Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $A_f$ be the abelian variety over $\mathbf{Q}$ arising as a $\mathbf{Q}$-simple factor of the Jacobian $J_0(N)$ of the modular curve associated to a normalized newform $f$ of weight $2$ on the congruence subgroup $\Gamma_0(N)$. There is a piece $\mathbf{T}_f$ of the Hecke ring acting on $A_f$. Such a ring is an order in the number field generated by the Fourier coefficients of $f$, whose degree equals the dimension of $A$. It is a result of Ribet (I think) that the natural map $\mathbf{T}_f\otimes\mathbf{Q}\rightarrow\textrm{End}(A)\otimes\mathbf{Q}$ is an isomorphism, where $\textrm{End}(A)$ is the ring of $\mathbf{Q}$-enndomorphisms of $A$ (I couldn't attach the index to End..).

I have two questions, please:

1) Can we find an abelian variety $A_f'$ over $\mathbf{Q}$, which is $\mathbf{Q}$-isogenous to $A_f$, and such that $\textrm{End}(A_f')$ is the maximal order of $\textrm{End}(A_f')\otimes\mathbf{Q}=\textrm{End}(A_f)\otimes\mathbf{Q}$? (Notice that in the one dimensional case $\textrm{End}(A_f)$ is already maximal, being it the ring of integers)

2) Does the fact that $A_f$ has such a large endomorphism ring imply that the mod $p$ reduction of $A_f$ be simple over the prime field with $p$ elements? Here $p$ is a prime of good reduction for $A_f$.

Thanks!

share|improve this question
1  
If your $A_f$ is absolutely simple, then this paper of Zywina might help with question 2: mast.queensu.ca/~zywina/papers/Splitting.pdf But I suspect that $A_f$ are usually not absolutely simple. –  Barinder Banwait Oct 10 '12 at 0:04
    
In the case of $A_f$, I think it's known that if $A_f$ is absolutely simple, then $A_f$ mod $p$ is simple for a set of primes $p$ of density 1. See Murty-Patankar's conjecture in Zywina's paper. –  François Brunault Oct 10 '12 at 9:30
    
@François: That would be true provided the Mumford-Tate conjecture was known for $A_f$, and the field $K_A^{conn}$ in Zywina's paper was $\mathbb{Q}$. I'm not sure about either of these claims. –  Barinder Banwait Oct 10 '12 at 11:31
1  
There is a canonical $A_f$ which is defined as the quotient $J_0(N)/I_f J_0(N)$ where $I_f$ is the kernel of the map $\mathbf{T} \to K_f$ sending $T_n$ to $a_n(f)$. This is called the optimal quotient. Using Magma, you can determine the order $\mathbf{T}_f = \mathbf{Z}[a_n(f),n \geq 1]$ inside $K_f$, and you can also determine the index of $\mathbf{T}_f$ inside $\operatorname{End}(A_f)$ (for this $A_f$), so you can deduce the index of the order $\operatorname{End}(A_f)$. It is not always the maximal order, the first counterexample appears at $N=69$ I think. (...) –  François Brunault Oct 21 '12 at 19:54
1  
(...) So optimal quotients are not always optimal from the point of view of the endomorphism ring. –  François Brunault Oct 21 '12 at 19:54
show 4 more comments

1 Answer

up vote 4 down vote accepted

I found a reference for the first question in the following PhD thesis :

J. Wilson, Curves of genus 2 with real multiplication by a square root of 5

The answer is yes. This is a consequence of the following general fact about abelian varieties (see Prop 2.5.4 in the thesis).

Let $A$ be an abelian variety over a field $k$. Let $R$ be an order in a number field $F$. Assume that $R$ embeds into $\operatorname{End}_k(A)$. Then there exists an abelian variety $B/k$ which is $k$-isogenous to $A$ and such that $\mathcal{O}_F$ embeds into $\operatorname{End}_k(B)$.

The idea is to take $B=A/G$ with $G=(n\mathcal{O}_F) A[n^2]$, where $n$ is the index of $R$ in $\mathcal{O}_F$.

The thesis also contains interesting examples of varieties $A_f$ with Hecke field $K_f=\mathbf{Q}(\sqrt{5})$. These are natural examples to try for Question 2 (although I have no idea how to compute the reduction of $A_f$ mod $p$).

EDIT. The answer to Question 2 is negative in general. There are newforms $f$ of weight $2$ on $\Gamma_0(N)$ such that $A_f$ splits over $\overline{\mathbf{Q}}$. This happens for example when $f$ has extra-twist. The first example appears at level $N=63$, see Table 1 p. 13 in

MR1933828 (2003i:11078) González-Jiménez, Enrique ; González, Josep. Modular curves of genus 2. Math. Comp. 72 (2003), no. 241, 397--418 (electronic).

Assume $A_f \sim E_1 \times E_2$ where everything is defined over some number field $K$. If $p$ is a prime of good reduction for $A_f$ which splits totally in $K$, then $A_f$ mod $p$ is $\mathbf{F}_p$-isogenous to a product of elliptic curves over $\mathbf{F}_p$, so it is not simple.

share|improve this answer
    
Thanks Francois, this is helpful! –  Tommaso Centeleghe Oct 21 '12 at 18:52
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.