Let $[X,\lambda]$ be a principally polarized abelian variety over finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the prime. Motivated by an equidistribution heuristic, we introduce factor $\nu_v([X,\lambda])$ for each place $v$ of $\mathbb Q$, and show product these factors essentially computes size isogeny class $[X,\lambda]$. The derivation thi...

We prove a certain Riemann-Roch type formula for symmetric powers of Galois modules on Dedekind schemes which, in the number field or function field case, specializes to a formula of Burns and Chinburg for Cassou-Noguès-Taylor operations. Introduction Let G be a finite group and E a number field. Let OE denote the ring of integers in E, Y := Spec(OE), and Cl(OYG) := ker(rank : K0(OEG) → Z) the ...

Let k be a finite field and let E be an elliptic curve over k. In this paper we describe, for each finite extension l of k, the structure of the group E(l) of points of E over l as a module over the ring R of endomorphisms of E that are defined over k. If the Frobenius endomorphism ? of E over k does not belong to the subring Z of R, then we find that E(l)$R R(?&1), where n is the degree of l o...

Abstract The endomorphism ring of the projective plane over a field F characteristic neither two nor three is slightly more complicated in Morel–Voevodsky motivic stable homotopy category than Voevodsky’s derived motives. In particular, it not commutative precisely if there exists square which does admit sixth root. A byproduct these computations proof Suslin’s conjecture on Suslin–Hurewicz hom...

The inner automorphisms of a group G can be characterized within the category of groups without reference to group elements: they are precisely those automorphisms of G that can be extended, in a functorial manner, to all groups H given with homomorphisms G→ H. (Precise statement in §1.) The group of such extended systems of automorphisms, unlike the group of inner automorphisms of G itself, is...

Elliptic curves over number elds with CM can be used to design non-isogenous elliptic cryptosystems over nite elds e ciently. The existing algorithm to build such CM curves, so-called the CM eld algorithm, is based on analytic expansion of modular functions, costing computations of O(2 5h=2 h 21=4 ) where h is the class number of the endomorphism ring of the CM curve. Thus it is e ective only i...

Depth three and finite depth are notions known for subfactors via diagrams and Frobenius extensions of rings via centralizers in endomorphism towers. From the point of view of depth two ring extensions, we provide a clear definition of depth three for a tower of three rings C ⊆ B ⊆ A. If A = EndBC and B |C is a Frobenius extension, this captures the notion of depth three for a Frobenius extensi...

We apply minimal weakly generating sets to study the existence of Add ( U R ) -covers for a uniserial module . If is right over ring , then S : = End has at most two maximal (right, left, two-sided) ideals: one set I all endomorphisms that are not injective, and other K surjective. prove if either finitely generated, or artinian, ? class covering only it closed under direct limit. Moreover, we ...