Question

Let $N$ be a prime number. Let $J(N)$ be the jacobian of $X_\mu(N)$, the moduli space of elliptic curves with $E[N]$ symplectically isomorphic to $Z/NZ \times \mu_N$. Over complex numbers we get that J(N) is isogeneous to product of bunch of irreducible Abelian varieties. Is there a way of describing these Abelian varieties using $J_1(M)$ and $J_0(M)$? Specifically, what can we say about the decomposition of $J(11)$?

Note that $X_\mu(N)$ is birationally isomorphic as a curve to the fibre product $X_0(N^2) \times_{X_0(N)} X_1(N)$. (This is because $\Gamma(N)$ is conjugate to $\Gamma_0(N^2) \cap \Gamma_1(N)$, and the group generated by $\Gamma_0(N^2)$ and $\Gamma_1(N)$ is $\Gamma_0(N)$.) Therefore, we have $J_1(N)$ and $J_0(N^2)$ are both some of the factors in $J(N)$. In fact, we know that $J(7)$ is three copies of $J_0(49)$. For N=11, the above fibre product to $X_0(121)$ is an unramified covering. If I was going to make a guess on what $J(11)$ going to decompose as, I would guess that it is five copies of $J_0^{new}(121)$ and six copies of $J_1(11)$. Is that reasonable? Is there a geometric way of arguing this?

Also, I'm guessing that the question about $SL_2(F_N)$ decompoposition of space of cusprforms is related to this, and Jared Weienstein's thesis will come into play here, but I'm not sure how.

Answer 1

The decomposition of $J(11)$ was known (at least over $\mathbf{C}$) to Hecke. It turns out that the Jacobian of the compactification of $\Gamma(11) \backslash \mathfrak{h}$ is isogenous to a product of 26 elliptic curves. All this is very well explained in the following article :

MR0463118 (57 #3079) Ligozat, Gérard . Courbes modulaires de niveau $11$. (French) Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), pp. 149--237. Lecture Notes in Math., Vol. 601, Springer, Berlin, 1977. http://www.springerlink.com/index/6722kj1764m8g50t.pdf

The idea is to look at the natural representation of the group $\mathrm{PSL}_2(\mathbf{F}_p)$ on the space of cusp forms $S_2(\Gamma(p))$. So, you're right that there is a geometric interpretation.

If I remember well, there are, among the factors of $J(11)$, elliptic curves of conductor $121$ which are $11$-isogenous to itself. These can be seen as rational points of the modular curve $X_0(11)$ which are not cusps (there are three such points).

EDIT : I remembered somewhat incorrectly. The three non-cuspidal points of $X_0(11)(\mathbf{Q})$ correspond to the elliptic curves 121B1, 121C1 and 121C2. The subgroups of order $11$ of these curves are described as follows : the elliptic curve 121B1 has CM by $\mathbf{Z}[\frac{1+i\sqrt{11}}{2}]$, so it is $11$-isogenous to itself, whereas 121C1 and 121C2 are $11$-isogenous to each other. Using the notations of Cremona's tables, the Jacobian of the compactification of $\Gamma(11)\backslash \mathfrak{h}$ is then isogenous to $(11A)^{11} \times (121B)^5 \times (121C)^{10}$.

Answer 2

Ernst Kani was very interested in this and related questions around 2000. I remember implementing an algorithm for him in around 2000 when I visited Essen to compute a basis of $S_2(\Gamma(p))$ in terms of $\Gamma_1(p^2)$. I'm sure Kani knows the decomposition of $J(N)$ for small $N$, since I vaguely remember talking about it with him, but I didn't explicitly see it in a cursory glance through the papers at http://www.mast.queensu.ca/~kani/. You may want to look at the papers up there from around 2000, since many mention X(11) explicitly. You might also just email Kani.