Sharedwww / people / dieulefait / emailOpen in CoCalc
Author: William A. Stein
1From [email protected]  Mon Nov 30 06:37:48 1998
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12Date: Mon, 30 Nov 1998 15:38:25 +0100 (MET)
13From: Luis Victor Dieulefait <[email protected]>
14To: [email protected]
15Subject: newforms
16Message-ID: <[email protected]>
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22William
23     Hello! I'm Luis from the University of Barcelona, we met at the
24ABC workshop (Arizona). Kevin Buzzard told me you may help me with
25some computations with newforms of high levels. I have visited your
26site on the web with tables of hecke polynomials, but I need values for
27higher levels, and I don't have linux (for the moment).
28     The example I have is a newform of level 8192. The number field
29generated by its coefficients is given by a root of:
30x^8 - 24 x^6 + 164 x^4 -240 x^2 +2 , which is  a factor of the
31T_3 of this level (please correct if there's some mistake)
32(sorry, I forgot to mention that we are in weight 2 and
33trivial nebentypus)
34This form seems to have an inner twist, and it also seems from the
35discussion between you and Kevin that this can be  proved (I have to take
36a look at Shimura's book...)
37If your algorithm works well for this level, please send me the a_5 and
38a_7 of this newform.
39
40One more thing: I've found in your table the T_2, T_3 , and T_5 for level
412048. Can you send me the T_7 and T_11 for this same level?
42
43Up to what level can you compute these Hecke polynomials?
4430000 would be too much?
45
46Thank you!
47                   Luis
48
49From [email protected]  Tue Dec  1 05:54:17 1998
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57Date: Tue, 1 Dec 1998 14:58:39 +0100 (MET)
58From: Luis Victor Dieulefait <[email protected]>
59To: [email protected]
60Subject: Re: newforms
62Message-ID: <[email protected]>
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67
68William
69  Thank you very much for your help! I will see wether or not I can manage
70to obtain the information I need from the polynomials reduced mod. some
71primes, but I guess that combining these reduced polynomials with the
72known bounds something can be done.
73  I still need some more details. For level 2048 I haven't been able to
74find out which are the eigenvalues I need. I'm interested in the newform
75whose a_3 is a root of x^4 -20* x^2 + 98 , please send me the a_5,
76a_7,...., a_17  of this form.
77
78Regarding level 8192, the form whose a_3 I sent you in the former mail is
79new and the degree 8 polynomial whose root is the a_3 is a simple factor
80of the new part of the T_3 of level 8192, and this a_3 generates the whole
81number field attached to this newform. With this information, theorem
823.64 of Shimura's book and the value of a_3 it follows that this form has
83an inner twist given
84by the mod 4 character  chi . (I think there is no need to look at the
85oldforms because if f is new of level 8192 and f*chi is old, then
86(f*chi)*chi=f would be old, thus giving a contradiction. So this form f
87has an inner twist.
88
89Thanks again,
90                           Luis
91
92From [email protected]  Wed Dec  2 05:25:54 1998
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100Date: Wed, 2 Dec 1998 14:29:39 +0100 (MET)
101From: Luis Victor Dieulefait <[email protected]>
102Reply-To: Luis Victor Dieulefait <[email protected]>
103To: [email protected]
104Subject: Re: newforms
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112William,
113       The relation between the number field corresponding to f := Q_f
114and the subfield fixed by the action of the inner twists:= F_f ,
115 gets into the picture of the abelian variety A_f. More precisely:
116THe endomorphism algebra of A_f (over Q) is a central simple algebra over
117F_f which contains Q_f as a maximal conmutative subfield. Its degree over
118Q is [Q_f : Q]*[Q_f : F_f] .
119
120This is proved in : K. Ribet: "Twists of Modular Forms and Endomorphisms
121of Abelian VArieties", Math. Ann. 253, 43-62 (1980)
122
123Regarding the coefficient a_5 and a_7 of the form of level 8192 that are
124"probably" 0, I think that, for example for the a_5, the upper bound for
125the absolute value of it, 2*sqrt(5), proves that it will be 0 when you can
126show this mod some primes whose product is greater that (2*sqrt(5))^8 =
127160000. The two primes you took are not enough, but if you do the same for
128one more prime greater than 25 then it will be enough to get a proof that
129a_5 = 0. If you think this argument is correct, please do this computation
130with such a prime, say 29.
131
132Regarding modular forms of very big level (more than 30000) I will like to
133discuss with you some other time the possibility of doing computations, at
134least mod some primes, maybe some previous information I have about the
135number field Q_f in some cases can make the computations easier....
136
137BEst Regards,
138                          Luis
139PD: During december when you mail me, please send a copy to
140[email protected]r.edu.ar , I will spend some weeks there.
141
142From [email protected]  Wed Dec  2 07:16:38 1998
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150Date: Wed, 2 Dec 1998 16:12:00 +0100 (MET)
151From: Luis Victor Dieulefait <[email protected]>
152To: [email protected]
153Subject: Re: newforms
155Message-ID: <[email protected]>
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161William,
162   As a matter of fact, I can prove already that a_5=0 for the newform of
163level 8192, because we can use the fact that this form has an inner
164twist given by the mod 4 character to see that a_5 belongs to F_f, the
165subfield of Q_f fixed by the automorphism giving the inner twist.
166The field F_f has degree 4, so that taking norms we get from the fact
167that the abs. value of a_5 is bounded by 2*sqrt(5) that its norm is
168smaller than 400. Then with the minimal polynomial mod p you computed
169for those two primes it's enough to conclude that the norm of a_5, and
170the a_5, is 0, becouse the product of the 2 primes is bigger that 400.
171For the a_7, more computation would be needed (the twist is no longer
172useful). Are you convinced with this proof that a_5=0 ?
173Regards,
174         Luis
175
176
177From [email protected]  Fri Dec  4 05:59:45 1998
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185Date: Fri, 4 Dec 1998 14:48:03 +0100 (MET)
186From: Luis Victor Dieulefait <[email protected]>
187To: [email protected]
188Subject: Re: newforms
190Message-ID: <[email protected]>
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196William,
197   I have just realize that in the 2 examples of newforms of level 2048
198and 8192, we can prove (using theorem 3.64 of Shimura's book) that the 2
199characters: chi: (Z/4Z)* -->{+-1} and
200psi: (Z/8Z)* -->{+-1} give the inner twist: chi*f = psi*f = f^(gamma),
201where gamma is the "real conjugation": a_3 --> - a_3 , of Q_f. (this
202automorphism has F_f = Q((a_3)^2) as its fixed field)
203  From this it follows that (in the 2 examples):
204                  a_p = 0 for every p congruent to 5 or 7 (mod 8)
205
206(Proof: take p congruent to 5 mod 8. If a_p is not 0, chi(p)=1 implies
207that a_p belongs to F_f;  psi(p)= -1 implies that a_p doesn't belong to
208F_f. Then  a_p =0  )
209
210In particular, in the example of level 8192, this proves that a_3 = a_5 =
211a_13 = 0 , as suggested by the computations you have done.
212
213           Luis
214
215From [email protected] Tue Jan 19 07:27:02 1999
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225Date: Tue, 19 Jan 1999 16:24:57 +0100 (MET)
226From: Luis Victor Dieulefait <[email protected]>
227To: [email protected]
228Subject: Re: newforms
230Message-ID: <[email protected]>
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235
236William,
237    So it seems that the examples of newforms we were working on have CM.
238I think that the argument of Shimura I thought was contradictory with this
239fact can only be used with odd primes in the level, but in the case of the
240power of 2 level I can't say nothing about CM (I mean "a priori").
241     I want examples similar to the ones we were working on , but without
242CM. I have a few candidates to check.
243     Can you send me the fourier expansion of the level 1024 newforms
244corresponding to the following factor of the characteristic polynomial of
245the T_3:   (x^4-8*x^2+8)^2
246Maybe we are lucky in this example and the forms don't have CM. The fact
247that the factor appears with mult. 2 makes posible that the form has one
248(and only one!) twist. With a few more coefficients I think this can be
249checked.
250     If this example doesn't work, a good place to look at is the space
251of newforms with level 3*1024 = 3072, where CM can not occur.
252
253In the case of level 8192, remember that there was another newform whose
254corresp. number field was of degree 8 (the a_3 is in fact the square root
255of an element belonging to the maximal real subfiel of the cyclotomic
256field of the 16-th roots of unity). Computing its a_p modulo some random
257prime, does this form also seem to have CM ??
258
259 By the way, your site in the internet is unreachable these days, do you
260have any idea of what can be the problem?
261Thank you a lot! Best wishes,
262                                     Luis
263PS: If you can,please send me the results of the computations this week,
264because I have a congress starting this sunday and it will be great if I
265have this results before leaving.
266
267
268From [email protected] Tue Jan 19 08:08:46 1999
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275Date: Tue, 19 Jan 1999 17:07:49 +0100 (MET)
276From: Luis Victor Dieulefait <[email protected]>
277To: William Arthur Stein <[email protected]>
278Subject: Re: newforms
280Message-ID: <[email protected]>
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286William,
287
288In case it helps you, here's the polynomial giving the a_3  of the level
2898192 newform whose coefficients I've asked you to compute:
290 578 -624*x^2 + 196*x^4 - 24*x^6 + x^8
291I'm reaching your site (with the new address you gave me) without
292problems. Thanks again, best regards,
293                                           Luis
294
295
296