Giada Grossi : "Higher Hida Theory, part 1 of 3". The goal of higher Hida theory is to define an “ordinary part” of higher coherent cohomology of Shimura varieties and interpolate it p-adically in the weight. We will define some higher coherent cohomological analogs of ordinary p-adic modular forms and prove control theorems comparing them to classical cohomology. These objects will be defined as some cohomologies with support conditions. One major source of technical difficulties is that we do not have well understood integral models of the Hecke correspondences arising in the theory. In the first lecture we will introduce some of the ideas in the simplest case of the modular curve, and in the subsequent lectures we will explain some developments in the Siegel and Hilbert cases.
› Admissible pairs, p-adic Hodge structures, and Ax-Lindemann for basic local Shimura varieties (or, a Tale of Two Analyticities) - Sean Howe, Department of Mathematics - University of Utah
15:00-16:00 (1h)
A. Pozzi : "Higher Elliptic Elements and a tame analogue of a conjecture of Perrin-Riou". A conjecture proposed by Harris and Venkatesh relates the derived Hecke algebra of weight one modular forms to a certain Stark unit. This conjecture can be formulated in terms of a pairing involving with the Shimura class, a class in the first etale cohomology group of the modular curve. Instances of this conjecture have recently been proved by Darmon, Harris, Rotger and Venkatesh. A key ingredient is the study of a generalised Eisenstein eigenspace for mod p modular forms. In this talk, I will discuss an analogue construction for generalised “elliptic" eigenspaces, which can be viewed as a tame refinement of a conjecture of Perrin-Riou. This is joint work in progress with Henri Darmon.
V. Pilloni : "Higher Hida Theory, part 2 of 3". The goal of higher Hida theory is to define an “ordinary part” of higher coherent cohomology of Shimura varieties and interpolate it p-adically in the weight. We will define some higher coherent cohomological analogs of ordinary p-adic modular forms and prove control theorems comparing them to classical cohomology. These objects will be defined as some cohomologies with support conditions. One major source of technical difficulties is that we do not have well understood integral models of the Hecke correspondences arising in the theory. In the first lecture we will introduce some of the ideas in the simplest case of the modular curve, and in the subsequent lectures we will explain some developments in the Siegel and Hilbert cases.
› Vanishing theorems for local and global Shimura varieties - Teruhisa Koshikawa, Research Institute for Mathematical Sciences, Kyoto University
09:00-10:00 (1h)
G. Boxer : "Higher Hida Theory, part 3 of 3". The goal of higher Hida theory is to define an “ordinary part” of higher coherent cohomology of Shimura varieties and interpolate it p-adically in the weight. We will define some higher coherent cohomological analogs of ordinary p-adic modular forms and prove control theorems comparing them to classical cohomology. These objects will be defined as some cohomologies with support conditions. One major source of technical difficulties is that we do not have well understood integral models of the Hecke correspondences arising in the theory. In the first lecture we will introduce some of the ideas in the simplest case of the modular curve, and in the subsequent lectures we will explain some developments in the Siegel and Hilbert cases.
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