An abelian surface A over a field K has potential quaternionic multiplication if the ring End K̄ (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is...

We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local k-algebra R, not necessarily commutative, where k is a field. The class of topological modules we consider include all those of finite rank over k and some of infinite rank as well, namely those with a Schauder basis in the sense of §1. This generalizes the resul...

For a finite group G, let E(G) denote the near-ring of functions generated by the semigroup, End(G), of endomorphisms of G. We characterize when E(G) is maximal as a subnear-ring of M0(G). A group G is an E-group if E(G) is a ring. We give a new characterization of finite Egroups and investigate the problem of determining, for finite E-groups, when E(G) is maximal as a ring in M0(G). 2000 Mathe...

We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the Bost-Connes (BC) quantum statistical mechanical system and the second on the Habiro ring, where the Habiro functions have, in addition to evaluations at roots of unity, also full Taylor expan...

As usual, we write Z,Q,Fp,C for the ring of integers, the field of rational numbers, the finite field with p elements and the field of complex numbers respectively. If Z is a smooth algebraic variety over an algebraically closed field then we write Ω(Z) for the space of differentials of the first kind on Z. If Z is an abelian variety then we write End(Z) for its ring of (absolute) endomorphisms...

We present a Mordell-Lang statement for Drinfeld modules. In the case of Drinfeld modules of finite characteristic, we show the connection between our Mordell-Lang statement and the study of the ring of quasi-endomorphisms for a certain minimal group in the theory of separably closed fields associated to the Drinfeld module.

The Virasoro operations in Witten’s theory of two-dimensional topological gravity have a homotopy-theoretic interpretation as endomorphisms of an ordinary cohomology theory with coefficients in a localization of I. Schur’s ring ∆ of Q-functions. The resulting theory has many of the features of a vertex operator algebra.

Let E be an elliptic curve, K1 its Kummer curve E/{±1}, E its square product, and K2 the split Kummer surface E /{±1}. The addition law on E gives a large endomorphism ring, which induce endomorphisms of K2. With a view to the practical applications to scalar multiplication on K1, we study the explicit arithmetic of K2.

The nilpotent endomorphisms over a finite free module over a domain with principal ideal are characterized. One may apply these results to the study of the maximal Cohen-Macaulay modules over the ring R := A[[x]]/(x), n ≥ 2, where A is a DVR. Subject Classification: 15A21, 13C14.