We define a quantum operation as a special type of endomorphism between spaces of matrices. Representations of endomorphisms are considered and an isomorphism between higher dimensional matrices and endomorphisms is derived. We then employ this isomorphism to prove various results for endomorphisms and quantum operations. For example, an endomorphism is completely positive if and only if its co...

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that one may compute the endomorphism ring of an elliptic curve defined over a finite field by using a depth first search algorithm in the graph. In dimension 2,...

Recently, A.A. Kirillov introduced an interesting class of associative algebras connected with the adjoint representation of G [Ki]. In our paper, such algebras are called g-endomorphism algebras. Each g-endomorphism algebra is a module over the algebra of invariants k[g]; furthermore, it is a direct sum of modules of covariants. Hence it is a free graded finitely generated module over k[g]. Th...

As mentioned several times in class, the arithmetic of quaternionic Shimura varieties is strongly controlled by the behavior of the class of CM points (just as in the case of modular curves). Just as for elliptic curves, if o is an order in an imaginary quadratic field K, one has a notion of a QM-surface with o-CM. Namely, let ι : O ↪→ A be a QM structure on an abelian surface (we are still wor...

We say the endomorphism problem is solvable for an element W in a free group F if it can be decided effectively whether, given U in F , there is an endomorphism φ of F sending W to U . This work analyzes an approach due to C. Edmunds and improved by C. Sims. Here we prove that the approach provides an efficient algorithm for solving the endomorphism problem when W is a twogenerator word. We sho...