above sum are each nomial f (X) over k to a polynomial over OK by lift- 1, so we get ing Peach coefficient to its Teichmuller lifting, say w X to Aw X , with each Aw its own q-th power. − − n (yf(x)) = qN1 (q 1) . The series Y y∈k∗ Θ w ∗ F(Y,X):= (Aw YX ) x∈k n w So far, nothing too exciting. But now come the has the property that for any point (y,x) in kn+1, fireworks, in three new ideas. The first is to express we have analytically a nontrivial additive character of k in (yf(x)) = F(Teich(y,x)). a way well adapted to the passage from k to kd. For this, Bernie now takes for K a large p-adic More generally, for any d ≥ 1, if we define field (large enough to contain the p-th roots of unity and all roots of unity of order prime to p) dY−1 qi qi and introduces the concept of a splitting function, Fd(Y,X):= F(Y ,X ), i.e., a power series in one variable over K, say i=0 Θ(T ), that converges in a disc |T | 1 such that the following two conditions hold: d

1) For a in k (= Fq), denote by Teich(a) d(yf(x)) = Fd(Teich(y,x)). its Teichmuller representative, the unique solution of Xq = X in the ring Let us recall how this relates to counting points. OK of integers in K that reduces We have to a in the residue field. Then X n a 7→ Θ(Teich(a)) is a nontrivial additive F(Teich(y,x)) = qN1 − (q − 1) , character of k, with values in K. y∈k∗ x∈k∗n 2) For each d ≥ 1, the nontrivial addi- and, more generally, for each d ≥ 1 we have tive character d of kd obtained by X d − d − n composing with the trace from kd to Fd(Teich(y,x)) = q Nd (q 1) . ∈ ∗ k is given as follows. For a in k , de- y kd d ∗ x∈k n note by Teich(a) the unique solution of d qd X = X in OK which reduces to a in The second new idea is to express a sum of the residue field. Then we have that form as the trace of a completely continuous dY−1 operator on a p-adic Banach space. On the space qi d(a)= Θ(Teich(a) ). of formal series over K in n +1variables, Bernie i=0 defines an operator Ψq by

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Prizes and Fellowships Awarded to (qd − 1)n+1Trace((Ψ ◦ F)d)=qdN − (qd − 1)n. Bernie q d Θ Sloan Fellowship, 1961 Because (T ) converges in a disc strictly bigger than the unit disc, F has good convergence prop- Cole Prize in Number Theory, 1962 erties. By restricting the action of Ψq ◦ F to a space Guggenheim Fellowship, 1964 of series with suitable growth conditions, one can Townsend Harris Medal, 1969 (from the make sense of Ψq ◦ F as a completely continuous Alumni Association of the City College of endomorphism of a p-adic Banach space. The Fred- − Ψ ◦ New York) holm characteristic series det(1 T q F) is an Guggenheim Fellowship, 1975 entire function of T, which as a formal power se- ries is given by

det(1 − T Ψq ◦ F) X − Ψ ◦ d d X X = exp Trace(( q F) )T /d . w w d≥1 Ψq Bw X := Bqw X . Denote by ∆(T ) the entire function Arguing heuristically,P one sees that for any formal det(1 − T Ψ ◦ F). Then from the identities w q series F = Cw X in the n +1 variables X0 := Y Ψ d n+1 d d d n and Xi for i =1to n, the composition of q with (q − 1) Trace((Ψq ◦ F) )=q Nd − (q − 1) multiplication by F has a trace, given by ≥ X for d 1 we get the identity of series Trace(Ψ ◦ F) = Trace Ψ ◦ C Xw q q w nY+1   (−1)n−i n+1 X ∆ i i w (q T ) = Cw Trace(Ψq ◦ X ) X i=0 Yn   = C(q−1)w . (−1)n−i n = Zeta(X/k, qT ) (1 − qiT ) i . The last equality is valid because in the “basis” i=0 Ψ ◦ w given by all monomials, the “matrix” of q X has Since ∆(T ) is p-adically entire, we see that the no nonzero terms on the diagonal unless w is zeta function is the ratio of two p-adically entire − (q 1)v for some v, in which case the (v,v) entry functions. is 1 and all other diagonal entriesP vanish. Still ar- To recapitulate, we now know that the zeta guing heuristically, weP interpret C(q−1)w in terms function as power series has integer coefficients w of values of F = Cw X at (n +1)-tuples of and that it is the ratio of two p-adically entire (q − 1)-th roots of unity in K by the identity functions. We also know the zeta function has a nonzero radius of archimedean convergence (since we have the trivial archimedean bound X X N ≤ (qd − 1)n). Bernie’s third new idea is to gen- − n+1 d (q 1) C(q−1)w = F(a) eralize a classical but largely forgotten result of a∈(µ − (K))n+1 Xq 1 E. Borel to show that any power series with these = F(Teich(y,x)) , three properties is a rational function. Thus he y∈k∗ proves the rationality of the zeta function. x∈k∗n Bernie then further developed his p-adic ap- proach and applied it to study in detail the zeta an identity that does hold, in fact, when F is a sin- function in the special case of a projective smooth gle monomial Xw or when F has coefficients tend- hypersurface X/k, say of dimension n and degree ing to zero. Thus we get a heuristic identity, using d. The Weil conjectures predicted that its zeta now the particular choice of F function should look like Y w  F(Y,X):= Θ(Aw YX ), Yn − n+1 w P(T )( 1) (1 − qiT ), namely, i=0 X n+1 with P a polynomial of known degree (namely, the (q − 1) Trace(Ψq ◦ F)= F(Teich(y,x)) ∗ middle “primitive” Betti number of any smooth pro- y∈k ∗ x∈k n jective hypersurface Hn,d over the complex num- n bers, of the same degree d and dimension n as X) = qN1 − (q − 1) . and that P(T ) should satisfy a certain functional The d-th iterate of Ψq ◦ F is easily checked to be equation. Bernie’s theory allowed him to confirm Ψ ◦ ≥ qd Fd, so for each d 1 we have a heuristic these predictions and to study the p-adic valua- identity tions of the reciprocal zeros of P(T ). He proved,

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for instance, that, provided p does not divide d, ilies as regular singular equations play for count- the Newton polygon of P(T ) always lies above the ing points in families. middle dimensional primitive “Hodge polygon” of Pursuing these ideas led Bernie to a long and Hn,d; cf. his Stockholm ICM talk, p. 259. This is the deep study of the p-adic and arithmetic properties first instance we know of a nontrivial relation be- of differential equations, both for their own sake tween Newton and Hodge polygons. Such relations and for their interaction with the arithmetic of va- were later established in great generality by Mazur. rieties over finite fields and with the algebraic Bernie also studied the way his p-adic con- geometry of families over C. He remained actively struction varied when the projective smooth hy- engaged in this study right up to his death. persurface varied in a family. The rich structure This is perhaps an appropriate point to com- ment on three early mathematical influences on he discovered was the first nontrivial instance of Bernie. what later came to be called an F-crystal. Roughly 1. We have already explained how it was an NYU speaking, an F-crystal is a differential equation evening course taught by Emil Artin in 1947 upon which a “Frobenius” operates. Bernie cor- which hooked Bernie on mathematics. rectly conjectured that the underlying differential 2. His interest in mod p and p-adic properties of equation to his F-crystal was the relative (primi- Picard-Fuchs equations probably dates from tive, middle dimensional) de Rham cohomology of the late 1950s at Johns Hopkins, when he the family, endowed with its Gauss-Manin con- learned from Igusa that the Hasse invariant for nection, i.e., the classical Picard-Fuchs equations the Legendre family of elliptic curves attached to the family. He also studied the varia- tion in a family of the Newton polygon attached y2 = x(x − 1)(x − λ) to an F-crystal, which led him to discover the in any odd characteristic p is a mod p solu- “slope filtration” of an F-crystal. It seems fair to tion of the Picard-Fuchs equation for that fam- say that a desire to understand Bernie’s results in ily (explicitly, the differential equation for the a more cohomological context was one of the main hypergeometric function F(1/2, 1/2, 1; λ)). motivations for the development, by Grothendieck 3. Where did Bernie get the idea that there could and Berthelot in the late 1960s, of crystalline co- be a connection between p-adic analysis and homology. zeta functions? It grew out of a letter from the One of Bernie’s key discoveries was that those second author of this article to Bernie, dated differential equations that “admit a Frobenius”, February 13, 1958, an extract of which is i.e., that underlie an F-crystal, have very special quoted on page 257 of Bernie’s ICM Stock- properties as p-adic differential equations (for in- holm talk. The letter contained a result on the stance, solutions have p-adic radius of conver- “unit root” of an ordinary elliptic curve, which gence 1 in generic discs). By crystalline theory, could be proved by using work of Michel any Picard-Fuchs equation underlies an F-crystal Lazard on formal groups to show that certain and hence has these special properties for almost p-adic power series have integral coefficients. all primes p. One enduring fascination of Bernie’s The letter writer, considering Bernie to be the was the still open problem of characterizing, by p- world’s leading expert in such matters, chal- adic conditions for almost all p, those differential lenged him to prove those results by p-adic equations over, say, number fields, that “come analysis. Bernie met the challenge almost by from geometry” (or “are motivic”, in the new ter- return mail and, going further, discovered the minology) in the sense, say, that they are succes- “close connection” that he mentions in the sive extensions of pieces, each of which is a sub- following quote from loc. cit., p. 250, “…but quotient of a Picard-Fuchs equation. using unpublished results of Tate and Lazard, we give indications of the existence of a de- Another of Bernie’s fundamental and icono- formation theory, involving a close connec- clastic discoveries concerns the arithmetic signif- tion between hypergeometric series and the icance of equations with irregular singular points. zeros of the zeta functions of elliptic curves. Picard-Fuchs equations are known to have regular We became aware of this connection in 1958; singular points, and for a long time it was gener- it was the first suggestion of a connection be- ally believed that the only differential equations rel- tween p-adic analysis and the theory of zeta evant to algebraic geometry were those with reg- functions.” ular singular points. But in the early 1970s Bernie Nurtured in Dwork’s amazingly original mind, achieved the remarkable insight that equations what marvelous fruit these three seeds bore. with irregular singular points (e.g., those for the hypergeometric function pFq for arbitrary p and q, q =6 p − 1) were not only not to be regarded as pathological, but they were in fact a fundamental feature of the p-adic algebro-geometric landscape, playing the same role for exponential sums in fam-

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