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We present a short proof that, for PEL-type Shimura varieties, subcanonical extensions of automorphic bundles, whose global sections over toroidal compactifications of Shimura varieties are represented by cuspidal automorphic forms, have no higher direct images under the canonical morphism to the minimal compactification, in characteristic zero or in positive characteristics greater than an explicitly computable bound.
We construct all irreducible cuspidal -modular representations of a unitary group in three variables attached to an unramified extension of local fields of odd residual characteristic with . We describe the -modular principal series and show that the supercuspidal support of an irreducible -modular representation is unique up to conjugacy.
Let be a finite group, let be an odd prime, and let . If , then there is a canonical correspondence between the irreducible complex characters of of degree not divisible by belonging to the principal block of and the linear characters of . As a consequence, we give a characterization of finite groups that possess a self-normalizing Sylow -subgroup or a -decomposable Sylow normalizer.
We study, from a combinatorial viewpoint, the quantized coordinate ring ofmatrices over an infinite field (often simply called quantum matrices).The first part of this paper shows that , which is traditionally defined by generators and relations, can be seen as a subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of as a sum over paths in the graph, each path being assigned an element of the quantum torus. The relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchon’s deleting derivations algorithm.
The second part of this paper applies the above to the theory of torus-invariant prime ideals of . We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only when and with transcendental over . Our strategy is to prove the stronger result that the quantum minors in a given torus-invariant ideal form a Gröbner basis.
In his reinterpretation of Gauss’s composition law for binary quadratic forms, Bhargava determined the integral orbits of a prehomogeneous vector space which arises naturally in the structure theory of the split group . We consider a twisted version of this prehomogeneous vector space which arises in quasisplit , where is an étale cubic algebra over a field . We classify the generic orbits over by twisted composition -algebras of -dimension .
We describe the structure of geometric quotients for proper locally triangulable -actions on locally trivial -bundles over a nœtherian normal base scheme defined over a field of characteristic . In the case where , we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank over . As a consequence, every proper triangulable -action on the affine four space over a field of characteristic is a translation with geometric quotient isomorphic to .
One of the many remarkable properties of the Apéry numbers , introduced in Apéry’s proof of the irrationality of , is that they satisfy the two-term supercongruences
for primes . Similar congruences are conjectured to hold for all Apéry-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Apéry numbers by showing that they extend to all Taylor coefficients of the rational function
The Apéry numbers are the diagonal coefficients of this function, which is simpler than previously known rational functions with this property.
Our main result offers analogous results for an infinite family of sequences, indexed by partitions , which also includes the Franel and Yang–Zudilin numbers as well as the Apéry numbers corresponding to . Using the example of the Almkvist–Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Apéry-like sequences.