Congruences between automorphic forms

证明了全实域上伽罗华群的全奇二维表示的强Artin猜想的新情形。更确切地说，我建议在惰性希尔伯特情形下建立“超收敛希尔伯特模本征式的解析延拓”，并将Buzzard-Taylor的主要定理推广到希尔伯特情形。仿照Taylor‘’Artin II‘’论文的方法，人们应该在某些局部条件下得到某些mod p Galois表示的模性，然后Artin猜想的许多情况应该是可理解的。其次，我建议证明在有理数Q的绝对Galois群的n维表示与Q上GL(N)的(尖角)自同构表示之间存在另一个由朗兰兹预测的对应的例子。遵循Hilbert模形式的Kisin-Lai的几何论点，应该可以通过“专门化”来构造自同构Galois表示和结合维Galois表示的p-进解析族到Q上的GL(N)的非正则代数自同构表示(其Hodge-Tate权‘在’无穷‘处不一定是不同的)。还可以通过特化Chenevier构造的Galois表示族来证明GL(N)的非正则自同构表示出现在Chenevier的本征簇和相关的Galois表示上。还将考虑类似于Deligne-Serre的另一种方法来研究“规则重量”和“非规则重量”的自同构形式之间的同余。

I propose to prove new cases of the strong Artin conjecture for totally odd two dimensional representations of the Galois group of a totally real field. More precisely, I propose to establish ``analytic continuation of overconvergent Hilbert modular eigenforms'' in the inert Hilbert case and generalise the main theorem of Buzzard-Taylor to the Hilbert case. Mimicking the approach of Taylor's ``Artin II'' paper, one should get modularity of certain mod p Galois representations subject to some local conditions, and many cases of Artin's conjecture should then become accessible. Secondly, I propose to show that there is another example of the correspondence predicted by Langlands between n-dimensional representations of the absolute Galois group of the rationals Q and (cuspidal) automorphic representations of GL(n) over Q. Following the geometric argument of Kisin-Lai for Hilbert modular forms, it should be possible to construct p-adic analytic families of automorphic Galois representations and associaten-dimensional Galois representations by ``specialisation'' to non-regular algebraic automorphic representations of GL(n) over Q (whose Hodge-Tate weights ``at infinity'' are not necessarily distinct). It may also be possible toprove that non-regular automorphic representations for GL(n) appear on the Chenevier's eigenvariety and associate Galois representations by specialising families of Galois representations Chenevier constructed. Another approach to look at congruences between automorphic forms of ``regular weight'' and ``non-regular weight'', analogous to Deligne-Serre, will also be considered.