The endomorphism ring of contains , where is the -th coefficient of the cusp form of , and the Frobenius endomorphism satisfies If is ordinary (i.e. has -rank ), then
where and is its maximal order. These reductions modulo are CM abelian varieties, but in general only the real subring generated by the trace terms lift back to the modular abelian variety over .
Note that the invariant is an invariant of the isomorphism class, but not the isogeny class, of . For instance the isogeny class of elliptic curves of conductor 57 denoted 57C by Cremona, consists of two curves:
such that there exists a -isogeny between them. This induces isogenies on the reductions , from which one concludes, for each , that either is a split or ramified prime in , or that divides the index , and the two local endomorphism rings differ by index 5:
If we consider among the first 1000 primes those for which is inert in , we can tabulate indices :
The primes for which is inert in are rare, and that there is no obvious preference for or to have the larger endomorphism ring. Can one determine a density of primes for which is inert in ?
Note that the condition is equivalent, up to isomorphism, to the action of on being:
The additional condition that is measured by the condition:
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.
For higher dimensional modular abelian varieties, it would be interesting to have algorithms to determine the exact endomorphism rings at , and to characterize the primes at which the reduction has -rank in .
Note that the endomorphism rings at ordinary primes are CM orders, and the canonical lift of the reduction 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).