Hardy-Ramanujan Journal 42 (2019), 4-11Hardy-Ramanujan 4-11 Remembrance of Alan Baker Gisbert W¨ustholz REMEMBRANCE OF ALAN BAKER It was at the conference on Diophantische Approximationen organized by Theodor Schneider by which took place at Oberwolfach Juli 14 - Juli 20 1974 when I first met Alan Baker. At the time I was together with Hans Peter SchlickeweiGISBERT a graduate WUSTHOLZ¨ student of Schneider at the Albert Ludwigs University at Freiburg i. Br.. We both were invited to the conference to help Schneider organize the conference in practical matters. In particular we were determined to make the speakers write down an abstract of their talks in the Vortagsbuch which was considered to be an important historical document for the future. Indeed if you open the webpage of the Mathematisches Forschungsinstitut Oberwolfach you find a link to the Oberwolfach Digital Archive where you can find all these Documents It was at the conference on Diophantische Approximationen organized by Theodor Schneider which took place at fromOberwofach 1944-2008. Juli 14 - Juli 20 1974 when I first met Alan Baker. At the time I was together with Hans Peter Schlickewei a graduateAt this student conference of Schneider I was at verythe Albert-Ludwig-University much impressed and at Freiburg touched i. Br.. by We seeing both were the invitedFields to Medallist the conference Baker whoseto help work Schneider I had to studied organize the in conference the past in year practical to matters.do the Inp-adic particular analogue we were determined of Baker's to make recent the versions speakers A write down an abstract of their talks in the Vortagsbuch which was considered to be an important historical document for sharpeningthe future. of Indeed the if bounds you open for the linear webpage forms of the Mathematisches in logarithms Forschungsinstitut I together with Oberwolfach the subsequent you find a linkSharpening to the II. ThisOberwolfach was done Digital previously Archive where by you Coates can find in all 1969 these but Documents based from on an 1944-2008. older and much less precise version. I did notAt this have conference much Iconversation was very much with impressed Baker and then. touched by seeing the Fields Medallist Baker whose work I had studied in the past year to do the p-adic analogue of Baker’s recent versions A sharpening of the bounds for linear forms inThis logarithms was at I together the time with when the subsequent the old historicalsharpening II building,. This was done an old previously castle, by of Coates the Forschungsinstitut in 1969 but based on an was stillolder there. and much less precise version. I did not have much conversation with Baker then. This was at the time when the old historical building, an old castle, of the Forschungsinstitut was still there. Later Baker told me many times that he misses the old building which was replaced in 1975 by the libraryLater Baker and told meetings me many buildingtimes that he of misses the theMFO, old building a gift which of the was VolkswagenStiftung. replaced in 1975 by the library He and liked meetings the old building of the MFO, a gift of the VolkswagenStiftung. He liked the old castle and the atmosphere very much. castle and the atmosphere very much. We thank episciences.org for providing open access hosting of the electronic journal Hardy-Ramanujan Journal Date: 09.08.2018. 1 Gisbert W¨ustholz, Remembrance of Alan Baker 5 2 GISBERT WUSTHOLZ¨ Baker gave a talk at this conference about joint work with Coates on an effective version of a result which has been 2 GISBERT WUSTHOLZ¨ Bakerobtained gave by Mahler a talk in at 1957. this He conference also makes the about remark joint that work an improvement with Coates of the on result an of effective a special version kind would of settlea result an whichBakeroutstanding has gave been a question talk obtained at this in connection conference by Mahler with about Waring’s in joint1957. problem. work He with also Coates makes on the an remark effective that version an of improvement a result which of has the been obtainedresult of by a Mahler special in kind 1957. would He also settle makes an the outstanding remark that an question improvement in connection of the result withof a special Waring's kindwould problem. settle an outstanding question in connection with Waring’s problem. FIGURE 1. Baker’s abstract 1974 Oberwolfach FBaker'sIGURE 1. abstract Baker’s abstract 1974 Oberwolfach 1974 Oberwolfach The conference in 1974 was not the first Oberwolfach Conference Baker attended. As far as I could trace he was at the Forschungsinstitut the first time in 1965 where he gave a talk on three different 66 topics. The first was on effective approximation of p3 2 by rationals, one of the famous results in the framework of Roth's theorem. The second topic was on linear forms with integer coefficients in exp(r0);:::; exp(rk) where r0; : : : ; rk are distinct positive rationals. The last topic was on a linear form with real coefficients in three rational primes. Gisbert W¨ustholz, Remembrance of Alan Baker 7 4 GISBERT WUSTHOLZ¨ After this 1965-conference he started to develop his theory on linear forms in logarithms. As an Afteroutcome, this 1965-conference he was able to he solve started a to sequence develop his of outstandingtheory on linear diophantine forms in logarithms. problems. As The an outcome talk was he one was of able to solvethe a sequence moments of outstanding of glory in diophantine diophantine problems. geometry The and talk also was oneat Oberwolfach. of the moments Another of glory in result diophantine which wasgeometry and alsonot at reported Oberwolfach. in Oberwolfach Another result was which the class was number not reported 1 problem. in Oberwolfach This was the class number 1 problem. This was attackedwas attacked before by before Hegner by in Heegner a paper in which a paper was not which accepted was not in the accepted community. in the Baker community. solved it using Baker linear solved forms in logarithmsit using but linear the situation forms with in logarithms Hegner’s proof but remained the situation unclear with at the Heegner's time. Schlickewei proof remained wrote his unclear Master at thesis the with Schneidertime. on Schlickewei Hegner’s proof wrote around his Masters that time. thesis with Schneider on Heegner's proof around that time. All these results stimulated very strongly research on linear forms in logarithms. As I mentioned earlier I decided to work on linearAll these forms results in p-adic stimulated logarithms. very As strongly far as I research can remember on linear this wasforms one in of logarithms. the topics for As a I PhD mentioned thesis which Schneiderearlier suggested I decided to Schlickewei to work on and linear myself forms when in p we-adic had logarithms. a first meeting As on far PhD as I problems. can remember I started this in was1973 and workedone on of it the before topics the 1974 for a Oberwolfach PhD thesis which conference. Schneider There suggested I met van der to Schlickewei Poorten whom and I reported myself when on the we status had of my work.a To first my meeting surprise onhe himself PhD problems. gave a talk I onstarted lower in bounds 1973 inand linear worked forms on in it p-adic before logarithms the 1974 and Oberwolfach claimed to have obtainedconference. the full p-adic There analogue I met van ofBaker’s der Poorten theorem. whom I was I reported very depressed on the because status of van my der work. Poorten To wasmy surprise considered as an authority and I gave up. Later it turned out that his proof contained serious mistakes which eventually were fixed in he himself gave a talk on lower bounds in linear forms in p-adic logarithms and claimed to have a first paper of my PhD student Kunrui Yu. I then changed my PhD problem and turned to linear forms in logarithms of U-numbersobtained and thegot the full PhD p-adic degree analogue at Freiburg. of Baker's theorem. I was very depressed because van der Poorten Afterwas the considered Oberwolfach as an conference authority in and1974 I Baker gave stoppedup. Later to attend it turned the subsequent out that his conferences proof contained and after1977 serious he also stoppedmistakes to work which in a serious eventually way onwere linear fixed forms in a in first logarithms. paper of The my PhD main student stream in Kunrui transcendence Yu. I then moved changed from linear formsmy which PhD seemed problem to have and run turned into a to dead linear end forms to group in logarithms varieties and of various U-numbers problems and in got that the direction PhD degree were attacked. at EventuallyFreiburg. I succeeded to prove the analytic subgroup theorem which put the Baker theory onto a much higher and general level. New methods from algebraic geometry were needed to replace the more adhoc tools in the Baker theory by multiplicityAfter estimates the Oberwolfach on group varieties. conference This in was 1974 partly Baker developed stopped jointly to attend with Masser. the subsequent As an easy conferences corollary Baker’ qualitativeand after theorem 1977 on helinear also forms stopped could to be work derived in from a serious the analytic way on subgroup linear formstheorem. in Around logarithms. 1985 I The realized main to very muchstream my surprise in transcendence that the very abstract moved theoryfrom linear was able forms to remove which seemed a second to order have factor run into from a the dead inequalities end to group that Baker established in 1977 which was considered as the ultimate result provable by the methods at the time.