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Author: William A. Stein
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From [email protected] Mon Nov 30 06:37:48 1998
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Date: Mon, 30 Nov 1998 15:38:25 +0100 (MET)
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From: Luis Victor Dieulefait <[email protected]>
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To: [email protected]
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Subject: newforms
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William
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Hello! I'm Luis from the University of Barcelona, we met at the
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ABC workshop (Arizona). Kevin Buzzard told me you may help me with
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some computations with newforms of high levels. I have visited your
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site on the web with tables of hecke polynomials, but I need values for
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higher levels, and I don't have linux (for the moment).
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The example I have is a newform of level 8192. The number field
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generated by its coefficients is given by a root of:
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x^8 - 24 x^6 + 164 x^4 -240 x^2 +2 , which is a factor of the
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T_3 of this level (please correct if there's some mistake)
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(sorry, I forgot to mention that we are in weight 2 and
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trivial nebentypus)
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This form seems to have an inner twist, and it also seems from the
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discussion between you and Kevin that this can be proved (I have to take
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a look at Shimura's book...)
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If your algorithm works well for this level, please send me the a_5 and
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a_7 of this newform.
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One more thing: I've found in your table the T_2, T_3 , and T_5 for level
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2048. Can you send me the T_7 and T_11 for this same level?
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Up to what level can you compute these Hecke polynomials?
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30000 would be too much?
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Thank you!
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Luis
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From [email protected] Tue Dec 1 05:54:17 1998
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Date: Tue, 1 Dec 1998 14:58:39 +0100 (MET)
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From: Luis Victor Dieulefait <[email protected]>
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To: [email protected]
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Subject: Re: newforms
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William
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Thank you very much for your help! I will see wether or not I can manage
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to obtain the information I need from the polynomials reduced mod. some
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primes, but I guess that combining these reduced polynomials with the
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known bounds something can be done.
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I still need some more details. For level 2048 I haven't been able to
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find out which are the eigenvalues I need. I'm interested in the newform
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whose a_3 is a root of x^4 -20* x^2 + 98 , please send me the a_5,
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a_7,...., a_17 of this form.
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Regarding level 8192, the form whose a_3 I sent you in the former mail is
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new and the degree 8 polynomial whose root is the a_3 is a simple factor
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of the new part of the T_3 of level 8192, and this a_3 generates the whole
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number field attached to this newform. With this information, theorem
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3.64 of Shimura's book and the value of a_3 it follows that this form has
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an inner twist given
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by the mod 4 character chi . (I think there is no need to look at the
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oldforms because if f is new of level 8192 and f*chi is old, then
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(f*chi)*chi=f would be old, thus giving a contradiction. So this form f
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has an inner twist.
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Thanks again,
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Luis
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From [email protected] Wed Dec 2 05:25:54 1998
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Date: Wed, 2 Dec 1998 14:29:39 +0100 (MET)
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From: Luis Victor Dieulefait <[email protected]>
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Reply-To: Luis Victor Dieulefait <[email protected]>
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To: [email protected]
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Subject: Re: newforms
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William,
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The relation between the number field corresponding to f := Q_f
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and the subfield fixed by the action of the inner twists:= F_f ,
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gets into the picture of the abelian variety A_f. More precisely:
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THe endomorphism algebra of A_f (over Q) is a central simple algebra over
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F_f which contains Q_f as a maximal conmutative subfield. Its degree over
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Q is [Q_f : Q]*[Q_f : F_f] .
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This is proved in : K. Ribet: "Twists of Modular Forms and Endomorphisms
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of Abelian VArieties", Math. Ann. 253, 43-62 (1980)
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Regarding the coefficient a_5 and a_7 of the form of level 8192 that are
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"probably" 0, I think that, for example for the a_5, the upper bound for
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the absolute value of it, 2*sqrt(5), proves that it will be 0 when you can
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show this mod some primes whose product is greater that (2*sqrt(5))^8 =
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160000. The two primes you took are not enough, but if you do the same for
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one more prime greater than 25 then it will be enough to get a proof that
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a_5 = 0. If you think this argument is correct, please do this computation
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with such a prime, say 29.
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Regarding modular forms of very big level (more than 30000) I will like to
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discuss with you some other time the possibility of doing computations, at
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least mod some primes, maybe some previous information I have about the
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number field Q_f in some cases can make the computations easier....
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BEst Regards,
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Luis
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PD: During december when you mail me, please send a copy to
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[email protected] , I will spend some weeks there.
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From [email protected] Wed Dec 2 07:16:38 1998
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Date: Wed, 2 Dec 1998 16:12:00 +0100 (MET)
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From: Luis Victor Dieulefait <[email protected]>
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To: [email protected]
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Subject: Re: newforms
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William,
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As a matter of fact, I can prove already that a_5=0 for the newform of
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level 8192, because we can use the fact that this form has an inner
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twist given by the mod 4 character to see that a_5 belongs to F_f, the
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subfield of Q_f fixed by the automorphism giving the inner twist.
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The field F_f has degree 4, so that taking norms we get from the fact
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that the abs. value of a_5 is bounded by 2*sqrt(5) that its norm is
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smaller than 400. Then with the minimal polynomial mod p you computed
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for those two primes it's enough to conclude that the norm of a_5, and
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the a_5, is 0, becouse the product of the 2 primes is bigger that 400.
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For the a_7, more computation would be needed (the twist is no longer
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useful). Are you convinced with this proof that a_5=0 ?
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Regards,
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Luis
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From [email protected] Fri Dec 4 05:59:45 1998
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Date: Fri, 4 Dec 1998 14:48:03 +0100 (MET)
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From: Luis Victor Dieulefait <[email protected]>
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To: [email protected]
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Subject: Re: newforms
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William,
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I have just realize that in the 2 examples of newforms of level 2048
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and 8192, we can prove (using theorem 3.64 of Shimura's book) that the 2
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characters: chi: (Z/4Z)* -->{+-1} and
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psi: (Z/8Z)* -->{+-1} give the inner twist: chi*f = psi*f = f^(gamma),
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where gamma is the "real conjugation": a_3 --> - a_3 , of Q_f. (this
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automorphism has F_f = Q((a_3)^2) as its fixed field)
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From this it follows that (in the 2 examples):
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a_p = 0 for every p congruent to 5 or 7 (mod 8)
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(Proof: take p congruent to 5 mod 8. If a_p is not 0, chi(p)=1 implies
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that a_p belongs to F_f; psi(p)= -1 implies that a_p doesn't belong to
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F_f. Then a_p =0 )
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In particular, in the example of level 8192, this proves that a_3 = a_5 =
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a_13 = 0 , as suggested by the computations you have done.
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Luis
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From [email protected] Tue Jan 19 07:27:02 1999
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Date: Tue, 19 Jan 1999 16:24:57 +0100 (MET)
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From: Luis Victor Dieulefait <[email protected]>
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To: [email protected]
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Subject: Re: newforms
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William,
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So it seems that the examples of newforms we were working on have CM.
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I think that the argument of Shimura I thought was contradictory with this
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fact can only be used with odd primes in the level, but in the case of the
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power of 2 level I can't say nothing about CM (I mean "a priori").
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I want examples similar to the ones we were working on , but without
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CM. I have a few candidates to check.
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Can you send me the fourier expansion of the level 1024 newforms
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corresponding to the following factor of the characteristic polynomial of
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the T_3: (x^4-8*x^2+8)^2
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Maybe we are lucky in this example and the forms don't have CM. The fact
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that the factor appears with mult. 2 makes posible that the form has one
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(and only one!) twist. With a few more coefficients I think this can be
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checked.
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If this example doesn't work, a good place to look at is the space
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of newforms with level 3*1024 = 3072, where CM can not occur.
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In the case of level 8192, remember that there was another newform whose
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corresp. number field was of degree 8 (the a_3 is in fact the square root
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of an element belonging to the maximal real subfiel of the cyclotomic
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field of the 16-th roots of unity). Computing its a_p modulo some random
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prime, does this form also seem to have CM ??
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By the way, your site in the internet is unreachable these days, do you
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have any idea of what can be the problem?
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Thank you a lot! Best wishes,
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Luis
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PS: If you can,please send me the results of the computations this week,
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because I have a congress starting this sunday and it will be great if I
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have this results before leaving.
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From [email protected] Tue Jan 19 08:08:46 1999
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Date: Tue, 19 Jan 1999 17:07:49 +0100 (MET)
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From: Luis Victor Dieulefait <[email protected]>
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To: William Arthur Stein <[email protected]>
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Subject: Re: newforms
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William,
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In case it helps you, here's the polynomial giving the a_3 of the level
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8192 newform whose coefficients I've asked you to compute:
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578 -624*x^2 + 196*x^4 - 24*x^6 + x^8
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I'm reaching your site (with the new address you gave me) without
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problems. Thanks again, best regards,
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Luis
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