3 A one line explanation on why the fiber product is biration to X(N)

Let $N$ be a prime number. Let J(N) $J(N)$ 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)?$J(11)$?

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.

2 Fixed the minor mistakes pointed out by BConrad

Let $N$ be a prime number. Let J(N) be the jacobian of X(N). $X_\mu(N)$, 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(N) $X_\mu(N)$ is birationally isomorphic as a curve to the fibre product $X_0(N^2) \times_{X_0(N)} X_1(N)$. 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.

1

# geometric decomposition of J(11),

Let $N$ be a prime number. Let J(N) be the jacobian of X(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(N) is isomorphic as a curve to the fibre product $X_0(N^2) \times_{X_0(N)} X_1(N)$. 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.