Wiles defect for Hecke algebras that are not complete intersections

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\bar {\rho }$ . If $\bar {\rho }$ is scalar at some primes dividing the discriminant of the quaternion algebra, then the Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight-2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda _f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda _f$ by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho _f$ which $f$ gives rise to by the Eichler–Shimura construction. One of the main tools used in the proof is Taylor–Wiles–Kisin patching.

GENERALISED QUANTUM DETERMINANTAL RINGS ARE MAXIMAL ORDERS

Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.

The p-adic Coates–Sinnott Conjecture over maximal orders

We prove the [Formula: see text]-adic version of the Coates–Sinnott Conjecture for all primes [Formula: see text], without assuming the vanishing of [Formula: see text]-invariants, for finite abelian extensions [Formula: see text] of a totally real number field [Formula: see text], where either the integral group ring [Formula: see text] of the Galois group [Formula: see text] is a maximal order in [Formula: see text] or [Formula: see text] is a CM-field of degree [Formula: see text] with [Formula: see text] odd and [Formula: see text], where the group ring [Formula: see text] is not a maximal order. The only assumption we have to make concerns the prime [Formula: see text], where for non-abelian fields we have to assume the Main Conjecture in Iwasawa theory and the equality of algebraic and analytic [Formula: see text]-invariants.

Computing embedding numbers and branches of orders via extensions of the Bruhat-Tits tree

Let [Formula: see text] be a local field and let [Formula: see text] be the two-by-two matrix algebra over [Formula: see text]. In our previous work, we developed a theory that allows the computation of the set of maximal orders in [Formula: see text] containing a given suborder. This set is given as a subgraph of the Bruhat–Tits (BT)-tree that is called the branch of the order. Branches have been used to study the global selectivity problem and also to compute local embedding numbers. They can usually be described in terms of two invariants. To compute these invariants explicitly, the strategy in our past work has been visualizing branches through the explicit representation of the BT-tree in terms of balls in [Formula: see text]. This is easier for orders spanning a split commutative subalgebra, i.e. an algebra isomorphic to [Formula: see text]. In fact, we have successfully used this idea in the past to compute embedding numbers for the split algebra. In the present work, we develop a theory of branches over field extensions that can be used to extend our previous computations to orders spanning a field. We use the same idea to compute branches for orders generated by arbitrary pairs of non-nilpotent pure quaternions, generalizing previous results due to the first author and Saavedra. We assume throughout that [Formula: see text].

Positively graded rings which are maximal orders and generalized Dedekind prime rings

Let [Formula: see text] be a positively graded ring which is a sub-ring of strongly graded ring of type [Formula: see text], where [Formula: see text] is a Noetherian prime ring. We define a concept of [Formula: see text]-invariant maximal order and show that [Formula: see text] is a maximal order if and only if [Formula: see text] is a [Formula: see text]-invariant maximal order. If [Formula: see text] is a maximal order, then we completely describe all [Formula: see text]-invertible ideals. As an application, we show that [Formula: see text] is a generalized Dedekind prime ring if and only if [Formula: see text] is a [Formula: see text]-invariant generalized Dedekind prime ring. We give examples of [Formula: see text]-invariant generalized Dedekind prime rings but neither generalized Dedekind prime rings nor maximal orders.