aa r X i v : . [ m a t h . HO ] O c t ALAN BAKER
DAVID MASSER
Abstract.
Alan Baker, Fields Medallist, died on 4th February 2018 in Cambridge Englandafter a severe stroke a few days earlier.In 1970 he was awarded the Fields Medal at the International Congress in Nice on thebasis of his outstanding work on linear forms in logarithms and its consequences. Since thenhe received many honours including the prestigious Adams Prize of Cambridge University,the election to the Royal Society (1973) and the Academia Europaea; and he was made anhonorary fellow of University College London, a foreign fellow of the Indian Academy ofScience, a foreign fellow of the National Academy of Sciences India, an honorary member ofthe Hungarian Academy of Sciences, and a fellow of the American Mathematical Society.In this article we survey Alan Baker’s achievements. Life and Career
Alan’s paternal grandparents were known as Marks and Mathilda Backer, married 1902 inLithuania. The surname presumably changed when children arrived. His parents were Barnetand Bessie (née Sohn). Alan was born on 19th August 1939 into this Jewish family. Hisearliest memories of wartime England were of evacuation to Camberley, Surrey. After the warthe family moved to Forest Gate in East London, where he spent most of his early life. Froma very early age he showed signs of mathematical brilliance (see the comments later about“brainbox”) and was encouraged by his parents. Already his father (who had been at schoolwith Jacob Bronowksi) was very gifted in this direction, but did not have the opportunityto develop, and became a tailor instead. That may have explained Alan’s clothes sense; hewas always well turned out with quality suits and tasteful ties (which were however not alwaysentirely appropriate to his later travelling, for example on the beach at Nice after being awardedthe Fields Medal or climbing a hill in the Australian Bush among snakes - such episodes mayhave contributed to his later investing in a distinctive yellow sun hat).Alan’s first education was at a Franciscan Convent (London E.7) 1945-49, followed by God-win County Primary School 1949-50 and Stratford Grammar School 1950-58. He then wentwith a State Scholarship to University College London (he recalls that the staff included J.W.Archbold, L.S. Bosanquet, P. Du Val, T. Estermann, C.A. Rogers, and K. Roth), where hestudied mathematics 1958-61, obtaining a first class honours B. Sc. (Special) degree.He then moved to Trinity College (where he would soon be based for the rest of his life) inCambridge to study 1961-64 with Harold Davenport, one of the leading number theorists atthe time with many international connections, of whom Alan writes:“
An excellent mathematician from whom I learnt a great deal; but I tended to follow my ownlines of research ”.Still, Chris Morley recalls a loud mathematical conversation between them in the Trinity Par-lour, and they did write a paper together (see later). It seems that in later life he enjoyedimitating Davenport’s Lancashire (Accrington) accent (and the writer has carried on this wor-thy tradition of imitating one’s supervisor). He obtained his Ph.D. 1965 and M.A. 1966, bywhich time he had already been awarded a Prize Fellowship 1964-68 also at Trinity. Duringthis period, which also included a year 1964-65 at University College, he took on John Coatesas a Ph.D. student and recalls also sharing responsibilities with Davenport in supervising T.W.Cusick, M.N. Huxley, H.L. Montgomery, and R.W.K. Odoni.In 1966 he was appointed as Assistant Lecturer, in 1968 as Lecturer, in 1972 promoted toReader in the Theory of Numbers, and finally in 1974 he was elected to a personal chair forPure Mathematics, all at Cambridge. Apparently he liked to draw attention to the unusualchronological order of Fields Medal then Fellow of Royal Society then Professor. During that period he supervised for Ph.D. also, in chronological order, the writer, Cameron Stewart, YuvalFlicker, Roger Heath-Brown, Richard Mason, Mark Coleman and E.J. Lee (whose full name Ihave been unable to find out). At the time of writing, the Mathematical Genealogy Projectlists 463 descendants.The writer recalls meeting Alan every two weeks or so, in his college rooms in Whewell’sCourt opposite the Great Gate. Actually these rooms (fitting C.P. Snow’s description “notspecially agreeable”) had been occupied previously by G.H. Hardy (about which the greatGerman mathematician David Hilbert - see later - indignantly wrote to the Master that Hardywas the best mathematician, not only in Trinity, but in England, and should therefore havethe best rooms). One had to ascend a slightly low and narrow spiral staircase, and then knockon the thick wooden door. Often in College une porte peut en cacher une autre and probablyhe did indeed have a second door immediately behind; at any rate he was sometimes a longtime coming and I supposed that he did not hear me through the two doors (also against theheavy traffic then in Trinity Street). On each visit I felt obliged to knock or bang harder andyet harder, sometimes bringing with me a heavy book to protect my hands.As mentioned, Alan Baker was firmly based in Cambridge; it seems that college life theresuited him especially in the style of Trinity, whose society he enriched for many years in anunspectacular way, for example being an interesting conversationalist (despite the impressiongiven by some newspaper obituaries). He also threw after-seminar parties fuelled by partic-ularly strong beer (on one occasion Frank Adams challenged others to climb around a table,underneath it, without the feet touching the ground, after he had himself demonstrated thatit could be done). In the Combination Room there was a bottle of Madeira kept specially forhim. He was a reasonable ballroom dancer, enthusiastically participating in the College MayBall. He played regularly on the Trinity Bowling Green, using an unconventional throw whichdelivered the ball from waist level instead of lower down.Outside the College, he enjoyed playing the slot machines in pubs, or playing table-tennisand snooker at the Graduate Centre. Eva McLean (née Gordon) writes of one occasion:“
In 1975 a vicious rapist was terrorising the female population of Cambridge. One eveningwhen I was leaving the Graduate Centre one of the porters who knew me well expressed greatconcern that I would be walking home alone. Not to worry, I assured him brightly, pointing toAlan who was going to escort me. The burly ex-policeman looked him up and down, all fivefoot six of him, grabbed his coat and joined us on our way. ”and draws attention to existential dangers of a different kind:“
Alan was once barricaded in his flat for several days and would not answer the door as acollector, all the way from America, kept coming back pleading, in vain, for a contribution tohis sperm bank. ”He had learnt to drive in America, and in Cambridge he bought a Rover car, apparentlyas an investment (“ you understand ”, as he insisted), although Eva McLean soon explained tohim that there were much better ways of investing (and he seems to have acted very well onthis advice). He drove with enthusiasm despite colliding with a stationary fire engine on hismaiden voyage, and despite being stopped for speeding which caused him to arrive late inchairing a session in London. William Chen recalls that it was actually the inaugural lectureof a colleague, and that Alan was very eager to explain that he had not exceeded 90mph, sohe was only speeding rather than driving recklessly. On one occasion the car got badly stuckin a driveway; Alan however remained cool and concentrated and succeeded in extricating itwithout damage. Later on the enthusiasm waned and a rusting hulk with flat tyres had to beremoved from New Court. He had a flat in Hendon and enjoyed life there in London too, forexample upmarket restaurants, or the theatre at which he always bought the best seats (as atthe Footlights Theatre in Cambridge). The latter interest may be due to his cousin HeatherRechtman, who writes:“
We lived near Stratford East (London) where the fledgeling company Theatre Workshop wasjust beginning to make a name for itself and as a stage struck teenager I dragged him along tomany of their plays ”. LAN BAKER 3
That was of course edgy stuff under Joan Littlewood; but he also enjoyed musicals. About onesuch trip to the theatre Eva McLean narrates:“ ... we stopped over at his Hendon flat. While there he decided to show me his mother’s minkcoat. When he opened the wardrobe door hundreds of moths awoke to greet us. His response tothis crisis? He shut the door firmly. ”as well as:“
On another eventful outing to the West End, we arrived in good time and right there by thetheatre came across a perfect parking place. Alan, however, decided it was too good to be trueand so the search continued. When we next passed the spot, the parking space was gone - aswas the first act of the play by the time we were inside. ”Her funeral tribute (23rd February 2018) sums up such anecdotes:“
So, in many ways, he was what Americans term ‘just a regular guy’. ”In his private life he was quite relaxed, never going beyond sometimes stamping his footwhen frustrated. However, in his professional life he was often reserved, even insecure in someways, and could on occasion be difficult. There were episodes, not just in Cambridge, whichperhaps still have not been entirely forgotten. But it is now impossible to give any balancedaccounts.He was enthusiastic about travel, which in America already started in 1969 with visitingprofessorships in Ann Arbor Michigan and Boulder Colorado and in 1970 membership of theInstitute for Advanced Study in Princeton New Jersey. He had at least three offers of chairs.One of these he turned down because the ivy-clad walls were too much like those in Cambridge,as was the weather. He preferred both modernity and sunshine. It was also thought that hewanted to look after his mother in London (this writer recalls around 1980 seeing two peopleslowly crossing a Maryland street, and recognizing one but not the other, and he followedthem into a restaurant where Alan introduced his mother). He made many American visits,and Eva McLean has shown me a lot of his postcards and other correspondence from abroad,for example from Texas (1984), New York (1987), and California (1989). Of course as hisreputation grew he was able to widen his horizons, for example to Australia, China/HongKong (missive 1988), India, Japan (postcard 1983), Russia, and many parts of Europe.In later life he made regular trips to Switzerland (postcards 1989 and 1990) to work withGisbert Wüstholz at ETH (a 1988 missive opens with “greetings from a gnome of Zürich”).There were social events too; and he thought nothing of occasionally bringing presents such asa frying pan to dinner invitations - unconventional to be sure, but with its own unassailablelogic. It was there, during a conference in honour of his 60th birthday, that he gave anentertaining and surprisingly candid speech about his life, starting with his recollections ofwartime London, also mentioning that he was regarded as the “ brainbox ” of the family, includingone more rendering of Davenport’s accent; and ending with his regrets about never marrying.Some people already knew that these regrets were not abstract but concerned specific ladies.I quote again from the funeral tribute:“
I first met Alan in 1975 playing table tennis in the Graduate Centre. I was immediatelytaken by his generosity of spirit. Unlike some of the others, he never minded being beaten. Heproved to be equally generous and attentive as a suitor, and later as an old friend. ”His last years were made more difficult by increasing deafness and a series of falls (he did notpay serious attention to medical advice on these). He had long since advanced from Whewell’sCourt to Great Court and indeed had been proud to have the very best set of rooms there, likea maisonette on two floors. It overlooked the Bowling Green where he now could no longerplay - see the cover of [58], the proceedings of the Zürich conference.During a Feast at Trinity in his memory he was described as “idiosyncratic” in place of wordslike “eccentric” or “enigmatic”.Other biographical articles about Alan Baker can be found in the Hardy-Ramanujan Journal[57].
DAVID MASSER
See also [63] for a scientific appraisal by Gisbert Wüstholz, which also contains a completelist of his publications. 2.
Mathematics - a preview
For a more structured narrative we shall divide Alan Baker’s work into eight categories (andin scientific detachment usually drop the first name):(a) Diophantine approximation.(b) Linear forms in logarithms.(c) Diophantine equations.(d) Elliptic functions.(e) Class numbers.(f) Abcology.(g) Miscellaneous.(h) Books.But before starting, we would like to mention perhaps the most easily stated of all his deepresults.When we make a list of the perfect squares , , , , , , , , , , , , . . ., , . . . we see that the gaps between consecutive members get larger and larger (and in a regular way).Similarly for the perfect cubes , , , , , , , , , , , , . . ., , . . .. But if we mesh the two lists together to get the “squbes” , , , , , , , , , , , . . ., , , . . . then it is not so clear that the gaps get large. Indeed it was proved by L.J. Mordell only in 1922;thus for example the gap 17 (twice above) or the gap 1621 (to take a year apparently at random)occurs at most finitely often. Unfortunately Mordell’s proof gave no way of determining allthe occurrences of a given gap.To do this amounts to specifying a non-zero k in the set Z of rational integers and findingall x, y in Z with(1) y = x + k. Baker achieved this in 1968 by showing that they all satisfy(2) max {| x | , | y |} ≤ exp { (10 | k | ) } . Despite (1) being around since at least the year 1621 (and the cases k = − , − were set byFermat in 1657 as a challenge to “you English”), there were no estimates at all for x, y until(2) nearly 350 years later.Thus for example to find all gaps 1621 one just has to examine all y , x with y, x between1 and say . This looks hopelessly impractical; yet we will see later that Baker (with Davenport) foundexceedingly efficient ways to do such things.There is an attractive single-sentence reformulation: for all positive integers x, y with x = y we have | x − y | > − (log x ) / . LAN BAKER 5 Diophantine approximation
It is classical that π = 3 . . . . and . . . . are suspiciously close. This raises the natural question: given a real number ξ , how well canwe approximate it by a rational number p/q ? An answer in convenient form was found byDirichlet: provided ξ is not already rational, we can find infinitely many p/q with(3) (cid:12)(cid:12)(cid:12)(cid:12) ξ − pq (cid:12)(cid:12)(cid:12)(cid:12) < q . We pause to give the simple proof, which involves the Box Principle or Pigeonhole Principle.Pick any Q ≥ in Z . For i = 0 , , . . . , Q we can find p i in Z with the Q + 1 pigeons θ i = iξ − p i in the interval from 0 to 1. We divide this interval into Q holes of length /Q . There are morepigeons than holes, so at least one hole must contain at least two pigeons. With say θ j and θ k ( j < k ) this leads at once to(4) | qξ − p | ≤ Q for p = p k − p j and q = k − j satisfying ≤ q ≤ Q . And now (3) follows; it is not too hard tosee that we get infinitely many p/q as Q varies.But the answer to the next natural question of whether we can beat (3) depends criticallyon the number ξ .For a class of ξ particularly interesting to number-theorists, this topic can be said to havebegun in earnest with Liouville in 1844, although it underlies the older concept of continuedfractions such as = 355113 or infinite ones like
11+ 1292+ ··· = π. More generally(5) a + 1 a + a + a
3+ 1 a ··· is usually linearized to just(6) x = [ a ; a , a , a , a , . . . ] . For example(7) α = [1; 1 , , , , . . . ] = 1 + √ , and this is actually an algebraic number in that α − α − . On the other hand Liouville’swork (see later) shows that if the positive integers a , a , a , a , a , . . . increase very rapidly,then x in (6) does not satisfy any equation(8) b x d + b x d − + · · · + b d = 0 for b , b , . . . , b d in Z not all zero. Thus by definition x is a transcendental number.In 1906 Maillet had given a different sort of transcendental continued fraction. A typicalexample is to take (7) and replace 1 by 2 in the positions k , k , . . . , where now these k , k , . . . increase very rapidly.In his very first paper [1] from 1962 Baker simplified and improved that work and also madethe estimates more explicit. A consequence here is that it suffices to take k n as small as n fortranscendence. For the proof one notes that something like x = [1; 1 , , , , , , , , , , , , . . . ] DAVID MASSER is rather close to α above, with similar approximations further along. In this case an easygeneralization of Liouville is applicable, but for other examples it is necessary to use KlausRoth’s “revolutionary improvement” (Ian Cassels) of Liouville’s result, or more precisely aconsequence by Davenport and Roth, and even LeVeque’s generalization of Roth.Such acknowledgements as Baker’s“ I should like to thank Professor Davenport for his valuable suggestions and help in preparingthe manuscript. ”can be seen quite often in the journal Mathematika around this time (which Davenportfounded).The paper [4] from 1964 can be considered as a sort of continuation of [1]. To describe someof its results we must recall that what Liouville proved is that for any algebraic number α ofdegree d ≥ (the smallest integer such that α satisfies an equation (8) above) there is c > such that(9) (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ cq d for all p and q ≥ in Z . When d ≥ Roth improved this to(10) (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ cq κ for any κ > , where now c is allowed to depend on κ . The Box Principle as in (3) shows thatthis is essentially best possible.Equivalently if for some ξ and κ > there is an infinite sequence of p i and q i ≥ in Z with p i /q i different and(11) (cid:12)(cid:12)(cid:12)(cid:12) ξ − p i q i (cid:12)(cid:12)(cid:12)(cid:12) < q κi ( i = 1 , , . . . ) then ξ must be transcendental. Baker then shows under an additional condition that ξ cannotbe too close to an algebraic number in the sense of what is since called a “transcendencemeasure”. The condition is that there should exist λ with q i +1 ≤ q λi for all i . In that case heshows that for each n there is µ n (possibly depending on ξ ) such that(12) | h ξ n + h ξ n − + · · · + h n | > H − µ n for all h , h , . . . , h n in Z not all zero, where H = max { , | h | , | h | , . . . , | h n |} .The significance of this is the following. From (12) by definition (due to Kurt Mahler) ξ cannot be a so-called U -number, even though (11) implies that the “partial quotients” a , a , . . . in (6) for x = ξ are unbounded. On the other hand the partial quotients for things like P ∞ k =1 − k ! are also unbounded and it is a U -number (for example (12) fails already for n =1 and h = 2 k ! with k large). Furthermore he shows that the method of [1] produces U -numbers with bounded partial quotients, and that a suitable generalization of the above resultto quadratic fields produces ξ that are not U -numbers but still with bounded partial quotients.More succinctly, there is no correlation between the properties of having bounded partialquotients and being U -numbers. In fact Baker is able to sharpen this to involve T -numbers and S -numbers (see just below) also in Mahler’s classification, and that was his main motivation.A spin-off is that either T -numbers exist or S -numbers “of type exceeding 1” exist. Thisforeshadows Wolfgang Schmidt’s breakthrough four years later that T -numbers exist.The proofs are rather formidable; indeed (11) and (12) lie close to a strengthening of Roth’sTheorem and accordingly Baker has to ramp up Roth’s entire machinery, even in the situationof LeVeque’s generalization.This time he writes only“ I am indebted to Professor Davenport for valuable suggestions in connection with the presentwork. ”In [14] from 1967 he returned to these themes, with generalizations to several numbers atleast in the case of bounded partial quotients. This property for a single ξ is equivalent to LAN BAKER 7 | ξ − p/q | ≥ c/q analogous to (9), and the natural extension to a pair ( ξ , ξ ) is max (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) ξ − p q (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) ξ − p q (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) ≥ cq / for all p , p and q ≥ in Z . He shows for example that there are U -numbers ξ , ξ satis-fying this, and also ξ , ξ that are not U -numbers. This time the proofs need no Roth-typeconsiderations but follow a still intricate “interval nesting” technique of Cassels and Davenport.In 1966 Baker [10] (published in the Proceedings of this Society) investigated certain “met-rical” properties of S -numbers. These are defined by refining (12) as follows. There is ω suchthat for each n and each κ > nω we can find c > (possibly depending on ξ, n, κ ) such that(13) | h ξ n + h ξ n − + · · · + h n | > cH − κ as in (12). For simplicity we stick to real ξ . Much as before, the Box Principle shows that ω ≥ . It had been conjectured by Mahler in 1932 that in fact we can take ω = 1 (“of typeexactly 1”) for almost all ξ in the sense of Lebesgue measure. This was proved by Sprindzhukin 1965. Having seen only preliminary announcements of that result, Baker was able to refine(13) even further to things like(14) | h ξ n + h ξ n − + · · · + h n | > cH − n (log H ) − λ for any λ > n .Later on in 1970 [29] with Schmidt he considered further refinements in terms of Hausdorffdimension; these are closer to Koksma’s classification into S ∗ -, T ∗ - and U ∗ -numbers basedinstead on the distance from ξ to algebraic numbers.Many of the results above can be expressed more concisely through the notation k x k = min m ∈ Z | x − m | for x in the field R of real numbers, meaning the distance to the nearest integer. For example(9) says that k qα k ≥ c/q d − for all q ≥ .This notation can be used in other contexts. Returning to 1964, we may cite his nice note[3], which shows that(15) | Q |k Q Θ kk Q Θ k ≥ e − for any non-zero polynomial Q in say R [ t ] , where Θ = e /t , Θ = e /t (same e but that isirrelevant) are interpreted as formal power series in the topological completion with respect tothe valuation | t | = e , and k X k = min M ∈ R [ t ] | X − M | measures the distance to the nearest polynomial. The significance here is that the analogue of(15) with R [ t ] replaced by Z (see just below) is thought to be false, by a famous conjectureof J.E. Littlewood (still unsolved). The note makes explicit an earlier result of Davenport andLewis and provides a simpler proof, essentially by differentiating and using a Padé approxima-tion (see later in this section). In fact the Padé element can be eliminated to give a yet simplerproof as follows.It can be checked that ∆ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q P P t Q ′ t P ′ + P t P ′ + 2 P t Q ′′ + 2 t Q ′ t P ′′ + (2 t + 2 t ) P ′ + P t P ′′ + (2 t + 4 t ) P ′ + 4 P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) remains unchanged on replacing P , P by E = P − Qe /t , E = P − Qe /t respectively.Now suppose Q, P , P , even in C [ t ] , are all non-zero. Inspecting the coefficients of smallestpowers of t in each entry shows that ∆ = 0 . Thus | ∆ | ≥ .Next for Q = 0 choose P , P so that |E | = k Qe /t k < , |E | = k Qe /t k < . Clearly P = 0 , P = 0 . Using | X ′ | ≤ e − | X | we get easily ≤ | ∆ | ≤ e | Q ||E ||E | giving a slight improvement of the result. DAVID MASSER
He evidently had this note in mind when writing [7] in 1965, back in R . A special resultthere is that for any ǫ > there is c > such that(16) q k qθ kk qθ k ≥ cq ǫ for any q ≥ in Z , where θ = e, θ = e .More generally he treats a product of several terms (as indeed in [3] above) involving variousnumbers θ = e φ with (different) rational φ (that then implies quite easily the transcendence of e with quite a good measure); and he is even able to refine the q ǫ . Of course one can no longerdo things just by differentiation; but that does play a role and the much more elaborate proofuses techniques introduced by Carl Ludwig Siegel in his work on the so-called E -functions,together with so-called transference principles.The paper [16] from 1967 is Baker’s only excursion into the Hardy-Littlewood Circle Method.Now one wants to know that certain approximations actually exist, and this time with primevalues of the variables. For example, he shows that given any κ there are infinitely manyprimes p , p , p with | p − √ p − √ p | < (log P ) − κ , where P = max { p , p , p } (with extension to arbitrary coefficients). This was the first suchresult with an explicit function of P on the right-hand side. As Bob Vaughan (who later greatlyimproved the result, to Baker’s surprise and dismay) has pointed out, he was also aware thata “localized” version in the style of (4), with P given in advance and just P ≥ max { p , p , p } ,is in principle impossible. We may remark that the result in the enigmatic footnote on thesecond page of this [16] had previously been proved by Vinogradov.Returning again to 1964 we examine [5] in some detail. The starting point is again (9) and(10). There is a fundamental difference between these two results. Namely, given α in (9), onecan calculate an explicit value of c (and in fact rather easily). One says informally that (9) is“effective”.In fact not so much attention had been paid to this concept, which dramatically increasedin importance thanks to the works of Alan Baker.On the other hand, given α and κ in (10), no-one knows how to calculate any value of c .This is true even for the simple-looking(17) (cid:12)(cid:12)(cid:12)(cid:12) / − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ cq . . The obstacle is a logical twist already in Thue’s improvement on (9) in 1909.Namely, his argument works very well if there happens to be already a very good rationalapproximation p ? /q ? to α . This sounds paradoxical, but it has the effect of “repelling” other p/q = p ? /q ? through the triangle inequality(18) (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) p ? q ? − pq (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) α − p ? q ? (cid:12)(cid:12)(cid:12)(cid:12) ≥ q ? q − (cid:12)(cid:12)(cid:12)(cid:12) α − p ? q ? (cid:12)(cid:12)(cid:12)(cid:12) . And if there are no such good approximations p ? /q ? that is also good news. However we haveno way of resolving this dichotomy; and in the first case c will depend on the uncontrollable p ? /q ? .In [5] Baker treats some numbers of this special form α = ( a/b ) /d for positive a, b in Z as in (17). He proves (10), but with different quantifiers. Namely, for each such α he findsan explicit κ such that (10) holds for some effective c > . Of course this already holds for κ = d by (9); but the new κ can sometimes be less than d . Furthermore, and this is Baker’skey observation, one can get κ arbitrarily close to 2 by choosing α carefully, usually with a, b approximately equal. This amounts to imposing p ? /q ? = 1 in advance. For example if b = 3 then(19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:18) b + 1 b (cid:19) / − pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ − q . , which is effective “but only just” (as Estermann’s Scotsman [54] said on p.33); and not aselegant as (17). LAN BAKER 9
Since Thue’s work it is known that any κ < d leads to consequences for diophantine equa-tions, and Baker presents some of these. More about that just below. The proofs follow theso-called hypergeometric method used by Thue in his earlier paper of 1908 (also see just below).Actually Thue in a later paper from 1918 had implicitly (in terms of diophantine equations)obtained results like (19). His work implies for example an effective (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ cq . for α = 17 / = 1 . . . . thanks to p ? /q ? = 3 / . ; a bit more elegant than (19).But the ultimate in elegance was then achieved by Baker [6] also in 1964: namely(20) (cid:12)(cid:12)(cid:12)(cid:12) / − pq (cid:12)(cid:12)(cid:12)(cid:12) > − q . . The consequence for diophantine equations is that all solutions x, y in Z of(21) x − y = m satisfy(22) max {| x | , | y |} ≤ (300000 | m | ) . Even though equations like (21) had been around since at least Thue’s work, this was the firsttime that (21) itself could be solved even in principle. It is similar to the situation around (1)and (2) above.Thus to solve completely for example(23) x − y = 1621 we only have to check all x, y with absolute value at most say . And with moderncomputers this is entirely feasible, using a trick found by Baker and Davenport a few yearslater (see section 5).Actually to prove these results one needs infinitely many elements like p ? /q ? in the set Q ofall rationals. An old idea, going back essentially to Hermite, for getting many approximations p/q in Q to a number ξ in R is to construct many approximations A/B in Q ( x ) (preferablyof so-called Padé type) to a suitable function f ( x ) . Then one specializes x .The choice f ( x ) = (1 − x ) / will do for (20). Thue had already used this, and found thathypergeometric functions turn up. Baker succeeded with (20) by means of a nice new twist.By easy linear algebra we see that for any r ≥ in Z there are polynomials A r ( x ) , B r ( x ) ,not both zero, such that(24) φ r ( x ) = A r ( x ) − (1 − x ) / B r ( x ) has a zero at x = 0 of order at least r + 1 (these are in fact Padé approximations). They areunique if we normalize B r (0) = 1 and then B r ( x ) = F (cid:18) − r, − r, − r, x (cid:19) = r X j =0 b rj x j for the Gauss function F (now a polynomial). Here the coefficients b rj = j − Y k =0 ((1 / − r + k )( − r + k )(1 + k )( − r + k ) are all rational, and seem to involve j in their denominators.As / = 1 . . . . the choice p ? /q ? = 5 / is tempting in the discussion above; and thistranslates into putting x = 3 / in (24), when (1 − x ) / = / / . As φ r ( x ) = cx r +1 + · · · is small, it seems plausible that we obtain a sequence of good rational approximations(25) B r (3 / A r (3 / r = 0 , , , . . . ) to / . For example with r = 0 , , , , we get , , , , (the last being accurate to 19 decimal places). The hope is that these will repel all others asin (18). But to check this we have to estimate denominators in (25). For example B r (cid:18) (cid:19) = r X j =0 b rj (cid:18) (cid:19) j ; and now we see that the j in the denominators of b rj are cancelled out. It is this “3-adic”feature that is Baker’s new twist (which led later to a minor industry). It would fail for theslightly more tempting p ? /q ? = 63 /
50 = 1 . .In the same paper he did similar things with a / for a = 17 , , , and . But, atleast up to 1988, the method fails for a = 5 and so does not lead to a complete solution of say x − y = 1621 (see section 4).In all Baker published no less than five papers in 1964, each highly non-trivial (he did evenbetter with six in 1967). We have already discussed four of them. The fifth is [2], whichfollows the broad principles of [5] and [6] above. Now he is dealing with numbers ξ = log α whose irrationality (or transcendence) is by itself somewhat deep. In fact for algebraic α = 0 , this was proved by Lindemann in 1882 (see just below in section 4). We indicate just oneconsequence in the style of (19): if b = 10 then (cid:12)(cid:12)(cid:12)(cid:12) log (cid:18) b + 1 b (cid:19) − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ − q . . But he is also able to extend to transcendence measures in the style of (13), with κ arbitrarilynear n in the spirit of (14). In the proofs the analogue of (24) is a system of forms(26) φ ( x ) = A (0 ( x ) + A (1) ( x ) log(1 − x ) + · · · + A ( n ) ( x )(log(1 − x )) n found by Mahler.This category of Baker’s work is rounded off by [17] in 1967, which extends [5] to linearforms in various ( a/b ) d /d , ( a/b ) d /d , . . . , ( a/b ) d n /d , again close to best possible in the spirit of (14). Instead of quoting an approximation resultwe mention an attractive application to diophantine equations. Namely if h = 10 + 1 and l = h − then all solutions x, y, z in Z of(27) x + ly + l z + 5 lxyz ( xz − y ) = m satisfy max {| x | , | y | , | z |} ≤ l m . This may not be quite so elegant as (21) and (22); but the exponent of m is much smaller, andcrucially this seems to be the first such example ever for equations in three variables (whereat the time of writing it is unknown if x + y + z = 114 - see [64] - has any solutions at all).Now another system like (26), also due to Mahler, plays a key role.This concludes the section on diophantine approximation. Baker did more very importantwork around (9), but that deserves an extra section, which now follows. In the sequenceof papers just described he seems to be exuberantly flexing his muscles on several differentmethods, (with hindsight) limbering up for his big achievements. In the early stages he waspossibly solving every problem Davenport threw at him; but pretty soon his own motivationtook over. 4. Linear forms in logarithms
This title hardly existed before Alan Baker.The classical theorem of Hermite-Lindemann (1873-1882) is equivalent to the fact that if α is a non-zero algebraic number, and log α is any non-zero choice of its complex logarithm, then and log α are linearly independent over the field Q of all complex algebraic numbers. Thisincludes the transcendence of e, π and so Z d xx + 1 = π , LAN BAKER 11 and of course the numbers log α mentioned in section 3, as well as the numbers e β for algebraic β = 0 .Similarly the classical theorem of A.O. Gelfond and T. Schneider (1929-1934) is equivalentto the fact that if α , α are non-zero algebraic numbers, and log α , log α are any choices oflogarithms which are linearly independent over Q , then they are even linearly independent over Q . This includes the transcendence of e π and most notably the numbers √ and α β mentionedby Hilbert (Seventh Problem) in his famous 1900 address to the International Congress ofMathematicians in Paris, as well as numbers like log 3 / log 2 .Before Baker, nothing was known about the independence of , log α , log α ; and practicallynothing about that of log α , log α , log α , despite the assertions of Gelfond and Linnik in 1948(repeated in Gelfond’s 1960 book) of the enormous importance of the latter problem (for whichsee later). There had been a paper by Gelfond and N.I. Feldman in 1949 about b log α + b log α + b log α for integers b , b , b ; but these coefficients were subject to a restrictive condition which essen-tially reduced the problem to two logarithms.Then Baker in a sequence of four papers 1966-68 (which we shall call the Linear FormsQuartet) went straight for any number of logarithms and proved (among much else) Theorem (Baker). If α , . . . , α n are non-zero algebraic numbers, and log α , . . . , log α n areany choices of logarithms which are linearly independent over Q , then (28) , log α , . . . , log α n are linearly independent over Q . The reader may easily construct simple examples of transcendental numbers like e β α β or α β α β not covered by Hermite-Lindemann or Gelfond-Schneider; less simple examples are(29) Z d xx + 1 = π √
39 + log 23 quoted by Siegel in his famous 1949 transcendence monograph, or Z d xx − x − β log α + β log α + β log α where α = 4 − α , α = 2 + α, α = − α + α ,β = − − α + 2 α , β = 2 − α − α , β = 2 + 2 α − α and α − α − (with say α > to provide unique real choices of logarithms).We proceed to examine these four papers I, II, III, IV in detail.Linear Forms I is [11] from 1966. It is one of the great number-theoretic papers of thetwentieth century.In fact the main result is a “linear independence measure” (analogous to transcendencemeasure in section 3) for just log α , . . . , log α n with a condition slightly stronger than that inthe above theorem. We return to such measures later; it is these refinements that are neededfor the applications suggested by Gelfond and Linnik.In an exemplary display of modesty, clarity and foresight, Baker writes“ Finally, as regards the proof of the theorem, our method depends on the construction of anauxiliary function of several complex variables which would seem to be the natural generalisationof the function of a single variable used in Gelfond’s original work. The subsequent treatmentemployed by Gelfond, however, is not applicable in the more general context and so it hasbeen necessary to devise a new technique. Nevertheless it will be appreciated that the argumentinvolves many familiar ideas. The method will probably be capable of considerable developmentfor it applies in principle to many other auxiliary functions apart from the one constructedhere. ”This is immediately followed by “ The author is grateful to Prof. H. Davenport, who read the original draft of this paper, forhis helpful criticism. ”which makes it clear that Davenport did not suggest the problem. Indeed the problem, despiteits great importance, seems not to have been very well known; for example there is no mentionof it in Lang’s book on transcendental numbers, which appeared in the same year 1966. Itis not referred to in the section “
Einige offene Fragestellungen ” in Schneider’s book of 1957;this section contains his celebrated Eight Problems. And in Siegel’s book from 1949 there isno mention of logarithms in connexion with (29). Also in Baker’s paper there is no seriousreference to any paper published after 1948. He sought out the problem himself, despite allprevious transcendence breakthroughs coming through French, Russian and German sources;and solved it in spectacular style.Let us give an idea of his “new technique”.We start with n = 2 and Gelfond-Schneider; slightly changing the notation we have todeduce a contradiction from a relation(30) β log α = log α ′ with α, α ′ , β algebraic and β irrational. Gelfond constructs an auxiliary function Φ( z ) whichis a non-zero polynomial of large degree in(31) e z , e βz . Note that from (30) the functions (31) take algebraic values at all points(32) z = s log α, s = 0 , , , . . . and this is true even of their derivatives, thanks to the differential equation for the exponentialfunction. That enables Φ to be constructed, with algebraic coefficients not all zero, such that(33) d t d z t Φ( s log α ) = 0 for all non-negative integers s, t in some large range(34) s ≤ S, t ≤ T. Thus Φ has many zeroes (with multiplicity) inside a large disc in C (say centred at the origin).Now well-known analytic techniques such as the Schwarz Lemma (or Maximum ModulusPrinciple - see (79) below) show that Φ must be very small on the whole of this disc, and evenon a slightly larger one; and this holds for the derivatives, and even slightly more of them.Then well-known arithmetic techniques show that (33) holds for say(35) s ≤ S, t ≤ T. This step could be then iterated, even indefinitely, to get infinitely many zeroes. In fact asingle zero of infinite multiplicity, say at z = 0 , suffices to prove that Φ must be identicallyzero, and this leads easily to the required contradiction.How did Baker adapt Gelfond’s proof? For simplicity we take n = 3 and ignore the extra in (28), so that now we have to deduce a contradiction from a relation(36) β log α + β log α = log α with β , β algebraic (with a suitable irrationality condition). The auxiliary function Φ( z , z ) is now a polynomial of large degree in(37) e z , e z , e β z + β z which indeed generalize in a fairly natural way (although no-one had previously written themdown) Gelfond’s Φ( z ) and (31). This time (36) shows that the functions (37) take algebraicvalues at all points(38) ( z , z ) = ( s log α , s log α ) , s = 0 , , , . . . LAN BAKER 13 and this is true also of their partial derivatives. That enables Φ to be constructed, withalgebraic coefficients not all zero, such that(39) ∂ t ∂z t ∂ t ∂z t Φ( s log α , s log α ) = 0 for all non-negative integers s, t , t in some large range(40) s ≤ S, t + t ≤ T. Thus Φ has many zeroes (still with a natural concept of multiplicity) inside a large disc nowin C .In 1941 Schneider had used several complex variables for transcendence purposes, and in C was able to go further because the set of zeroes was part of a structure like Z . (Later Bakerexpressed it “ ... this type of argument requires that the points in question form a cartesianproduct, a condition that can apparently be satisfied only with respect to particular multiply-periodic functions. ”) But in (38) the structure is only Z ; and indeed the points all lie ona line. This is too thin a set for the usual type of Schwarz Lemma. Indeed the function ( z log α − z log α ) T has the same zeroes as Φ but it need not be small even on a disc ofradius 1 in C .And to this day, no-one knows how to increase T in (40) to T as in (35).Here comes Baker’s decisive innovation. It does seem fairly natural to restrict to the line.But Baker considered also the partial derivatives Φ t ,t = Φ t ,t ( z ) = ∂ t ∂z t ∂ t ∂z t Φ( z log α , z log α ) however keeping only t + t ≤ T .Clearly d t Φ t ,t / d z t can be expressed as a linear combination of the Φ τ ,τ with τ + τ ≤ T + t. So as long as we keep also t ≤ T , then τ + τ ≤ T and we see from (39) that d t d z t Φ t ,t ( s ) = 0 for s ≤ S, t ≤ T. These are very similar to (33) and (34), firmly back in C .Now the Schwarz Lemma shows that the Φ t ,t are very small on a slightly large disc; andso the arithmetic gives us (39) for(41) s ≤ S, t + t ≤ T. We can iterate but not indefinitely in any profitable way, due to the loss of multiplicity.About this Martin Huxley has commented (1967):“
Where GEL’FOND had one exponential,ALAN BAKER foresaw the potential.A strange iterationGives zero-inflation.The right place to start is essential. ”Already this limited iteration was a new sort of difficulty, which Baker overcame by gettingjust as many zeroes as are needed for the contradiction (along the principle that a polynomialof degree D cannot have D + 1 zeroes). Here the non-vanishing of a certain Vandermondedeterminant suffices.Of course all this extends to any number of variables and(42) ∂ t ∂z t · · · ∂ t n ∂z t n n Φ( s log α , . . . , s log α n ) = 0 . So Baker’s main achievement was to introduce several complex variables, reduce them toa single variable along a line, introduce the magic trick with multiplicities, and supply themissing estimates for zeroes.Actually he was helped in this last “zero estimate” by Gelfond himself. In a 1935 paper hehad obtained a linear independence measure for two logarithms. This amounts to assuming that β log α − log α ′ is very small rather than zero as in (30). That adds some technical complicationsbut more significantly it means that here too one cannot iterate (35) indefinitely; hence theneed for some sort of zero estimate.To state Gelfond’s result we have at last to introduce heights. Recall that the degree ofan algebraic number α is the smallest d in any equation (8) satisfied by x = α . We can thenassume the coefficients have no common factor, and we define the height by H ( α ) = max {| b | , | b | , . . . , | b d |} (we are using H to distinguish it from a more modern version that need not concern us here -see however the end of this section).Thus Gelfond was able to show that for any κ > (and later for any κ > ) there is effective c > depending only on α , α and the degrees of β , β such that(43) | β log α + β log α | ≥ ce − (log B ) κ , where B = max { , H ( β ) , H ( β ) } , provided log α , log α are linearly independent over Q andtheir principal values are chosen.Accordingly in Linear Forms I Baker could prove an analogous(44) | β log α + · · · + β n log α n | ≥ ce − (log B ) κ for any κ > n +1 , where now B = max { , H ( β ) , . . . , H ( β n ) } , except that now log α , . . . , log α n , πi have to be linearly independent over Q .In the second paper Linear Forms II [12] from 1967 Baker obtained (44) for any κ > n + 1 when only log α , . . . , log α n are linearly independent over Q . This was crucial to an applicationto an effective improvement of Liouville’s (9) (see later).In Linear Forms III [13], also from 1967, Baker improved the above result of [12] for any κ > n . He also made a start on lower bounds for the “inhomogeneous form” β + β log α + · · · + β n log α n . This finally yields the theorem quoted above.In the last paper of the Quartet, Linear Forms IV [20] of 1968, Baker starts by noting thatthe various positive constants c in his lower bounds, although of course effective, are verysmall indeed, which could lead to practical computing difficulties in the applications. He givesthe following version leading to explicit c which are not so small. Suppose for some δ with < δ ≤ that there exist b , . . . , b n in Z with absolute values at most H such that(45) < | b log α + · · · + b n log α n | < e − δH . Then(46)
H < (4 n δ − d n log A ) (2 n +1) where d ≥ is an upper bound for the degrees of α , . . . , α n and A ≥ an upper bound fortheir heights.Implicit in this paper is the following associated result (see as well [42] p.3 and p.8), alsothe first of its kind. If log α , . . . , log α n are this time linearly dependent over Q then there isa “relatively bounded” relation(47) b log α + · · · + b n log α n for b , . . . , b n in Z with(48) < max {| b | , . . . , | b n |} < (4 n d n log A ) (2 n +1) . The idea here was greatly exploited in later work on isogenies between elliptic curves andbetween abelian varieties and consequences of that (see (91) below).
LAN BAKER 15
Shortly afterwards Baker wrote two important papers [18] and [19] which improve furtherthe Quartet results; but since the main emphasis there is on diophantine equations we shallpostpone our description to the corresponding category. It is also [18] that contains the firsteffective improvement on Liouville’s (9).Similar remarks hold for his 1971 paper [30], published in the top journal Annals of Math-ematics, in connexion with class numbers.We turn now to the later developments of (44). These may seem rather technical, but theyare specifically aimed at marvellous applications, to be described in sections 5 and 7.We start with his paper [33] with Harold Stark, also in the Annals and 1971.Already in [18] Baker had observed that the term (log A ) (2 n +1) in (46) could be replacedby C (log A n ) κ for any κ > n + 1 , where A n ≥ is an upper bound for H ( α n ) alone, and C (effective) is now allowed to depend on δ, κ, d as well as α , . . . , α n − . In [33] they replace (45)by(49) < | β log α + · · · + β n log α n | < e − δH where now β , . . . , β n are algebraic numbers also of degree at most d . The analogue of (46) is(50) H < max { e √ log B , C ′ (log A n ) κ } now for any κ > , where B ≥ is an upper bound for the heights of β , . . . , β n and C ′ hasthe same nature as C above.This indeed technical result could be applied to class number problems (see section 7); butalso to general effective improvements of Liouville’s (9), also described later on. But actuallymuch interest attaches to the proof. Already in [27] (see section 6) Baker had the nice idea oftaking s in (32) and (38) as rational not just integral, but now they combine this with anotherimportant idea, which for simplicity we illustrate with (32). Namely the functions (31) take at z = (1 /p ) log α the values α /p , α ′ /p and the vanishing of Φ there implies an algebraic relation between these. Such things can beruled out by Kummer theory (based on Galois theory) if p is for example a sufficiently largeprime (thus , /p , /p are linearly independent over Q for p > ); and this gives anotherapproach to the zero estimates.It is not difficult to guess which author wrote the sentence“ However, the latter work involves an appeal to Galois theory which we prefer to avoid. ”in a footnote.We now come to the “Sharpening Trio” I,II,III (changed at the last moment from “strength-ening”).In Sharpening I [35] (in memory of Davenport and Sierpiński) from 1972 Baker is able toimprove (50) for δ = 1 to H < C ′′ log B log A n provided β , . . . , β n are in Z , where C ′′ is allowed to depend only on d, α , . . . , α n − . Now themain motivation has become effective improvements of Liouville, and a little competition opensup between Baker and Feldman. Indeed the much sharper dependence on B (now essentiallybest possible) comes from a 1968 paper [55] of Feldman (who realized how to avoid certainfactorials - Thue himself had to do this as well). Also Baker concedes in a footnote thatFeldman has recently obtained those effective improvements (for which having κ = 1 in (50) isessential), but immediately adds that he can do it himself with a “slight generalization” of this[35].This slight generalization is Sharpening II [36], from 1973 (dedicated to Siegel). It containsa partial improvement of (46) when b n = − in (45); namely H < C ′′′ log A n , where the ever-effective C ′′′ is allowed to depend only on d, δ, α , . . . , α n − . Indeed this doesdo the trick for Liouville, and it is customary to declare the competition a draw. Thus we have Theorem (Baker-Feldman).
Given any algebraic number α of degree d ≥ there is effective κ < d and effective c > such that (51) (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ cq κ for all p and q ≥ in Z . Regarding this κ , we pause to mention the 1988 work [44] of Baker and Cam Stewart. Theygave explicit values of κ and c for every irrational α = a / with a ≥ in Z . We alreadymentioned a problem with a = 5 in connexion with (20); and this they solve with (cid:12)(cid:12)(cid:12)(cid:12) / − pq (cid:12)(cid:12)(cid:12)(cid:12) > − q . (which would have left Estermann’s Scotsman without words). The proofs use cubic units(here
41 + 24 α + 14 α ) and an ad hoc linear form in three logarithms.The Trio was interrupted with [37] in 1973. Here Baker introduces the now-familiar notation(52) Λ = β + β log α + · · · + β n log α n (back to inhomogeneous) and shows that if Λ = 0 then(53) | Λ | > ( B log A n ) − C ′′′′ log A n with A n as above, and now B ≥ is an upper bound for the heights of β , β , . . . , β n and C ′′′′ is allowed to depend effectively on d, α , . . . , α n − . This starts to look more like the modernversions. He gives the application (cid:12)(cid:12)(cid:12)(cid:12) e π − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ q C log log q for all p and q ≥ in Z , with C absolute (that is, depending on no additional parameters);this has never been improved.The Trio ends with Sharpening III [39] (in memory of Linnik and Mordell) from 1975. Thereis now a mini-competition with Stark. Baker shows that when β = 0 and β , . . . , β n are in Z in (52) still with Λ = 0 , then(54) | Λ | > B − C Ω log Ω . Here
Ω = log A · · · log A n where each A i ≥ is an upper bound for the height H ( α i ) . Thingsseem to be getting more complicated; but now the effective C depends only on n and d , so weare heading back in the direction of totally explicit estimates like (46). And the temporaryneglect of α , . . . , α n − has ended (this had been done by Stark two years earlier).Baker gives no applications; but a footnote “added in proof” mentions that the result has“recently” been used by Robert Tijdeman to deliver what is perhaps the most spectacularapplication of linear forms in logarithms to diophantine equations; more of that in the nextsection.With [42] from 1977 we are finally back to totally explicit estimates. After a detailed accountof the history (up to then), Baker shows firstly that for Λ = 0 as in (52) we have(55) | Λ | > ( B Ω) − C Ω log Ω ′ with Ω ′ = Ω / log A n and C = (16 nD ) n ; now D = [ Q ( α , . . . , α n , β , β , . . . , β n ) : Q ] measures degrees in a different way. And secondly that when β = 0 and β , . . . , β n are in Z the bracketed factor Ω in (55) can be eliminated to yield(56) | Λ | > B − C Ω log Ω ′ . The proof supplements his own formidable techniques with ideas of Tijdeman and Alf vander Poorten (about the new Ω ′ ) and Tarlok Shorey (about C ). LAN BAKER 17
And there Baker’s extraordinary solo work on linear forms in logarithms ended. We havealready mentioned a paper with Stewart in 1988 but it was not until 1993 that he did any morework on (56).This was [45] with Gisbert Wüstholz. The main result was expressed with a more modernversion of height and also took into account non-principal values of the logarithms, but we giveonly the consequence for principal values. Again we are in the so-called “rational case” when β = 0 and β , . . . , β n are in Z . Then they improve (56) to(57) | Λ | > B − C Ω with C = (16 nD ) n +4 . The elimination of log Ω ′ is due mainly to the zero estimates of Wüstholz arising from his moregeneral work on group varieties.Incredibly sharp as all these results are, we should note that there is much room for conjec-tural improvement; for example one believes that the product of logarithms Ω can be replacedby the sum (see section 8).Much more recently Vesselin Dimitrov and Philipp Habegger [53] have made a conjecturewhich would have valuable consequences in the theory of dynamical systems of algebraic origin.In our present framework it runs as follows. Given ǫ > and T ≥ , there exists C = C ( ǫ, T ) such that | b log α + b log( − | > exp( − CD ( DB ) ǫ ) for all algebraic α , not a root of unity, with | α | = 1 and H M ( α ) ≤ T . Here the notation isby now familiar except for H M , which is the standard modern version of height (or Mahlermeasure) defined using the conjugates of α . The crux is the very good dependence on D .The theory of Linear Forms in Logarithms (or logarithmic forms for short) continues to beone of the two main tools for solving diophantine problems in number theory (or indeed severalother branches of mathematics), with a key emphasis on effectivity. The other is the SubspaceTheorem due to Schmidt with important extensions by Hans Peter Schlickewei, which is partlynon-effective. 5. Diophantine equations
We already had several examples of these, such as (23), and we even mentioned the year1621; but the so-called Pell equation(58) x − y = 1 was encountered in the solution of the “Cattle Problem” of Archimedes (see [62] for example).In general it is a question of finding a solution with the variables in Z (which any idiot cando for Pell above) or more ambitiously finding all solutions (there are infinitely many for Pellabove, all but two of them safely out of the reach of idiots).To warm up let us show that there are at most finitely many positive integers r, s with(59) r − s = 1621 (the equation r − s = 1 , with a musicological background, was already solved by Levi benGershon in 1343 - see [59] for example). Assume H = max { r, s } is large, say H ≥ H . Then r is relatively close to s and of course(60) r − s = 1 + 16212 s . Taking logarithms shows that r log 3 − s log 2 is very small, and we get without trouble (45)with n = 2 , α = 3 , α = 2 , b = r, b = − s, δ = 1 / for an easily computable H . So (46) with d = 4 , A = 4 gives H < and ben’s your uncle.But for most of this section we shall consider only polynomial equations. It was Thue in1909 who first proved that equations like x − y = 1621 in (23) have at most finitely manysolutions, as a consequence of his improvement on Liouville’s (9). But we already observed the ineffectivity in that his method did not allow all the solutions to be found. We also saw Bakerovercome this problem with (22) for x − y = m in (21). However we noted that his methoddid not work for x − y = m .It was in [18] (part I of a Duo published in the Philosophical Transactions of this Society)from 1968 that Baker was able to treat the general equation considered by Thue. Namely, if f is a homogeneous polynomial in x, y of degree d ≥ (so Pell is out), irreducible over Q andwith coefficients in Z , and m = 0 is in Z , then all solutions of f ( x, y ) = m satisfy(61) max {| x | , | y |} < Ce (log | m | ) κ for any κ > d + 1 , where C depends only on f and κ .He deduces easily the very first effective improvement on (9); namely(62) (cid:12)(cid:12)(cid:12)(cid:12) α − pq (cid:12)(cid:12)(cid:12)(cid:12) ≥ ce (log q ) /κ q d , so he is already on his way to (51).The proof uses a new estimate for linear forms in logarithms (what else?), which we alreadycommented on. We sketch the method, because it shows that arbitrarily many logarithms maybe needed.For simplicity we take the equation(63) x d − y d = 1 with d ≥ an arbitrary prime. Pick any α with α d = 2 . Then factorizing the left-hand side of(63) shows that x − αy is a unit in the field Q ( α, ζ ) with ζ = e πi/d . Dirichlet’s Theorem showsthe unit group to have rank t = d ( d − / − , generated by say η , . . . , η t modulo torsion.Thus we may write x − αy = θη a · · · η a t t with integer exponents and a root of unity θ .We have d choices for α at our disposal; but it suffices to take just two more, say β, γ . Wededuce analogously x − βy = φη b · · · η b t t , x − γy = ψη c · · · η c t t . Now “Siegel’s Identity” ( β − γ )( x − αy ) + ( γ − α )( x − βy ) + ( α − β )( x − γy ) = 0 leads to(64) δη b − c · · · η b t − c t t = 1 + ǫ (a bit as in (60) above) with δ = − ( γ − α ) φ ( α − β ) ψ , ǫ = ( β − γ )( x − αy )( α − β )( x − γy ) . Choosing α as the real /d it may be seen that if x, y are very large then x − αy and so ǫ is very small. Thus taking logarithms in (64) gives an integer a such that log δ + ( b − c ) log η + · · · + ( b t − c t ) log η t + 2 πia is also very small (note that πi is a logarithm of 1). This turns out to be more than enoughto contradict (44).A similar treatment of x d − y d = m leads to (62) with α = 2 /d .In [19] (part II) also 1968 he solves the Mordell equation y = x + k in (1) as we alreadydescribed in section 2. He also makes (61) completely explicit with(65) C = exp( d ν H νd ) where H is the maximum of the absolute values of the coefficients of f and ν = 32 κ d/ ( κ − d − .The proofs rely on an explicit version of the linear forms estimate in [18]. The Mordellequation leads to a Thue equation by the classical method of reduction theory of binary cubic LAN BAKER 19 forms (which he also has to make explicit along the way, itself of no small interest), and thenthe general Thue equation needs some simple estimates for units (which were immediatelyimproved by Siegel himself).And thick and fast they came at last. Mathematically next comes [21] in 1967 (dedicatedto Mordell on his 80th birthday). There Baker treats the “elliptic” equation(66) y = ax + bx + cx + d for a, b, c, d in Z such that the cubic has distinct zeroes. He proves max {| x | , | y |} < exp { (10 H ) } where H = max {| a | , | b | , | c | , | d |} . This looks a bit like (2) but cannot be deduced from it.Instead Baker uses now explicit versions of reduction theory of binary quartic forms and thenapplies (65).Then in [22] from 1969 he generalizes to the “hyperelliptic” equation(67) y = a x n + · · · + a n with n ≥ and the polynomial on the right having at least three simple zeroes, and the“superelliptic” equation(68) y m = a x n + · · · + a n with m ≥ , n ≥ and the polynomial on the right having at least two simple zeroes; wherethe coefficients a , . . . , a n are in Z again with H as the maximum of their absolute values.For (68) he proves the doubly exponential max {| x | , | y |} < exp exp { (5 m ) ( n n H ) n } and for (67) the triply exponential “finite but only just”(69) max {| x | , | y |} < exp exp exp { ( n n H ) n } (at which the lecture audience always laughs). We emphasize though that these were the firstever bounds of their kind.The proofs rely on the calculations of [18] and [19] in suitably generalized form.In [28] with Coates from 1970 the bounds get even bigger, but the problem is somewhatdifferent. They take now a polynomial F in x, y with coefficients in Z which is irreducible over C . Then F ( x, y ) = 0 defines a curve C in C which has a certain geometric genus g ≥ in Z . They assume in fact g = 1 . This does not restrict the total degree n of F . Then for all points ( x, y ) in Z on C theyprove max {| x | , | y |} < exp exp exp { (2 H ) n } (audience goes crazy). The proof reduces to (66) (but now with algebraic coefficients) usingRiemann-Roch in the classical way. Of course this step must be made effective and explicit;and this is done by Coates in a separate paper of great independent value. They can thenfollow the arguments of [22] just above.They remark of their result:“ It does not seem to extend easily, however, to curves of genus > , and an effective proofin the general case remains an important quest. ”And today no-one has the slightest idea how to extend this even to genus g = 2 , even thoughthe finiteness of integral points is known since Siegel in 1929. Of course the standard formsare (67) for n = 5 , so already covered. But there are many other models; for example withUmberto Zannier we found the simple family x − y − axy = 0 which does not seem to be effectively solvable.We said back in section 2 that bounds like the above can be efficiently dealt with. This wasdone for the first time in 1969 by Baker and Davenport [25]. The numbers , , , have the property that the product of any two, increased by 1, isa square. Van Lint asked if this can be extended to , , , , N for some integer N > .Plainly then the property holds just for , , , N ; and he was able to show that if N exists,then(70) N ≥ which makes its existence unlikely. In [25] they prove that indeed there is no N .Again Martin Huxley (around 1969):“ Each product is one from a square.How many such numbers be there?The known set of fourCannot take any more,As DAVENPORT, BAKER declare. ”It seems worthwhile to give some details, as they also illustrate the raw power of linear formsin logarithms.Clearly if N exists it has the form x − for x in Z ; and now the other conditions amountto(71) y − x = − , z − x = − for y, z in Z . These imply ( yz ) = 24 x − x + 14 , of the form (67); but the bound (69)really may be too big.Instead we use the Pell theory behind things like (58) to see that the first equation of (71)forces(72) y + x √ √ √ r for some r in Z ; and the second(73) z + x √ √ √ s for some s in Z (there is actually a similar second possibility here which we shall ignore).It is easy to see that (72) is very near x √ and (73) is very near x √ . Eliminating x wesee that √ √ √ r is very near √ √ √ s . This is also like (59). Takinglogarithms in the usual way we see that b log α + b log α + b log α is very small for α = 2 + √ , α = 3 + √ , α = √ √ √ √ b = r, b = − s, b = − . In fact we have (45), and then (46) leads easily to(74) r < . But as N = x − grows exponentially in r by (72), this far from contradicts (70).More explicitly, (45) leads to (a stronger form of) say(75) | rθ − s + φ | < − r for certain real irrational θ, φ whose exact form we can now forget.Here comes the new idea. Suppose in place of (74) we have obtained(76) r < R for some huge integer R . We pick a largish integer K and then by Dirichlet’s (4) we find q with k qθ k ≤ KR , q ≤ KR.
It follows easily from (75) and (76) that(77) k qφ k ≤ q r + 1 K .
LAN BAKER 21
Now q was chosen specifically to make qθ very near an integer, and there seems no reason tothink that qφ is also near an integer (shades of Littlewood). For example we have k qφ k ≥ / with “probability” / . If we could have picked K such that k qφ k ≥ /K then by (77) weget r ≤ qK ≤ K R. For fixed K this gives a new bound for r that is logarithmic in R , and so it is plausible that(76) will be improved.And indeed in the stronger form of (75) they pick K around , find q efficiently from thecontinued fraction of θ , and end up with r < (thanks to the Atlas Computer Laboratory -and presumably not to the pocket calculator that Baker bought after seeing that Eva McLeanhad one). This does contradict (70) and the thing is done.This section would be incomplete without a mention of the Catalan equation(78) x r − y s = 1 this time to be solved for x, y, r, s in Z all bigger than 1. Thus it is a combination of (23) and(59). It is looking for all gaps 1 when we throw in fourth powers, fifth powers etc with thesqubes of section 2.In 1842 Catalan conjectured that the only solution is − = 1 . Progress was very slowuntil a result of Cassels, who also earlier in 1953 had made the weaker conjecture that thereare only finitely many solutions. Despite all the work of Baker, it came as a sensation whenTijdeman in 1976 proved the conjecture of Cassels with an ingenious application of linear formsin logarithms. He used Sharpening III (54), together with his own sharpening (now the knivesare really out) of Sharpening I. That was something like (54) with a slightly better dependenceon log A n and a worse dependence on the other factors of Ω (but not as good as (55) whichappeared soon after).At the time it was considered as a pinnacle of “Bakery” (van der Poorten) applied to dio-phantine equations. Since then Preda Mih˘ailescu proved the original Catalan conjecture in2004. Shortly afterwards in 2006 he found another proof not using linear forms in logarithms(and no computers either). See for example [51].6.
Elliptic functions
This section is short but sweet.We have seen that the Hermite-Lindemann Theorem essentially concerns the exponentialfunction E = e z , and the simple differential equation E ′ = E plays a role in the proofs. Itwas Siegel in 1921 who first investigated what happens for a Weierstrass elliptic function ℘ = ℘ ( z ) . This too satisfies a differential equation, namely ℘ ′ = 4 ℘ − g ℘ − g , and since weare working with algebraic numbers it is customary to assume that the “invariants” g , g arealgebraic. Minor technical problems are caused by the non-linearity; but a much more seriousdifficulty comes from analytic growth. The Schwarz Lemma in its simplest form says that if f is an entire function with f (0) = 0 , then(79) | f ( z ) | ≤ | z | R sup | z | = R | f ( z ) | for any z with | z | ≤ R . However ℘ is far from entire. We can make it entire by multiplyingby the square of the Weierstrass sigma function σ ( z ) . Unfortunately sup | z | = R | σ ( z ) | growsexponentially in R , while sup | z | = R | e z | = e R ; and this restricts the iteration procedure in theproof (there is a similar problem with “arithmetic growth”).Nevertheless Schneider in 1937 could prove the elliptic analogues of Hermite-Lindemann andGelfond-Schneider. We formulated those in terms of logarithms(80) log α = Z α d xx , and the elliptic analogue is(81) Z α ∞ d x p x − g x − g for which there is no standard “principal value” (and not even a standard notation).Thus it was natural that Baker should start thinking about elliptic analogues of his Theoremquoted in section 4. If we take α = 1 in (80), then there is a loop of integration which gives πi , a “fundamental period” of e z . Similarly if we take α = ∞ in (81), then there are two loopswhich give “fundamental periods” ω , ω of ℘ ( z ) , so that(82) ℘ ( z + ω ) = ℘ ( z ) , ℘ ( z + ω ) = ℘ ( z ) . In [27] from 1970, also a major work with far-reaching consequences later on, he succeededin proving that if β , β are algebraic, then the linear form in elliptic logarithms β ω + β ω is either zero or transcendental.The analogues of the functions (37) are(83) ℘ ( z ) , ℘ ( z ) , β z + β z and the analogues of the zeroes (38) are(84) ( z , z ) = (cid:18)(cid:18) s + 12 (cid:19) ω , (cid:18) s + 12 (cid:19) ω (cid:19) , s = 0 , , , . . . (the / just to avoid poles).However the growth problems mentioned above mean that these zeroes are not sufficientlynumerous to do the iterative steps. As hinted in section 4, Baker was forced to use more s + pq (for p = 1 , . . . , q − ) in (84); such things had not occurred in any earlier transcendence proofs.Then not only must the range of s be increased step by step, as in (41); but simultaneouslyalso the range of q .This brought new problems; for example ℘ ( ω /q ) , although still an algebraic number, willhave large degree growing like q as q increases. Nevertheless Baker was able to get enoughzeroes for a contradiction, although the Vandermonde argument had to be ratcheted up a bit.In fact he proved something more general, with ω as a period of a Weierstrass ℘ and ω a period of a second Weierstrass ℘ . Schneider had already determined when β ω + β ω canbe zero.Then in [24] published in 1969 he considered also not just periods of ℘ but also the corre-sponding quasi-periods η , η of the associated Weierstrass zeta function ζ ( z ) ; these satisfy ζ ( z + ω ) = ζ ( z ) + η , ζ ( z + ω ) = ζ ( z ) + η like (82). He showed that if β , β , γ , γ are algebraic then β ω + β ω + γ η + γ η is eitherzero or transcendental. Already Schneider in 1937 had considered the numbers β ω + γ η and β ω + γ η separately. Now Baker had to deal with new algebraic numbers like ζ ( ω /q ) − η /q .He could treat ℘ , ℘ also here. In the paper he points out that this implies the transcendenceof the sum of the circumferences of two ellipses with algebraic “axes-lengths”.Finally the note [26] from 1970 gives a lower bound for | β + β ω + β ω | like (44) whichhas the amusing consequence | ℘ ( n ) | ≤ Ce (log n ) κ for any n ≥ in Z . LAN BAKER 23 Class numbers
Now we come to more pinnacles of Bakery.Inside the field Q we have the ring Z , and inside Z sit the prime numbers, which it is nowconvenient to take as ± , ± , ± , ± , ± , ± , ± , ± , ± , ± , . . ., ± , . . . , ± , . . . ; and in the present context one calls them “irreducibles”. Euclid more or less proved that thereis unique factorization in the sense that(1) Every n = 0 , ± in Z is a product of irreducibles.(2) If two such products p · · · p k , q · · · q l are equal then k = l .(3) And in (2), possibly after a permutation, we have p i = ± q i for all i .Inside a number field K we have similarly a natural ring O together with irreducibles, andone can hope for the analogues of (1),(2),(3), except that the “ ± ” must be interpreted in termsof units as in the discussion around (63). And indeed if this were true for all K then “Fermat’sLast Theorem” would be relatively easy to prove. But it fails already for Q ( √− .The nearest approach to these analogues is through ideal classes. One works with ideals A , which are just (non-zero) O -modules in O . Two ideals A , A ′ are said to be equivalent ifthere are non-zero α, α ′ in O with α ′ A = α A ′ . The set of equivalence classes is finite, and itscardinality is the “class number” h of K .It is elementary that h = 1 is the same as the analogues of (1),(2),(3).When K is quadratic over Q and imaginary, all this was known to Gauss, who in his famous Disquisitiones Arithmeticae of 1801 calculated many K with small given class number (he ac-tually worked with quadratic forms instead). He conjectured that his lists were complete, butthat rigorous proofs seem to be very difficult: “ Demonstrationes autem rigorosae harum obser-vationum perdifficiles esse videntur. ”. Now K = Q ( √− d ) for some unique positive squarefreeinteger d , and we write h ( d ) for the class number. His list of d for h ( d ) = 1 was , , , , , , , , so nine fields K . The last d here corresponds to Euler’s discovery that the expression x + x +41 takes prime values for x = 0 , , , . . . , , and also to the fact that e π √ = 262537412640768743 . . . . is so close to an integer.In 1934 Heilbronn and Linfoot proved that there is at most one more value of d ; and if itexists then(85) d > exp(10 ) . But it was not until Baker’s work in 1966 that this could be in principle ruled out. In hisgreat paper [11] he remarked that Gelfond and Linnik in 1948 had reduced the problem usingKronecker’s Limit Formula to a weaker form of his (44) with n = 3 logarithms. So that did it.On 22nd December 1966 J-P. Serre [52] wrote to John Tate:“ Et Davenport dit qu’un jeune Anglais a prouvé l’inexistence du 10-ième corps quadratiqueimaginaire principal. Pas mal d’un coup! ”to which Tate replied on 13th January 1967 in apparent contradiction:“
The guy who proved the inexistence of the 10th imaginary quadratic field with h = 1 is American , not English. A student of Lehmer, whose thesis proved something like | D | ≥ if D < − and h D = 1 . Now he has done better! He talked recently at MIT about it. I won’ttake time to try to give the idea now, although I think I could if I tried. Stark (that’s his name)says he sees no way to generalize the thing, in order to determine all D with h ( D ) = 2 , forexample. Too bad. ”Then Serre on 24th June 1967:“ Je reviens à Baker: les anglais sont très excités par son travail, notamment par le fait qu’ildonne des bornes calculables pour les problèmes d’approximation diophantienne qu’il considère. ” Actually in [31] from 1971 he improved the work of Gelfond and Linnik by showing thatthere is effective C such that if d > C exists (he later calculated C = 10 ), then d is alsosquarefree and(86) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h (21 d ) log √ ! − π √ d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < e − π √ d/ . This involves just two logarithms (including πi ), and classical estimates for h (21 d ) then showthat the inequality can be dealt with by Gelfond’s much older bound (43). A strange turn ofevents! Later Baker produced a second inequality like (86) and then eliminated π √ d to obtain(45) for n = 2 for which (46) leads to d < . So by (85) “the tenth man” d does not exist(except as a novel by Graham Greene set in Paris not Vienna).As hinted above, events turn even stranger when it is realized that Stark [60] in 1967 hadgiven an independent proof that the tenth man does not exist; that was based on work ofHeegner from 1952 which was, at least in some circles, also accepted as a proof – and thenmore widely discounted until it was finally recognized as essentially correct.Tate refers to difficulties with class number h = 2 , which for general K is the same as sayingthat the analogue of (1) or (3) fails but (2) survives (Carlitz). And indeed doing h ( d ) = 2 wasmuch tougher. Here Gauss’s list was , , , , , , , , , , , , , , , , , . We have already referred to Baker’s paper [30] from 1971; and indeed here is where heshowed how to settle this in principle.Already in [23] from 1969 he had used two logarithms to prove that d < if it is not 3modulo 8. So he could assume d is 3 modulo 8. In that case Gauss’s theory of genera showsthat d = pq for primes p congruent to 1 modulo 4 and q congruent to 3 modulo 4. If q ≤ d / then also [23] suffices. So he could also assume that q > d / .He then proves an analogue of (86). Namely there is effective C such that if d > C exists,then d and q are squarefree and(87) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h (21 d ) log √ ! + h ′ h (21 q ) log η − π √ d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < e − π √ d/ , where h ′ is the class number of the real quadratic field Q ( √ p ) and η is a fundamental unitthere. An essential difference with (86) is that one of the logarithms, namely log η , is not fixed.So nothing from the Quartet can be used, except possibly IV. But there the coefficients wererational; and anyway the exponent (2 n + 1) = 49 in (46) is far too large. The troublesome η might have H ( η ) as large as p √ p so log H ( η ) as large as d / . As H is around d / we wouldneed an exponent κ < / . Baker was able to get any κ > , also for algebraic coefficients β , β , β (compare (50) above), thereby settling h ( d ) = 2 . This was another motivation forthe Sharpening Trio. It may be noted that Stark [61] independently settled the problem, andtheir papers could not have appeared more simultaneously, with Baker’s occupying pages 139to 152 and Stark’s 153 to 173 in volume 94 of the Annals.To this day no-one has been able to treat h ( d ) = 3 by similar methods. However thesubsequent work of Goldfeld, Gross and Zagier (see [56] for example) enables in principle anyclass number to be treated effectively.We already referred to the note [23]. This was used in a paper [32] with Andrzej Schinzel (inmemory of Davenport), also from 1971, where they make a contribution to the (still unsolved)problem of showing that Euler’s list , , , , , , , , , , , , , , , , , , . . ., , , , , , , , of 65 “numeri idonei” is complete. These were defined in terms of binary quadratic forms butthey are also the positive integers not of the form xy + yz + zx for distinct positive integers x, y, z . LAN BAKER 25 Abcology
This refers to the abc
Conjecture, formulated in 1985 by Joseph Oesterlé and the writer.Namely, for any κ > there is C = C ( κ ) such that for any non-zero a, b, c in Z with no commonfactor and a + b + c = 0 we have(88) max {| a | , | b | , | c |} ≤ CS κ for the “squarefree kernel” S = Y p | abc p ≥ , the product being taken over all primes p > .In [46] from 1998 Baker proposes a version(89) max {| a | , | b | , | c |} ≤ C S (log S ) s s ! where s = P p | abc and now C is absolute. It is easily seen that s ≤ L = [ C ′ log S/ log log S ] (if S ≥ ) for C ′ > absolute, and then Stirling gives (for L ≤ log S ) (log S ) s s ! ≤ (log S ) L L ! ≤ (cid:18) e log SL (cid:19) L ; so (89) strengthens (88) - but not so much as to contradict known lower bounds.He suggested an even stronger version and showed that it is equivalent to a certain lowerbound for min { , | Λ |} Y p min { , p | Λ | p } for the linear form in logarithms Λ = b log a + · · · + b n log a n with b , . . . , b n , a ≥ , . . . , a n ≥ in Z . Here | Λ | p is defined in a natural p -adic way. Heremarks that this lower bound suggests that the expression Ω = log A · · · log A n in (57) couldbe replaced by Γ = log A + · · · + log A n . He even speculates that | Λ | ≥ e − C Γ B − C ω , where ω = X p | a ··· a n ≤ C ′ Γlog Γ for C , C ′ absolute.But despite the extraordinarily precise works of Yu Kunrui on p -adic linear forms in loga-rithms it seems that this approach to abc remains for future generations to explore.Later on in [47] from 2004 Baker stuck his neck out by conjecturing (on extensive computerevidence) that (89) holds for C = 6 / .See also his paper [48] from 2007 for related remarks.9. Miscellaneous
Here we comment on five papers that do not so neatly fall into the above categories.In [8] from 1965 he considers algebraic functions f = f ( x ) satisfying P ( x, f ) = 0 , where P ( x, y ) has the special form P ( x, y ) = 1 + (cid:18) pq x (cid:19) y + xQ ( x, y ) for Q ( x, y ) in Z [ x, y ] and coprime p, q in Z with q ≥ . As P (0 , −
1) = 0 but ( ∂P/∂y )(0 , − = 0 ,imposing f (0) = − gives a unique power series f ( x ) = − a x + a x + a x + · · · . He shows that q n a n is in Z and also prime to q . This is in accordance with the well-knownEisenstein Theorem about denominators; but the point is that there is no q ? ≥ in Z with q n − a n in Z for all n ≥ , so that the Theorem is best possible in this respect, however wechoose Q . The simplest choice Q = 0 gives of course f ( x ) = − pq x − p q x + p q x + · · · . In [9] from 1966 he extends the Padé systems used in [5], [6] (see (24) above), [2] (see (26)above) and (later) [17]. Namely let ω , . . . , ω k be in C distinct modulo Z and let ρ rs ( r =0 , . . . , k ; s = 0 , . . . , l ) be in Z with ≤ ρ rl ≤ · · · ≤ ρ r ( r = 0 , . . . , k ) . Then if P rs ( x ) is in C [ x ] with degree at most ρ rs − , the function k X r =0 l X s =0 P rs ( x )(1 − x ) ω r (log(1 − x )) s cannot have a zero of order at least P kr =0 P ls =0 ρ rs at x = 0 unless all P rs ( x ) = 0 . He mentions(without proof) that such things can be used to prove the transcendence of e π (see section 4above).In [15] from 1967 he extends a famous result of Pólya to functions of several variables. Asthe former played an important part in the history of transcendence, and the latter a key rolein linear forms in logarithms, we should say a few words.Pólya shows that if f ( z ) is an entire function such that f (0) , f (1) , f (2) , . . . are all in Z , thena relatively slow growth rate | f ( z ) | ≤ Ce θ | z | for some θ < log 2 and some C forces it to be a polynomial. This led about 20 years later tothe Gelfond-Schneider Theorem (compare (31) and (32) above). The example f ( z ) = 2 z showsit would be false for θ = log 2 .Baker proves a similarly sharp result for f ( z , . . . , z n ) with values in Z at z , . . . , z n =0 , , , . . . . Here it suffices that | f ( z , . . . , z n ) | ≤ Ce θ ( | z | + ··· + | z n | ) , again for some θ < log 2 , to force a polynomial.He refers to Schneider’s 1941 paper mentioned in section 4. He also gives generalizationsinvolving partial derivatives ∂ t ∂z t · · · ∂ t n ∂z t n n f ( z , . . . , z n ) as in (42), so this paper seems inextricably linked with Linear Forms I (which appeared justone year earlier).The paper [38] from 1973, with Bryan Birch and Eduard Wirsing, uses Linear Forms II andIII to prove several results about a problem of Chowla, for example that the values of L (1 , χ ) = ∞ X n =1 χ ( n ) n , as χ runs over all non-principal characters modulo q , are linearly independent over Q provided q is prime to its image φ ( q ) under the Euler function.The paper [40] from 1975 with Coates is diophantine approximation of a different sort. Let p, q be coprime in Z with p > q ≥ . It is trivial that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) pq (cid:19) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ q k for all k ≥ in Z . Mahler showed that for any ǫ > there is c = c ( p, q, ǫ ) > such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) pq (cid:19) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ cq ǫk . LAN BAKER 27
For example there are at most finitely many k with (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) (cid:19) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < k + 2 k k − k ; this inequality is significant as (for k ≥ ) it is implied by the number g ( k ) in Waring’s ProblemNOT being given by(90) g ( k ) = 2 k + "(cid:18) (cid:19) k − . However the c above is “Thue-ineffective”, so these exceptional k cannot be determined (it ispresumed they do not exist).Baker and Coates show that there are effective η = η ( p, q ) < and c = c ( p, q ) > such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:18) pq (cid:19) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ cq ηk for all k . However they admit that getting η (3 , < log(4 / . . . . , as would be required to find all exceptions to (90), may need fundamentally new ideas.The proofs use a refinement of the methods of Linear Forms I, Sharpening I,II and Coates’searlier work on p -adic linear forms in logarithms (but not with p as above).10. Books
Before Baker there were three famous books on transcendental numbers written by leaders inthe field: Siegel’s “
Transcendental numbers ” (1949), Schneider’s “
Einführung in die transzen-denten Zahlen ” (1957), and Gelfond’s “
Transcendental and algebraic numbers ” (1960). Thedates reflect a certain irregular growth in the subject corresponding to various breakthroughs.Also Lang’s “
Introduction to transcendental numbers” (1966) was influential.His own “
Transcendental number theory ” [41] from 1975 is a worthy successor to these(although the writer well remembers Baker’s wail of agony over a particularly visible printingerror on page 85). It formed the substance of an essay for the Adams Prize (1972) of CambridgeUniversity.After an introductory chapter containing probably some of the world’s shortest proofs (onepage for the transcendence of e , two pages for Lindemann-Weierstrass) he launches straightinto linear forms in logarithms, and the subsequent four chapters give an excellent account ofthis topic and its various applications. They are followed by a chapter on elliptic functionsincluding his own work. The next chapter describes Schmidt’s subspace extension of Roth’sTheorem. There follows a chapter on Mahler’s Classification including Schmidt’s beautifulproof that “ T -numbers do exist ”. The next two chapters contain some of his own work in [10]and [7], and the book finishes with two chapters about the topic of algebraic independence.One is an account of Shidlovsky’s fundamental work on E -functions and the other contains forexample the proof, due independently to Dale Brownawell and Michel Waldschmidt, that atleast one of e e and e e is transcendental.Nearly all the chapters contain the first account in book form of major results.The book was eagerly devoured as soon as it appeared. The concision of the first chaptersets the tone for the rest, which makes it not so easy for beginners. It ran into a second editionin 1990, and at the time of writing a reprint is planned.Some of the material, now updated, found its way into the first three chapters of his book“ Logarithmic forms and diophantine geometry ” [49] with Wüstholz in 2007.The other chapters are then concerned more with developments for general commutativealgebraic groups G . The simplest examples of these are the additive group G a and the mul-tiplicative group G m . The Quartet results can be naturally formulated in terms of G n m and G a × G n m . The next simplest example is an elliptic curve E , and Baker’s result in [27] involves in a similar way G a × E or even G a × E × E . And given E there is a unique G = G ( E ) sitting inside an exact sequence −→ G a −→ G −→ E −→ which is however not isomorphic to G a × E ; similarly his paper [24] involves (implicitly) G ( E ) × G ( E ) and even a quotient by a line inside.The fourth chapter supplies the basic theory of such general G .As already mentioned, Baker had to work a bit in [27] to get enough zeroes for a finalcontradiction. This indeed was the main obstacle to generalizing even to three elliptic curves. Itwas overcome by Wüstholz, and the fifth chapter is devoted to the zero estimate or “multiplicityestimate” that did the trick. The resulting extension of Baker’s works on linear independenceto general G was Wüstholz’s “Analytic Subgroup Theorem”; and this is the topic of the sixthchapter.The seventh chapter is concerned with linear independence measures, both improved versionsfor linear forms in logarithms (as in [45] above) and versions for general G . Some consequencesof the latter more in the realm of diophantine geometry (due to Wüstholz and the writer)are also discussed, such as the existence of “small” isogenies between abelian varieties andpolarizations of abelian varieties, as well as effective versions of Serre’s Open Image Theoremfor elliptic curves, the Tate Conjecture for abelian varieties and some of Faltings’s FinitenessTheorems.We pause to make clearer the connexion between isogenies and (47),(48). Suppose forexample that E and E ′ are elliptic curves with periods ω , ω and ω ′ , ω ′ respectively as in(82). An isogeny between E, E ′ leads to relations(91) βω ′ = b ω + b ω , βω ′ = b ω + b ω as in (47). The correct analogue of (48) then leads to an upper bound on the degree of aconnecting isogeny.The book closes with concise accounts of other diophantine topics such as Schmidt’s Sub-space Theorem and the André-Oort Conjecture; the latter takes the reader very near to a greatdeal of current work.Baker had already published his “ A concise introduction to the theory of numbers ” [43] in1984. This attractive volume is much more elementary and suitable for a first course in NumberTheory in general.He then developed [43] into “
A comprehensive course in number theory ” [50] in 2012. Thisis over twice as long and goes up to the graduate level.Two conferences that Baker organised led to published proceedings: the first (Cambridge1976) edited by Baker and the writer (and between the two of us we refereed practically allthe 16 papers there), mentioned in [42]; and the second (Durham 1986) edited by Baker alone(where a less intensive programme than usual produced more mathematical advances thanusual), mentioned in [44]. 11.
Fields Medal
This medal was for many years the supreme prize in mathematics and still manages verywell to hold its own among various lifetime awards. It is given only to people aged at most 40,two or three or four of them every four years at the International Congress. At the time ofwriting, there have been 60 winners since it began in 1936.Baker won the prize at Nice in 1970, just 13 days after his 31st birthday. See [34] for theProceedings Volume containing his address. The citation was:“
Generalized the Gelfond-Schneider theorem (the solution to Hilbert’s seventh problem).From this work he generated transcendental numbers not previously identified.” (and indeed at the time of writing that is the only accomplishment listed in his somewhatminimal Wikipedia entry). Paul Turán in his talk at Nice wrote of this:“
The analytic prowess displayed by Baker could hardly receive a higher testimonial. ” LAN BAKER 29
However any reader of this memoir will see that Baker went far beyond that. And Turánhimself recognized the fact in his closing words:“
To conclude, I remark that his work exemplifies two things very convincingly. Firstly,that beside the worthy tendency to start a theory in order to solve a problem it pays also toattack specific difficult problems directly. Particularly is this the case with such problems whererather singular circumstances do not make it probable that a solution would fall out as an easyconsequence of a general theory. Secondly, it shows that a direct solution of a deep problemdevelops itself quite naturally into a healthy theory and gets into early and fruitful contact withother significant problems of mathematics. So, let the two different ways of doing mathematicslive in peaceful coexistence for the benefit of our science. ”Baker was presented to Président Pompidou at the Elysée Palace and later wrote:“ ... and was quite impressed by his feat of memory when making a welcoming scientificaddress ”.He was the third British winner of the Fields Medal (after Klaus Roth 1958 and MichaelAtiyah 1966).Alan Baker single-handedly transformed several areas of number theory. He achieved amajor breakthrough in transcendence and applied it to obtain a new and important large classof transcendental numbers (opening the way to the subsequent discovery of several other suchclasses); developed quantitative versions and applied them to the effective solutions of manyclassical diophantine equations as well as the first effective improvement on Liouville’s 1844result on diophantine approximation and the resolution of the celebrated Gauss Conjecturesof 1801 on class numbers, not only h = 1 but also h = 2 , of imaginary quadratic fields; andstarted the study of extensions to elliptic curves (opening the way to later generalizations toabelian varieties and commutative group varieties and in turn their applications to old andnew problems in diophantine geometry).Despite this, his cousin describes him as “extremely modest”. This is confirmed by anotherextract from the funeral tribute of Eva McLean:“ His most striking characteristic, however, was a genuine modesty. He really believed thathis brilliant achievement was merely down to hard work and determination. At the same timehe spoke in great awe of those mathematicians whom he regarded as the truly greats. ”ACKNOWLEDGEMENTSI thank the Royal Society for allowing me to see their files. I am grateful to Trinity College,particularly to the Librarian Nicolas Bell about Whewell’s Court accommodation; he also notesthat Alan Baker’s personal papers and Fields Medal have been deposited in the College Library.In addition I have much profited from conversations and correspondence with Sheldon Baker,Béla Bollobás, William Chen, Yuval Flicker, Martin Huxley, Chris Morley, Sir Michael Pepper,Heather Rechtman, Klaus Schmidt, Cam Stewart, Sir Martin Taylor, Rob Tijdeman, MichelWaldschmidt, Michael Grae Worster, and Gisbert Wüstholz (also for sending me a completelist of papers) on the personal aspects of Alan Baker’s life. But in this respect my greatestdebt is of course to Eva McLean.And on the scientific side I thank too Roger Baker, Enrico Bombieri, Bob Vaughan, andUmberto Zannier, as well as two referees.
References [1] Continued fractions of transcendental numbers, Mathematika (1962), 1–8.[2] Approximations to the logarithms of certain rational numbers, Acta Arithmetica (1964), 315–323. [3] On an analogue of Littlewood’s Diophantine approximation problem, Michigan Math. J. (1964), 247–250.[4] On Mahler’s classification of transcendental numbers, Acta Math. (1964), 97–120.[5] Rational approximations to certain algebraic numbers, Proc. London Math. Soc. (1964), 385–398.[6] Rational approximations to √ and other algebraic numbers, Quarterly. J. Math. (1964), 375–383.[7] On some Diophantine inequalities involving the exponential function, Canadian J. Math. (1965), 616–626.[8] Power series representing algebraic functions, J. London Math. Soc. (1965), 108–110.[9] A note on the Padé table, Indagationes Math. (1966), 596–601.[10] On a theorem of Sprindˇ z uk , Proc. Royal Soc. London A292 (1966), 92–104.[11] Linear forms in the logarithms of algebraic numbers, Mathematika (1966), 204–216.[12] Linear forms in the logarithms of algebraic numbers II, Mathematika (1967), 102–107.[13] Linear forms in the logarithms of algebraic numbers III, Mathematika (1967), 220–228.[14] On Mahler’s classification of transcendental numbers II: Simultaneous Diophantine approximation. ActaArithmetica (1967), 281–288.[15] A note on integral integer-valued functions of several variables. Proc. Cambridge Phil. Soc. (1967),715–720.[16] On some diophantine inequalities involving primes, J. reine angew. Math. (1967), 166–181.[17] Simultaneous rational approximations to certain algebraic numbers, Proc. Cambridge Phil. Soc. (1967),693–702.[18] Contributions to the theory of Diophantine equations. I. On the representation of integers by binary forms,Phil. Trans. Royal Soc. London A263 (1968), 173–191.[19] Contributions to the theory of Diophantine equations. II. The Diophantine equation y = x + k , Phil.Trans. Royal Soc. London A263 (1968), 193–208.[20] Linear forms in the logarithms of algebraic numbers IV, Mathematika (1968), 204–216.[21] The diophantine equation y = ax + bx + cx + d , J. London Math. Soc. (1968), 1–9.[22] Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Phil. Soc. (1969), 439–444.[23] A remark on the class number of quadratic fields, Bull. London Math. Soc. (1969), 98–102.[24] On the quasi-periods of the Weierstrass ζ -function, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II(1969), 145–157.[25] (with H. Davenport) The equations x − y and x − z , Quarterly J. Math. (1969), 129–137.[26] An estimate for the ℘ -function at an algebraic point, Amer. J. Math. (1970), 619–622.[27] On the periods of the Weierstrass ℘ -function, in Symposia Mathematica, Vol. IV (INDAM, Rome 1968/69),Academic Press, London 1970 (pp. 155-174).[28] (with J. Coates) Integer points on curves of genus 1, Proc. Cambridge Phil. Soc. (1970), 595–602.[29] (with W. M. Schmidt) Diophantine approximation and Hausdorff dimension, Proc. London Math. Soc. (1970), 1–11.[30] Imaginary quadratic fields with class number 2, Annals of Math. (1971), 139–152.[31] On the class number of imaginary quadratic fields, Bull. Amer. Math. Soc. (1971), 678–684.[32] (with A. Schinzel) On the least integers represented by the genera of binary quadratic forms, Acta Arith-metica (1971), 137–144.[33] (with H. M. Stark) On a fundamental inequality in number theory, Annals of Math. (1971), 190–199.[34] Effective methods in the theory of numbers, in Actes du Congrès International des Mathématiciens I (Nice,1970), Gauthier-Villars, Paris 1971 (pp. 19–26).[35] A sharpening of the bounds for linear forms in logarithms, Acta Arithmetica (1972), 117–129.[36] A sharpening of the bounds for linear forms in logarithms II, Acta Arithmetica (1973), 33–36.[37] A central theorem in transcendence theory, in Diophantine approximation and its applications (ed. C.F.Osgood), Academic Press, New York 1973 (pp. 1–23).[38] (with B. J. Birch and E. A. Wirsing) On a problem of Chowla, J. Number Theory (1973), 224–236.[39] A sharpening of the bounds for linear forms in logarithms III, Acta Arithmetica (1975), 247–252.[40] (with J. Coates) Fractional parts of powers of rationals, Math. Proc. Cambridge Phil. Soc. (1975),269–279.[41] Transcendental number theory , Cambridge Mathematical Library. Cambridge University Press, Cambridge1975; second edition 1990.[42] The theory of linear forms in logarithms, in
Transcendence theory: advances and applications (eds. A.Baker, D.Masser), Academic Press, London 1977 (pp.1–27).[43]
A concise introduction to the theory of numbers , Cambridge University Press, Cambridge 1984.[44] (with C. L. Stewart) On effective approximations to cubic irrationals, in
New advances in transcendencetheory (ed. A. Baker), Cambridge University Press, Cambridge 1988 (pp. 1–24).[45] (with G. Wüstholz) Logarithmic forms and group varieties, J. reine angew. Math. (1993), 19–62.[46] Logarithmic forms and the abc -conjecture, in
Number theory (eds. K. Györy, A. Pethö, V.T. Sos), deGruyter, Berlin 1998 (pp.37-44).[47] Experiments on the abc -conjecture, Publ. Math. Debrecen (2004), 253–260.[48] On an arithmetical function associated with the abc -conjecture, in Diophantine geometry (ed. U. Zannier),CRM Series , Ed. Norm., Pisa 2007 (pp. 25–33).[49] (with G. Wüstholz) Logarithmic forms and Diophantine geometry , New Mathematical Monographs ,Cambridge University Press, Cambridge 2007. LAN BAKER 31 [50]
A comprehensive course in number theory , Cambridge University Press, Cambridge 2012.OTHER REFERENCES[51] Y. Bilu, Y. Bugeaud, M. Mignotte,
The Problem of Catalan , Springer 2014.[52]
Correspondance Serre-Tate I (eds. P. Colmez, J-P. Serre), Documents Mathématiques , Soc. Math. deFrance 2015.[53] V. Dimitrov, P. Habegger, Galois orbits of torsion points near atoral sets, https://arxiv.org/abs/1909.06051(September 2019).[54] T. Estermann, Introduction to modern prime number theory , Cambridge Tracts , Cambridge UniversityPress, Cambridge 1961.[55] N.I. Feldman, Improved estimate for a linear form of the logarithms of algebraic numbers, Math. USSR-Sb. (1968), 393–406.[56] D.M. Goldfeld, Gauss’ class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. (1985), 23–37.[57] Hardy-Ramanujan J. (2019), Special commemorative volume in honour of Alan Baker.[58] A Panorama of Number Theory or The View from Baker’s Garden (ed. G. Wüstholz), Cambridge Uni-versity Press, Cambridge 2002 .[59] S. Simonson, The Mathematics of Levi ben Gershon, the Ralbag,https://u.cs.biu.ac.il/ ∼ tsaban/Pdf/MathofLevi.pdf[60] H.M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math.J. (1967), 1–27.[61] H.M. Stark, A transcendence theorem for class-number problems, Annals of Math. (1971), 153–173.[62] I. Vardi, Archimedes’ Cattle Problem, Amer. Math. Monthly (1998), 305–319.[63] G. Wüstholz, Obituary of Alan Baker FRS, Acta Arithmetica (2019), 309–345.[64] https://en.wikipedia.org/wiki/Sums of three cubes Department of Mathematics and Computer ScienceUniversity of BaselSpiegelgasse 1, 4051 Basel, Switzerland
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