Timeline for Fourier coefficient of a modular form
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2010 at 4:05 | vote | accept | schur | ||
| Oct 21, 2010 at 6:11 | answer | added | user631 | timeline score: 9 | |
| Oct 19, 2010 at 15:44 | comment | added | Tony Scholl | @Jared: good point. I had in mind trying to find directly a Hecke character with the right local factor at $p$ - which would give an AV over $\mathbb{Q}$ with real multiplication by some tot.real field containing $F^+$. But I haven't had time to give it serious thought and maybe there is a trivial obstruction. | |
| Oct 19, 2010 at 14:29 | comment | added | Jared Weinstein | @Tony: I tried thinking CM forms. Let's say $a_p$ is totally real, let $\pi$ be a root of $X^2+a_pX+p$, let $F^+=\mathbf{Q}(a_p)$ and let $F=F^+(\pi)$. Then $F/F^+$ is CM. Now $\pi$ is a Weil $p$-integer: the proof of Honda-Tate supplies us with an abelian variety $A$ over $\mathbf{F}_p$ with CM by $F$ on which Frob is $\pi$ (up to a root of unity). The problem is lifting $A$ to an abelian variety over $\mathbf{Q}$. (It lifts to some undetermined number field; restricting scalars to $\mathbf{Q}$ results in something too big.) I don't see any way around this. | |
| Oct 19, 2010 at 11:33 | comment | added | François Brunault | @Olivier: yes, I think so. In fact if $f$ is a newform of level $N$ and Nebentypus character $\psi$ modulo $N$ then $\overline{f} = f \otimes \overline{\psi}$ so that $\overline{a_n(f)} = a_n(f) \overline{\psi}(n)$ for any $n$ prime to $N$. In particular we have the condition that $x/\overline{x}$ should be a root of unity. I suggest first looking at the case of a totally real algebraic integer. | |
| Oct 19, 2010 at 10:37 | comment | added | Olivier | And if it is not totally real, by the stability of the Hecke algebra under the Rosatti involution and its positivity, the field generated by the coefficients of $f$ is a CM field, isn't it? | |
| Oct 19, 2010 at 10:25 | comment | added | François Brunault | There might also be some restriction on the number field generated by $x$. For example, if $f$ is a weight $k$ newform on $\Gamma_0(N)$, then the number field gen'd by the coeffs of $f$ is totally real (this is because of the Atkin-Lehner involution). | |
| Oct 19, 2010 at 10:03 | comment | added | François Brunault | In the case $x$ is an integer, I think it is known that there exists an elliptic curve $E$ over $\mathbf{F}_p$ with $a_p(E)=x$. So in this case the answer to Ian's question is yes. | |
| Oct 19, 2010 at 9:51 | history | edited | Tony Scholl |
edited tags
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| Oct 19, 2010 at 7:46 | comment | added | Tony Scholl | Have you thought about finding a CM form with $a_p=x$? | |
| Oct 19, 2010 at 5:10 | comment | added | Jared Weinstein | I think you have to impose the condition that all of the Q-conjugates of $a_p$ also have to lie inside the Hasse-Weil bound. (In this context I'd call it the Ramanujan bound, but I know what you mean.) For if $f$ is a normalized newform, then so is $\sigma(f)$, where $\sigma$ is any automorphism of $\overline{\mathbf{Q}}$. | |
| Oct 19, 2010 at 2:26 | comment | added | schur | Dear Ben and Alex, that's right, I was being imprecise. I meant normalized newform. | |
| Oct 19, 2010 at 2:24 | history | edited | schur | CC BY-SA 2.5 |
Normalized newform is what I meant.
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| Oct 19, 2010 at 2:13 | comment | added | user1073 | Alex - Even if the modular form had to be normalized, the question would still follow more or less trivially from the theory of newforms (in fact, you could let x be any complex number and get the same result). Perhaps he requires the modular form to be a normalized newform? | |
| Oct 19, 2010 at 1:34 | comment | added | Alex B. | I assume that Ian means a normalised modular form. | |
| Oct 19, 2010 at 1:15 | comment | added | user1073 | Because if it is, I'm afraid that I don't really understand the question; why couldn't I just take any modular form whose p-th fourier coefficient is nonzero and multiply it by a suitable scalar? | |
| Oct 19, 2010 at 1:10 | comment | added | user1073 | Is a_p supposed to be the p-th fourier coefficient of your modular form? | |
| Oct 19, 2010 at 0:54 | history | asked | schur | CC BY-SA 2.5 |