This result was first conjectured by Gauss in Section 303 of his Disquisitiones Arithmeticae (1798). It was essentially proven by Kurt Heegner in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until Harold Stark gave a complete proof in 1967, which had many commonalities to Heegner's work, but sufficiently many differences that Stark considers the proofs to be different.[3] Heegner "died before anyone really understood what he had done".[4] Stark formally filled in the gap in Heegner's proof in 1969 (other contemporary papers produced various similar proofs by modular functions, but Stark concentrated on explicitly filling Heegner's gap).[5]
Alan Baker gave a completely different proof slightly earlier (1966) than Stark's work (or more precisely Baker reduced the result to a finite amount of computation, with Stark's work in his 1963/4 thesis already providing this computation), and won the Fields Medal for his methods. Stark later pointed out that Baker's proof, involving linear forms in 3 logarithms, could be reduced to only 2 logarithms, when the result was already known from 1949 by Gelfond and Linnik.[6]
Stark's 1969 paper (Stark 1969a) also cited the 1895 text by Heinrich Martin Weber and noted that if Weber had "only made the observation that the reducibility of [a certain equation] would lead to a Diophantine equation, the class-number one problem would have been solved 60 years ago". Bryan Birch notes that Weber's book, and essentially the whole field of modular functions, dropped out of interest for half a century: "Unhappily, in 1952 there was no one left who was sufficiently expert in Weber's Algebra to appreciate Heegner's achievement."[7]
Deuring, Siegel, and Chowla all gave slightly variant proofs by modular functions in the immediate years after Stark.[8] Other versions in this genre have also cropped up over the years. For instance, in 1985, Monsur Kenku gave a proof using the Klein quartic (though again utilizing modular functions).[9] And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline).[10]
The work of Gross and Zagier (1986) (Gross & Zagier 1986) combined with that of Goldfeld (1976) also gives an alternative proof.[11]
Real caseedit
On the other hand, it is unknown whether there are infinitely many d > 0 for which Q(√d) has class number 1. Computational results indicate that there are many such fields. Number Fields with class number one provides a list of some of these.
Notesedit
^Elkies (1999) calls this the Stark–Heegner theorem (cognate to Stark–Heegner points as in page xiii of Darmon (2004)) but omitting Baker's name is atypical. Chowla (1970) gratuitously adds Deuring and Siegel in his paper's title.
Chen, Imin (1999), "On Siegel's Modular Curve of Level 5 and the Class Number One Problem", Journal of Number Theory, 74 (2): 278–297, doi:10.1006/jnth.1998.2320
Darmon, Henri (2004), "Preface to Heegner Points and Rankin L-Series" (PDF), MSRI Publications, 49: ix–xiii
Elkies, Noam D. (1999), "The Klein Quartic in Number Theory" (PDF), in Levy, Silvio (ed.), The Eightfold Way: The Beauty of Klein's Quartic Curve, MSRI Publications, vol. 35, Cambridge University Press, pp. 51–101, MR 1722413
Stark, H. M. (2011), The Origin of the "Stark" conjectures, vol. appearing in Arithmetic of L-functions
December 18, 2023
stark, heegner, theorem, number, theory, baker, heegner, stark, theorem, establishes, complete, list, quadratic, imaginary, number, fields, whose, rings, integers, principal, ideal, domains, solves, special, case, gauss, class, number, problem, determining, nu. In number theory the Baker Heegner Stark theorem 1 establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains It solves a special case of Gauss s class number problem of determining the number of imaginary quadratic fields that have a given fixed class number Let Q denote the set of rational numbers and let d be a square free integer The field Q d is a quadratic extension of Q The class number of Q d is one if and only if the ring of integers of Q d is a principal ideal domain The Baker Heegner Stark theorem can then be stated as follows If d lt 0 then the class number of Q d is one if and only if d 1 2 3 7 11 19 43 67 163 displaystyle d in 1 2 3 7 11 19 43 67 163 These are known as the Heegner numbers By replacing d with the discriminant D of Q d this list is often written as 2 D 3 4 7 8 11 19 43 67 163 displaystyle D in 3 4 7 8 11 19 43 67 163 Contents 1 History 2 Real case 3 Notes 4 ReferencesHistory editThis result was first conjectured by Gauss in Section 303 of his Disquisitiones Arithmeticae 1798 It was essentially proven by Kurt Heegner in 1952 but Heegner s proof had some minor gaps and the theorem was not accepted until Harold Stark gave a complete proof in 1967 which had many commonalities to Heegner s work but sufficiently many differences that Stark considers the proofs to be different 3 Heegner died before anyone really understood what he had done 4 Stark formally filled in the gap in Heegner s proof in 1969 other contemporary papers produced various similar proofs by modular functions but Stark concentrated on explicitly filling Heegner s gap 5 Alan Baker gave a completely different proof slightly earlier 1966 than Stark s work or more precisely Baker reduced the result to a finite amount of computation with Stark s work in his 1963 4 thesis already providing this computation and won the Fields Medal for his methods Stark later pointed out that Baker s proof involving linear forms in 3 logarithms could be reduced to only 2 logarithms when the result was already known from 1949 by Gelfond and Linnik 6 Stark s 1969 paper Stark 1969a also cited the 1895 text by Heinrich Martin Weber and noted that if Weber had only made the observation that the reducibility of a certain equation would lead to a Diophantine equation the class number one problem would have been solved 60 years ago Bryan Birch notes that Weber s book and essentially the whole field of modular functions dropped out of interest for half a century Unhappily in 1952 there was no one left who was sufficiently expert in Weber s Algebra to appreciate Heegner s achievement 7 Deuring Siegel and Chowla all gave slightly variant proofs by modular functions in the immediate years after Stark 8 Other versions in this genre have also cropped up over the years For instance in 1985 Monsur Kenku gave a proof using the Klein quartic though again utilizing modular functions 9 And again in 1999 Imin Chen gave another variant proof by modular functions following Siegel s outline 10 The work of Gross and Zagier 1986 Gross amp Zagier 1986 combined with that of Goldfeld 1976 also gives an alternative proof 11 Real case editOn the other hand it is unknown whether there are infinitely many d gt 0 for which Q d has class number 1 Computational results indicate that there are many such fields Number Fields with class number one provides a list of some of these Notes edit Elkies 1999 calls this the Stark Heegner theorem cognate to Stark Heegner points as in page xiii of Darmon 2004 but omitting Baker s name is atypical Chowla 1970 gratuitously adds Deuring and Siegel in his paper s title Elkies 1999 p 93 Stark 2011 page 42 Goldfeld 1985 Stark 1969a Stark 1969b Birch 2004 Chowla 1970 Kenku 1985 Chen 1999 Goldfeld 1985 References editBirch Bryan 2004 Heegner Points The Beginnings PDF MSRI Publications 49 1 10 Chen Imin 1999 On Siegel s Modular Curve of Level 5 and the Class Number One Problem Journal of Number Theory 74 2 278 297 doi 10 1006 jnth 1998 2320 Chowla S 1970 The Heegner Stark Baker Deuring Siegel Theorem Journal fur die reine und angewandte Mathematik 241 47 48 doi 10 1515 crll 1970 241 47 Darmon Henri 2004 Preface to Heegner Points and Rankin L Series PDF MSRI Publications 49 ix xiii Elkies Noam D 1999 The Klein Quartic in Number Theory PDF in Levy Silvio ed The Eightfold Way The Beauty of Klein s Quartic Curve MSRI Publications vol 35 Cambridge University Press pp 51 101 MR 1722413 Goldfeld Dorian 1985 Gauss s class number problem for imaginary quadratic fields Bulletin of the American Mathematical Society 13 23 37 doi 10 1090 S0273 0979 1985 15352 2 MR 0788386 Gross Benedict H Zagier Don B 1986 Heegner points and derivatives of L series Inventiones Mathematicae 84 2 225 320 Bibcode 1986InMat 84 225G doi 10 1007 BF01388809 MR 0833192 S2CID 125716869 Heegner Kurt 1952 Diophantische Analysis und Modulfunktionen Diophantine Analysis and Modular Functions Mathematische Zeitschrift in German 56 3 227 253 doi 10 1007 BF01174749 MR 0053135 S2CID 120109035 Kenku M Q 1985 A note on the integral points of a modular curve of level 7 Mathematika 32 45 48 doi 10 1112 S0025579300010846 MR 0817106 Levy Silvio ed 1999 The Eightfold Way The Beauty of Klein s Quartic Curve MSRI Publications vol 35 Cambridge University Press Stark H M 1969a On the gap in the theorem of Heegner PDF Journal of Number Theory 1 1 16 27 Bibcode 1969JNT 1 16S doi 10 1016 0022 314X 69 90023 7 hdl 2027 42 33039 Stark H M 1969b A historical note on complex quadratic fields with class number one Proceedings of the American Mathematical Society 21 254 255 doi 10 1090 S0002 9939 1969 0237461 X Stark H M 2011 The Origin of the Stark conjectures vol appearing in Arithmetic of L functions Retrieved from https en wikipedia org w index php title Stark Heegner theorem amp oldid 1185985230, wikipedia, wiki, book, books, library,
