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
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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",
)
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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
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