PLANNING MEETING TUESDAY SEPTEMBER 6
The course is officially scheduled Tuesday and Thursday 1:10-2:25 and the first meeting will be Tuesday
September 6 at 1:10, room 622.
This time is convenient for those who plan to attend Hélène Esnault's Eilenberg lectures on Tuesday
at 2:40 (starting September 13). Students in my course are strongly encouraged to attend the Eilenberg lectures, which will be on the closely related topic of local systems in arithmetic and algebraic geometry.
Hodge-Tate theory and p-adic automorphic forms
Let S(G,X) be the Shimura variety attached to the reductive group G and the hermitian symmetric space X.
The Hodge-Tate period map, introduced by Scholze in his work on Galois representations attached to torsion cohomology classes, takes the version of S(G,X) at infinite p-adic level to the flag variety which is the algebraic version of the compact dual of X. The map, whose definition is based on p-adic analytic geometry, has since found many applications to the p-adic variation of automorphic forms. Its role is analogous to that of complex analysis on the symmetric space X in the classical theory of Shimura varieties.
An especially fruitful direction was initiated with Lue Pan's work on the completed cohomology of modular curves, which uncovered unexpected relations between the p-adic Simpson correspondence, D-modules, and p-adic Hodge theory, pointing toward a direct role for the representation theory of enveloping algebras in the p-adic theory of automorphic forms. Pilloni's reinterpretation of Lue Pan's work, and its generalization to all Shimura varieties by Rodríguez Camargo, represent important steps toward developing a geometric theory of p-adic automorphic forms comparable to that already known for the complex theory.
The course will focus on the articles cited below of Lue Pan and of Pilloni, with the ultimate aim of understanding Rodríguez's more complete but much more technically demanding article. Necessary notions from p-adic geometry and functional analysis, p-adic Hodge theory, and the localization theory of D-modules on flag varieties will be introduced as necessary.
Here is the tentative schedule for the first part of the course.
Some references
As much as possible the course will focus on geometric constructions that apply to automorphic forms. Basic notions of p-adic Hodge theory and p-adic analytic geometry will be introduced along with their relevant properties but these will not be proved. The book Perfectoid Spaces is an excellent introduction to this material; Scholze's introduction and the chapter by Weinstein are especially valuable for this course.
Perfectoid Spaces (Lecture notes from the 2017 Arizona Winter School), Mathematical Surveys and Monographs, AMS, Volume 242, 2019.
George Boxer and Vincent Pilloni. Higher Coleman Theory. https://arxiv.org/abs/2110.10251, 2021.
Ana Caraiani. Lecture notes on perfectoid Shimura varieties. https://swc-math.github.io/aws/2017/2017CaraianiNotes.pdf.
Ana Caraiani and Peter Scholze. On the generic part of the cohomology of compact
unitary Shimura varieties. Ann. of Math., 186(3):649–766, 2017.
Sean Howe, Overconvergent modular forms are highest weight vectors in the Hodge-Tate weight zero part of completed cohomology, https://arxiv.org/abs/2008.08029, 2022.
Lue Pan. On locally analytic vectors of the completed cohomology of modular curves. Forum of Mathematics, Pi, 10:e7, 2022.
Lue Pan. On locally analytic vectors of the completed cohomology of modular curves, II. manuscript, 2022.
Vincent Pilloni, Faisceaux equivariants sur ℙ^1 et faisceaux automorphes,
https://arxiv.org/abs/2202.13112, 2022.
Juan Esteban Rodríguez Camargo. Locally analytic completed cohomology of Shimura varieties and
overconvergent BGG maps, https://arXiv:2205.02016, 2022.
Peter Scholze. Perfectoid Shimura varieties. Japan J. Math., 11 15-32, 2016.
Peter Scholze. On torsion in the cohomology of locally symmetric varieties. Ann. of Math. , 182(3):945–1066, 2015.