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# Endomorphism ring of

Let be a modular abelian variety over associated to a newform in . Let be a prime of good reduction for (so ). Let be the reduction of the modulo , which is an abelian variety over .

Problem 8.1.1   Compute the endomorphism ring .

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.

Problem 8.1.2   Implement in SAGE an algorithm to compute for 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 , and to characterize the primes at which the reduction has -rank in .

Problem 8.1.3   Let be an abelian variety of dimension attached to a newform and let be a prime of good reduction. Find an algorithm to compute the exact endomorphism ring .

Problem 8.1.4   Let be an abelian variety of dimension attached to a newform. Give an algorithm to compute set of primes at which the reduction has -rank with .

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).

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

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