Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:24:39Z http://mathoverflow.net/feeds/question/77449 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77449/elliptic-curves-over-mathbfq-with-isogenous-mod-ell-reductions-for-sever Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$ Tommaso Centeleghe 2011-10-07T13:31:55Z 2011-10-07T14:54:59Z <p>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!).</p> <p>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 $\ell\neq p$.</p> <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.</p> <p>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 $7$, $31$, $41$, $101$, $137$, $173$, $179$,..$39769$).</p> <p>$Q1$: 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?</p> <p>$Q2$: 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 $Q$?</p> <p>$Q3$: Can we say why we do not see the above phenomenon in level one?</p> http://mathoverflow.net/questions/77449/elliptic-curves-over-mathbfq-with-isogenous-mod-ell-reductions-for-sever/77457#77457 Answer by JSE for Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$ JSE 2011-10-07T14:54:59Z 2011-10-07T14:54:59Z <p>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.</p> <p>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.</p>