# Geometric decomposition of J(11)

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 is related to this, and Jared Weienstein's thesis will come into play here, but I'm not sure how.

• The curve $X(N)$ in its standard definition as a moduli space is not geometrically connected over $\mathbf{Q}$). You must mean to use the variant $X_{\mu}(N)$ classifying "twisted" full level-$N$ structures of type $\mathbf{Z}/(N) \times \mu_N$ as symplectic spaces? And are you interested in the simple isogeny factors over $\mathbf{Q}$, or working geometrically (which might come to the same thing, but only after the fact)? Also, the "fiber product" description looks non-smooth near the cusps. It is at best birational, no? Anyway, representation theory should be better than geometry here. Sep 28, 2010 at 1:50
• You are definitely right. I meant X(N) to be the twisted full level N structure, and I changed the wording to reflect that. On the other hand, I think the fiber product is actually smooth at the cusps, since the covering of X_0(N) by X_1(N) is unramified at the cusps. I think for primes N larger than 5, that is an actual isomorphism. I also agree with you that representation theory is probably better than geometry for this type of problems. However, I was trying to figure out what's happening geometrically, and there seems to be something there. Sep 29, 2010 at 19:39
• Also, I'm mostly interested in this geometrically. So, I want to know the simple isogeny factors over C. Sep 29, 2010 at 21:10
• Could you give some details on how you get $X_{\mu}(N)$ as a fibered product ? Sep 30, 2010 at 1:32
• Thanks for the details on the fibered product. I agree that $J_1(N)$ is a factor of $J(N)$, but I am a little bit confused about $J_0(N^2)$ because for $N=11$ not all elliptic curves of conductor $121$ appear in $J(11)$. Maybe this is because we are looking things over $\mathbf{C}$ and not over $\mathbf{Q}$ ? Oct 2, 2010 at 15:17

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

• The link to springerlink.com is broken. I'm also unable to find any snapshot saved on the Wayback Machine. May 3 at 19:06
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.