A trace formula approach to control theorems for overconvergent automorphic forms

We present an approach to proving control theorems for overconvergent automorphic forms on some Harris-Taylor unitary Shimura varieties based on a comparison between the rigid coho- mology of the multiplicative ordinary locus and the rigid cohomology of the overlying Igusa tower, the latter which may be computed using the Harris-Taylor version of the Langlands-Kottwitz method. We also prove a higher level version, generalizing work of Coleman.Comment: 25 pages. Main results strengthened, higher level version include

A remark on a conjecture of Buzzard-Gee and the cohomology of Shimura varieties

We compare the conjecture of Buzzard-Gee on the association of Galois representations to C-algebraic automorphic representations with the conjectural description of the cohomology of Shimura varieties due to Kottwitz, and the reciprocity law at infinity due to Arthur. This is done by extending Langlands's representation of the L-group associated with a Shimura datum to a representation of the C-group of Buzzard-Gee. The approach offers an explanation of the explicit Tate twist appearing in Kottwitz's description.Comment: 10 pages, minor corrections and changes to presentatio

On the Sato-Tate conjecture for non-generic abelian surfaces

We prove many non-generic cases of the Sato-Tate conjecture for abelian surfaces as formulated by Fite, Kedlaya, Rotger and Sutherland, using the potential automorphy theorems of Barnet-Lamb, Gee, Geraghty and Taylor.Comment: 21 pages. Minor changes and corrections. With an appendix by Francesc Fit\'e. Essentially final version, to appear in Transactions of the AM

Event Prediction and Object Motion Estimation in the Development of Visual Attention

A model of gaze control is describes that includes mechanisms for predictive control using a forward model and event driven expectations of target behavior. The model roughly undergoes stages similar to those of human infants if the influence of the predictive systems is gradually increased

On subquotients of the etale cohomology of Shimura varieties

We study the conditions imposed by conjectures of Arthur and Kottwitz on the Galois representations occurring in the cohomology of Shimura varieties.Comment: Accepted manuscript. To appear in Shimura Varieties volum

Irreducible components of extended eigenvarieties and interpolating Langlands functoriality

We study the basic geometry of a class of analytic adic spaces that arise inthe study of the extended (or adic) eigenvarieties constructed byAndreatta--Iovita--Pilloni, Gulotta and the authors. We apply this to prove ageneral interpolation theorem for Langlands functoriality, which works forextended eigenvarieties and improves upon existing results in characteristic 0.As an application, we show that the characteristic p locus of the extendedeigenvariety for GL(2)/F, where F is a cyclic extension of the rational numbersQ, contains non-ordinary components of dimension at least [F:Q]