Pseudorepresentations of weight one are unramified

Deformation Conditions for Pseudorepresentations

Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur’s Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna...

Ordinary Pseudorepresentations and Modular Forms

In this short note, we observe that the techniques of [WWE15] can be used to provide a new proof of some of the residually reducible modularity lifting results of Skinner and Wiles [SW99]. In these cases, we have found that a deformation ring of ordinary pseudorepresentations is equal to the Eisenstein local component of a Hida Hecke algebra. We also show that Vandiver’s conjecture implies Shar...

Primer of Unramified Principal Series

An important but seldom-heard remark is that there is essentially just one irreducible representation of a p-adic group. Thus, rather than opening a Pandora’s box of disastrous complication, introduction of representation-theoretic methods in study of automorphic forms and L-functions is exactly the right thing: to the extent that there can be simplicity here, it is in the context of representa...

Unramified Cohomology of Quadrics, Ii

We compute the unramified cohomology of quadrics of dimension 4 in degree 4 over an arbitrary field of characteristic different from 2. We find that it is related to classical invariants of a more elementary nature, such as the group of spinor norms and the projective special orthogonal group modulo Manin’s R-equivalence.

Primer of Unramified Principal Series