From: bobs@mathworks.com (Bob Silverman) Newsgroups: sci.math Subject: Re: Quadratic fields Date: 24 Oct 1995 22:39:10 -0400 In article , Dik T. Winter wrote: >I was reading through Weiss' Algebraic Number Theory today and again >stumbled on the statement that there are 9 complex quadratic fields >with class number 1 and with discriminant > -520. Also there is at most >one more with discriminant (according to Lehmer) < -5.10^9. A similar >statement can be found in Hardy & Wright. As both books are quite old I >want to know whether there is more known about this currently. Also, is There are only 9. The largest absolute discriminant is -163. For class number 2 there are 18. -427 is the last. For class number 3 there are 16. -907 is the last. For 4, the last is -1555. See: D. Goldfeld The class number of quadratic fields and the conj. of Birch & Swinnerton-Dyer, Ann. Sc. Norm. Super. Pisa vol 3. 1976 and Gross & Zagier, Heegner points and the derivatives of L-series. Invent. Math. 84 (1986) >it already known whether there are an infinite number of real quadratic >fields with class number 1? >-- >dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924098 >home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: dik@cwi.nl -- Bob Silverman The MathWorks Inc. 24 Prime Park Way Natick, MA ============================================================================== From: rgep@pmms.cam.ac.uk (Richard Pinch) Newsgroups: sci.math Subject: Re: Quadratic fields Date: 27 Oct 1995 12:14:50 GMT In article , dik@cwi.nl (Dik T. Winter) writes: |> I was reading through Weiss' Algebraic Number Theory today and again |> stumbled on the statement that there are 9 complex quadratic fields |> with class number 1 and with discriminant > -520. Also there is at most |> one more with discriminant (according to Lehmer) < -5.10^9. A similar |> statement can be found in Hardy & Wright. As both books are quite old I |> want to know whether there is more known about this currently. The list is complete, by results of Baker and Stark: see Baker \cite{Bak:transc}. The complete list of discriminants of class-number $H$ has been computed for a number of small values of $H$: see, for example, \cite{Arn:classno4}. Richard Pinch; Queens' College, Cambridge @BOOK(Bak:transc, AUTHOR= "Baker, Alan", TITLE= "Transcendental number theory", PUBLISHER= "Cambridge University Press", YEAR= 1975, ) @ARTICLE(Arn:classno4, AUTHOR= "Arno, S.", TITLE= "The imaginary quadratic fields of class number 4", JOURNAL="Acta Arithmetica", VOLUME= 60, NUMBER= 4, YEAR= 1992, PAGES= "321--334", ) ============================================================================== From: adam@abel.harvard.edu (Adam Logan) Newsgroups: sci.math Subject: Re: Quadratic fields Date: 27 Oct 1995 18:58:21 GMT In article <46qomq$704@puff.mathworks.com>, Bob Silverman wrote: > >Richard, do you know how far this has been pushed? I know that results >are complete for h = 2,3,4 as well, and that the work of Goldfeld, Gross, >& Zagier showed how (in principle) to find all fields for a particular h, >but I don't know whether their method (based on some kind of L-series >calculation??) is effective. Can you clarify? [hoping Richard Pinch won't mind this interruption...] A paper on the algebraic number theory preprint server gives the complete result for odd h with 5 <= h <= 23. Adam ==============================================================================