AF AND MSOGR · Automorphic Forms and Moduli Spaces of Galois Representations
7РП — „Хора“ (Действия „Мария Кюри“)
- Период
- 2012-04-02 → 2016-04-01
- Финансиране от ЕС
- 100 000 €
- Участници
- 1
- Схема
- MC-CIG
Линиите свързват координатора с партньорите.
Накратко на български
Автоморфните форми и пространствата на Галоа свързват теорията на числата с други области на математиката. Работата по тях помага за разбирането на p-адичната програма на Лангландс и доказва части от хипотези като тази на Сер.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
Automorphic Forms and Moduli Spaces of Galois Representations
The project results in outputs in three areas: on moduli spaces of mod p representations, on the p-adic Langlands program, and on the Gouvea-Mazur conjecture. These are all part of the Langlands program, a web of conjectures relating number theory to other areas of mathematics. One part of the project concerned the construction of moduli spaces. The first paper on this subject (with Emerton) appeared during the project, and a detailed study of the 2-dimensional spaces is being carried out with Caraiani, Emerton and Savitt; this paper is over 100 pages and almost complete, and another 50 page preprint with Emerton should appear very soon. This work was one of the main focuses of a conference in Pisa in June 2016 organised by Michael Harris and Peter Schneider. My joint work on the p-adic Langlands program with Caraiani-Emerton-Geraghty-Paskunas-Shin, and on the Breuil-M\'ezard conjecture with Kisin, during the project, has considerably advanced the state of the art; in particular, it has made it clear that there will be a p-adic Langlands correspondence with considerable links to automorphy lifting theorems. The concrete output is a proof of the weight part of Serre's conjecture for Hilbert modular forms, and a construction of a candidate for the general n-dimensional p-adic Langlands correspondence. Finally, considerable numerical data was gained towards a better understanding of the project on the Gouvea-Mazur conjecture, and a survey article on the problem was written with Kevin Buzzard.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
I propose to solve three problems. The first is to prove Serre’s conjecture for real quadratic fields. I will do this by using automorphic induction to transfer the problem to U(4) over the rational numbers, where I will use automorphy lifting theorems and results on the weight part of Serre's conjecture that I established in my earlier work to reduce the problem to improving results in small weight and level. I will prove these base cases via integral p-adic Hodge theory and discriminant bounds.The second problem is to develop a geometric theory of moduli spaces of mod p and p-adic Galois representations, and to use it to establish the Breuil–Mezard conjecture in arbitrary dimension, by reinterpreting the conjecture in geometric terms, independently of any fixed mod p Galois representation. This will transform the subject by building the first connections between the p-adic Langlands program and the geometric Langlands program, providing an entirely new world of techniques for number theorists. As a consequence of the Breuil-Mezard conjecture, I will be able to deduce far stronger automorphy lifting theorems (in arbitrary dimension) than those currently available.The third problem is to prove a strengthened version of the Gouvea–Mazur conjecture, by completely determining the reduction mod p of certain two-dimensional crystalline representations. I will do this by means of explicit computations with the p-adic local Langlands correspondence for GL_2(Q_p), as well as by improving existing arguments which prove multiplicity one theorems via automorphy lifting theorems. This work will show that the existence of counterexamples to the Gouvea-Mazur conjecture is due to a purely local phenomenon, and that when this local obstruction vanishes, far stronger conjectures of Buzzard on the slopes of the U_p operator hold.
Оригинален текст от CORDIS (на английски).
Участници
- IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE · LondonКоординаторОбединеното кралство
Връзки
Данни: CORDIS, © Европейски съюз