Congruences between automorphic forms

我提议证明一个完全实数场的伽罗瓦群的完全奇二维表示的强阿廷猜想的新情况。更准确地说，我建议在惰性希尔伯特情况下建立“过收敛希尔伯特模特征型的解析延拓”，并将Buzzard-Taylor的主要定理推广到希尔伯特情况。模仿泰勒的“Artin II”论文的方法，在某些局部条件下，我们应该得到某些模p伽罗瓦表示的模块化，并且Artin猜想的许多情况应该是可访问的。其次，我提出了另一个由朗兰兹所预测的对应关系的例子，即：绝对伽罗瓦群Q的n维表示与GL(n)在Q上的（倒丘）自同构表示之间的对应关系。通过“专门化”GL(n)在Q上的非正则代数自同构表示（其“在无穷远处”的霍奇-泰特权不一定不同），应该可以构造自同构伽罗瓦表示和共轭维伽罗瓦表示的p进解析族。也有可能证明GL(n)的非正则自同构表示出现在Chenevier的特征变量和相关的伽罗瓦表示上，通过特殊化Chenevier构造的伽罗瓦表示族。另一种研究“规则权重”和“非规则权重”自同构形式之间的同余性的方法，类似于delign - serre，也将被考虑。

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.