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Endomorphism ring of $ \bar{A}/\mathbb{F}_p$

Let $ A=A_f$ be a modular abelian variety over $ \mathbb{Q}$ associated to a newform in $ S_2(\Gamma_0(N))$ . Let $ p$ be a prime of good reduction for $ A$ (so $ (N,p) = 1$ ). Let $ \bar{A} = A_{\mathbb{F}_p}$ be the reduction of the $ A$ modulo $ p$ , which is an abelian variety over $ \mathbb{F}_p$ .

Problem 8.1.1   Compute the endomorphism ring $ {\mathrm{End}}(\bar{A}/\mathbb{F}_p)$ .

The endomorphism ring of $ \bar{A}/\mathbb{F}_p$ contains $ \mathbb{T}[\pi_p] =
\mathbb{Z}[\{\alpha_n\}][\pi_p]$ , where $ \alpha_n$ is the $ n$ -th coefficient of the cusp form $ f$ of $ A$ , and the Frobenius endomorphism $ \pi_p$ satisfies $ \pi_p^2 - \alpha_p\pi_p + p = 0. $ If $ \bar{A}$ is ordinary (i.e. has $ p$ -rank $ g = \dim(A)$ ), then

$\displaystyle \mathbb{T}[\pi_p] \subseteq {\mathrm{End}}(\bar{A}) \subseteq \mathcal{O}_K
$

where $ K = \mathbb{T}[\pi_p] \otimes\mathbb{Q}$ and $ \mathcal{O}_K$ is its maximal order. These reductions modulo $ p$ are CM abelian varieties, but in general only the real subring $ \mathbb{T}$ generated by the trace terms lift back to the modular abelian variety over $ \mathbb{Q}$ .

Note that the invariant $ {\mathrm{End}}(\bar{A})$ is an invariant of the isomorphism class, but not the isogeny class, of $ A$ . For instance the isogeny class of elliptic curves of conductor 57 denoted 57C by Cremona, consists of two curves:

\begin{displaymath}
\begin{array}{ll}
E_1: y^2 + y = x^3 + x^2 + 20x - 32,\\
E_2: y^2 + y = x^3 + x^2 - 4390x - 113432,
\end{array}\end{displaymath}

such that there exists a $ 5$ -isogeny $ \phi: E_1 \rightarrow E_2$ between them. This induces isogenies on the reductions $ \phi: \bar{E}_1 \rightarrow \bar{E}_2$ , from which one concludes, for each $ p$ , that either $ 5$ is a split or ramified prime in $ \mathcal{O}_K$ , or that $ 5$ divides the index $ [\mathcal{O}_K:\mathbb{Z}[\pi_p]]$ , and the two local endomorphism rings differ by index 5:

$\displaystyle \frac{[\mathcal{O}_K:{\mathrm{End}}(\bar{E}_1)]}{[\mathcal{O}_K:{\mathrm{End}}(\bar{E}_2)]} \in \{5^{-1},5\}.
$

If we consider among the first 1000 primes those for which $ (5)$ is inert in $ \mathcal{O}_K$ , we can tabulate indices $ m_i = [\mathcal{O}_K:{\mathrm{End}}(\bar{E}_i)]$ :

\begin{displaymath}
\begin{array}{c*{20}c@{}}
p &
521 & 1171 & 1741 & 2081 & 2...
...\\
m_2 &
5 & 1 & 1 & 10 & 1 & 5 & 12 & 1 & 1 & 2
\end{array}\end{displaymath}

The primes for which $ (5)$ is inert in $ \mathcal{O}_K$ are rare, and that there is no obvious preference for $ \bar{E}_1$ or $ \bar{E}_2$ to have the larger endomorphism ring. Can one determine a density of primes $ p$ for which $ (5)$ is inert in $ \mathcal{O}_K$ ?

Note that the condition $ [\mathcal{O}_K:\mathbb{Z}[\pi_p]] \equiv 0 \bmod 5$ is equivalent, up to isomorphism, to the action of $ \pi_p$ on $ \bar{E}_i[5]$ being:

$\displaystyle \pi_p \equiv \left(\begin{array}{cc}\mu&* 0&\mu\end{array}\right)\bmod 5.
$

The additional condition that $ [{\mathrm{End}}(\bar{E}):\mathbb{Z}[\pi_p]] \equiv 0 \bmod 5$ is measured by the condition:

$\displaystyle \pi_p \equiv \left(\begin{array}{cc}\mu&0 0&\mu\end{array}\right)\bmod 5.
$

Note that there a similar number of primes of supersingular reduction among the first 1000 primes, yet they are known to form a set of density zero.

Problem 8.1.2   Implement in SAGE an algorithm to compute $ {\mathrm{End}}(\bar{E})$ for $ \bar{E}$ an elliptic curve over a finite field. (Does this problem make sense for the special fiber of a Néron model as well?)

For higher dimensional modular abelian varieties, it would be interesting to have algorithms to determine the exact endomorphism rings at $ p$ , and to characterize the primes at which the reduction $ \bar{A}$ has $ p$ -rank $ r$ in $ 0 \le r \le g = \dim(A)$ .

Problem 8.1.3   Let $ A$ be an abelian variety of dimension $ \geq 1$ attached to a newform and let $ p$ be a prime of good reduction. Find an algorithm to compute the exact endomorphism ring $ {\mathrm{End}}(\bar{A}/\mathbb{F}_p)$ .

Problem 8.1.4   Let $ A$ be an abelian variety of dimension $ \geq 1$ attached to a newform. Give an algorithm to compute set of primes at which the reduction $ \bar{A}/\mathbb{F}_p$ has $ p$ -rank $ r$ with $ 0 \le r \le g = \dim(A)$ .

Note that the endomorphism rings at ordinary primes are CM orders, and the canonical lift of the reduction $ \bar{A}$ is a CM abelian variety. A database of invariants of CM moduli for small genus would aid in classifying these endomorphism rings (at small primes).

Problem 8.1.5   Create a database of invariants of CM moduli for small genus.


next up previous contents
Next: Endomorphism Rings over Number Up: Invariants of Modular Abelian Previous: Invariants of Modular Abelian   Contents
William Stein 2006-10-20