CoCalc Public Fileswww / papers / thesis / cover / wasOpen with one click!
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
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From [email protected] Fri Apr 28 04:35 PDT 2000
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Date: Fri, 28 Apr 2000 13:34:45 +0200 (CEST)
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From: Kevin Buzzard <[email protected]>
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To: "William A. Stein" <[email protected]>
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Subject: Re: Modular Symbols handbook
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Oh, PS:
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> eps:=DirichletCharacter(9,[3]);
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> M:=ModularSymbols(eps,10,-1);
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Creating M_10(Gamma_1(9),eps;F_0)^-
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, 1.269 seconds.
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> T:=Tn(Sk(M),3);
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Tn(
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M: Full Vector space of degree 8 over Cyclotomic Field of order...,
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n: 3
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)
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HeckeOperator(
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M: Full Vector space of degree 8 over Cyclotomic Field of order...,
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n: 3
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)
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HeckeOperator(
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M: Full Vector space of degree 8 over Cyclotomic Field of order...,
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Heil: [ [ 1, 0, 0, 3 ], [ 3, 1, 0, 1 ], [ 1, 0, 1, 3 ], [ 3, 0, 0,...
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)
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In file "/home/buzzard/maths/computers/magma/magma/Hecke/hecke.m", line
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1036,
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column 14:
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>> F := M`F;
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^
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Runtime error in `: Attribute 'F' for this structure is valid but not
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assigned
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>
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From [email protected] Fri Apr 28 04:38 PDT 2000
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Date: Fri, 28 Apr 2000 13:33:58 +0200 (CEST)
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From: Kevin Buzzard <[email protected]>
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To: "William A. Stein" <[email protected]>
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Subject: Re: Modular Symbols handbook
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I'll try and take a look at it. I don't have 1/10th the time I used to
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have though.
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Ironically, having 1/10th the time I used to have has made me more
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focussed, ignoring emails that can safely be ignored and putting more time
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into work when I have time. So I sat down and read through all of the
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artin thing. I want to finish it before I do anything else. I will try and
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type up the final version within the next few days. I have some questions
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but they're at home and I'm at work.
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Kevin
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From [email protected] Fri Apr 28 04:39 PDT 2000
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Date: Fri, 28 Apr 2000 13:38:48 +0200 (CEST)
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From: Kevin Buzzard <[email protected]ihp.jussieu.fr>
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To: "William A. Stein" <[email protected]>
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Subject: Re: Modular Symbols handbook
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PPS I gave a talk here at IHP about patterns in slopes and now people are
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pestering me to do lots of things that I can't do immediately because I
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can't work out newton polys of polys defined over the integers of
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cyclotomic fields. If there's a way of
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(a) persuading them into Q_p(zeta) from Q(zeta)
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(b) accessing a valuation function in Q_p(zeta)
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then presumably NewtonSlopes would be easy to adapt so that it works in
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this case. Can this be done? I guess I should find out myself...
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Kevin
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PS I remember one of the questions I had about Buzzard-Stein: if you
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believe this "Buzzard Bound" then how many of those A_5 extns can we
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unconditionally prove come from modular forms?
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Subject: !
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Actually I just discovered that the Valuation command is pretty flexible.
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All I have to do is to rewrite NewtonSlopes a little, so it doesn't
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want p as an input. It's quite easy.
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> K<zeta>:=CyclotomicField(3);
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> IsPrime(zeta-1);
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false
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Is that right???
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Kevin
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From: Kevin Buzzard <[email protected]>
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Subject: :-)
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Now I've spent 2 minutes writing a NewtonSlopes myself, we see already
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something that nobody seems to have realised: the eigencurve for p=3
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is also etale round the edges. Not only that, it appears to be a disjoint union
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of copies of weight space. How else could one explain:
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> eps:=DirichletCharacter(27,[9]);
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> K<z>:=CyclotomicField(9);
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> M:=ModularSymbols(eps,6,-1);
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Creating M_6(Gamma_1(27),eps;F_0)^-
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, 12.839 seconds.
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> T:=Tn(M,3);
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> f:=CharacteristicPolynomial(T);
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> NewtonSlopes(f,z-1);
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[* 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 *]
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(note that I've increased the scope of NewtonSlopes: the fact that I was
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too lazy to do this before meant that I'd never spotted this). So I reckon
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that for general p this will also hold! It's hard to say when this idea
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would have come to light without Hecke as a resource :-) Now I have my
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new toy I can check some more.
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Kevin
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From: "Maggie Wong" <[email protected]>
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To: [email protected]
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Subject: See ya....
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Date: Fri, 28 Apr 2000 14:28:15 GMT
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Hi William
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See you tomorrow!!
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Mag.
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________________________________________________________________________
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Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com
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