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Characteristic polynomials of Hecke operators $T_\ell$, with $\ell$ prime, acting on cusp forms $S_k$ of level one and weight $k$ "appear to be" squarefree (even irreducible!).

This can be interpreted as follows: if $p$ is any prime, then the $p$-adic Galois representations $\rho_i$, where $1\leq i\leq \dim(S_k)$, attached to eigenforms in $S_k$ "appear to be" pairwise non isomorphic ${\it locally}$ at primes $\ell\neq p$.

This is completely false for other levels. For example in the two dimensional space $S_2(\Gamma_0(37))$, I learn from Magma and Cremona's tables that there are exactly two normalized eigenforms, $f_1$ and $f_2$, with rational coefficients, corresponding to two elliptic curves $E_1$ and $E_2$ defined over $\mathbf{Q}$, of conductor $37$, and uniquely determined up to $Q$-isogeny.

Looking at some Hecke operators on this space, one easily finds examples of $T_\ell$ acting diagonally on $S_2(\Gamma_0(37))$, i.e., examples of primes $\ell\neq 37$ for which the two elliptic curves have $p$-adic Tate modules isomorphic as local Galois modules at $\ell$ (some of the $\ell$'s for which this happens are $7$, $31$, $41$, $101$, $137$, $173$, $179$,..$39769$).

$Q1$: Is it reasonable to suspect that $E_1$ and $E_2$ become isogenous over an extension $F$ of $Q$? If this were the case, then one should see the phenomenon described above for primes $\ell$ that are split in $F$, right?

$Q2$: On the other hand, given an elliptic curve $E$ over $\mathbf{Q}$, what are the known ways to construct more elliptic curves $A$, defined over $\mathbf{Q}$, possibly of the same conductor as $E$, which are not $\mathbf{Q}$-isogenous to $E$ but such that they become so over a non-trivial extension of $Q$?

$Q3$: Can we say why we do not see the above phenomenon in level one?

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Interesting. Can you tell me more about the phenomenon in level 1? Did you observe it yourself? is tab a well-known observation? Do you have references? –  Joël Oct 7 '11 at 14:07
Re. Q2: Twisting is the only way to do this. Use Faltings's proof of the Tate conjecture plus Chebotarev: if $E_1,E_2$ are defined over $\mathbf{Q}$ and $Hom_{G_K}(\rho_1,\rho_2)$ is nonempty for some $K/\mathbf{Q}$ of finite degree, then $\rho_1$ and $\rho_2$ are twist-equivalent by Frobenius reciprocity. –  David Hansen Oct 7 '11 at 14:55
I think that people do not have any example where $T_p$ acting on level ONE and weight k has a reducible characteristic polynomial (let alone its squarefreeness). I've always heard of computations aimed to check this, but never did it myself. Google points me towards this paper: www.math.clemson.edu/~kevja/PAPERS/FarmerJames2000.pdf. I am not sure about any kind of theoretical evidence. Maeda's conjecture, saying that in level one there is only one galois orbit of forms, addresses a related issue. –  Tommaso Centeleghe Oct 7 '11 at 15:02
@David: thanks, that's helpful. The twist is a quadratic one, right? –  Tommaso Centeleghe Oct 7 '11 at 15:06
@Tommaso: Yup! –  David Hansen Oct 7 '11 at 15:07
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1 Answer

up vote 4 down vote accepted

I imagine this is a weight issue, not a level issue. Let f and g be the two weight-2 newforms in $S_2(\Gamma_0(37))$. Then a "random" coefficient of f-g is going to have size about $p^{1/2}$, so there should be about $X^{1/2-\epsilon}$ primes $p$ less than $X$ such that $a_p(f) = a_p(g)$, just by chance. When the weight is larger, the Fourier coefficients are bigger, and it is much more surprising to see coincidences of Fourier coefficients.

Try other weight 2 cases, and try some higher weight cases in level 37, and I'll bet you'll see that your phenomenon happens in weight 2 and not in weight 4.

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Thanks, this clearly answers my questions. –  Tommaso Centeleghe Oct 7 '11 at 15:13
By the way, this is somewhat badly written: I should have written "E_1 and E_2" since I'm really using the fact that the coefficients of f and g are rational integers. –  JSE Oct 7 '11 at 15:21
Ok, thanks. I guess your argument can be adapted to the case where there is no form with rational coefficients in $S_2(\gamma_0(N))$. At least one can observe a similar phenomenon as that above even for values of $N$ (e.s $23$) for which there is not elliptic curve of that conductor. –  Tommaso Centeleghe Oct 7 '11 at 15:30
Try it for a case where the space has dimension higher than 2 and the forms are all Galois conjugate! –  JSE Oct 7 '11 at 16:12
Nice answer, but does it mean that you guys think that the observation that in level 1 it never happens that two eigenforms have the same T_l eigenvalues for any l is false? But then it would be nice to have a counter-example. –  Joël Oct 7 '11 at 16:22
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