It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL2type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of S...

Elliptic curves have a well-known and explicit theory for the construction and application of endomorphisms, which can be applied to improve performance in scalar multiplication. Recent work has extended these techniques to hyperelliptic Jacobians, but one obstruction is the lack of explicit models of curves together with an efficiently computable endomorphism. In the case of hyperelliptic curv...

We prove a noncompact Serre-Swan theorem characterising modules which are sections of vector bundles not necessarily trivial at infinity. We then identify the endomorphism algebras of the resulting modules. The endomorphism results continue to hold for the generalisation of these modules to noncommutative, nonunital algebras. Finally, we apply these results to not necessarily compact noncommuta...

The paper studies the following question: Given a ring R, when does the zero-divisor graph (R) have a regular endomorphism monoid? We prove if R contains at least one nontrivial idempotent, then (R) has a regular endomorphism monoid if and only if R is isomorphic to one of the following rings: Z2 × Z2 × Z2; Z2 × Z4; Z2 × (Z2[x]/(x)); F1 × F2, where F1, F2 are fields. In addition, we determine a...

A group is called morphic if for each normal endomorphism α in end(G),there exists β such that ker(α)= Gβ and Gα= ker(β). In this paper, we consider the case that there exist normal endomorphisms β and γ such that ker(α)= Gβ and Gα = ker(γ). We call G quasi-morphic, if this happens for any normal endomorphism α in end(G). We get the following results: G is quasi-morphic if and only if, for any ...

Given a maximal rigid object T of the cluster tube, we determine the objects finitely presented by T . We then use the method of Keller and Reiten to show that the endomorphism algebra of T is Gorenstein and of finite representation type, as first shown by Vatne. This algebra turns out to be the Jacobian algebra of a certain quiver with potential, when the characteristic of the base field is no...

Let S be the left bialgebroid End BAB over the centralizer R of a right D2 algebra extension A | B, which is to say that its tensor-square is isomorphic as A-B-bimodules to a direct summand of a finite direct sum of A with itself. Without an antipode, we prove that the left endomorphism algebra is a left S-Galois extension of A, and find a formula for the inverse Galois mapping. As a corollary,...

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq . Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q , while our bound for the second algorithm depends primarily on log |DE |, where DE is the discriminant of the order isomorphic to En...

In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For arbitrary self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism...

We study the scaling function of a C1+h expanding circle endomorphism. We find necessary and sufficient conditions for a Hölder continuous function on the dual symbolic space to be realized as the scaling function of a C1+h expanding circle endomorphism. We further represent the Teichmüller space of C1+h expanding circle endomorphisms by the space of Hölder continuous functions on the dual symb...