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 were going to make a guess on what $J(11)$ going to decompose as, I would guess that it is five copies of $J\_0^\text{new}(121)$ 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
 [$\operatorname{SL}\_2(F\_N)$ decompoposition of space of cuspforms](https://mathoverflow.net/questions/4763/sl2-z-n-decomposition-of-space-of-cusp-forms-for-gamman)  is related to this, and Jared Weienstein's thesis will come into play here, but I'm not sure how.



Geometric decomposition of J(11)
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.


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 $(11A)^{11} \times (121B)^5 \times (121C)^{10}$.


