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

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

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:

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

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:

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

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

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

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:

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

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

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

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.