Consider an absolutely simple abelian variety X over a number field K. We show that if the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then Xp is absolutely simple for almost all primes p. Conversely, if the absolute endomorphism ring of X is noncommutative, then Xp is reducible for p in a set of positive density. An absolutely simple abelian variety ...

Irreducible complex representations of GL(2) and SL(2) over a finite field can be classified by methods useful for p-adic reductive groups and real Lie groups. (See Piatetski-Shapiro’s inspiring A.M.S. Memoir on this subject.) That is, we first see that most principal series representations are irreducible. We determine the irreducible constituents of the irregular principal series. We prove th...

Let K,S,D be a division ring an endomorphism and a S-derivation of K, respectively. In this setting we introduce generalized noncommutative symmetric functions and obtain Viète formula and decompositions of differential operators. W -polynomials show up naturally, their connections with P -independency, Vandermonde and Wronskian matrices are briefly studied. The different linear factorizations ...

We present a new algorithm to compute the integral closure of a reduced Noetherian ring in its total ring of fractions. We proceed, as in the classical case, by constructing an ascending chain of endomorphism rings of test ideals. However, our approach avoids the increasing complexity when enlarging the rings by doing most computations over the original ring. A modification, applicable in posit...

Let X be a k-vector space, and U a maximal proper filter of subspaces of X. Then the ring of endomorphisms of X that are ‘‘continuous’’ with respect to U modulo the ideal of those that are ‘‘trivial’’ with respect to U forms a division ring E(U ). (These division rings can also be described as the endomorphism rings of the simple left End( X )-modules.) We study this and the dual construction, ...

A non-commutative space X is a Grothendieck category ModX. We say X is integral if there is an indecomposable injective X-module EX such that its endomorphism ring is a division ring and every X-module is a subquotient of a direct sum of copies of EX . A noetherian scheme is integral in this sense if and only if it is integral in the usual sense. We show that several classes of non-commutative ...

We generalize the notion of a jet scheme (truncated arc space) to arbitrary fat points via adjunction, and show that this yields for each fat point, an endomorphism on each schemic Grothendieck ring as defined in [17]. We prove that some of the analogues for linear jets still hold true, like locally trivial fibration over the smooth locus. In this formalism, we can define several generating zet...

We will use commutators to provide decompositions of 3×3 matrices as sums whose terms satisfy some polynomial identities, and we apply them bounded linear operators endomorphisms free modules infinite rank. In particular it is proved that every operator an infinite-dimensional complex Hilbert space a sum four automorphisms order 3 simple ring obtained quotient the endomorphism vector modulo its...