Congruences between automorphic forms

自同构形式之间的同余

• 批准号: EP/G050511/1
• 负责人: Shu Sasaki
• 金额: $ 29.17万
• 依托单位: King's College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2009
• 资助国家: 英国
• 起止时间: 2009 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG050511%2F1
• 关键词: Congruences; between; automorphic; forms

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.

我建议证明新的情况下，强阿廷猜想完全奇的二维表示的伽罗瓦群的一个完全真实的领域。更确切地说，我建议建立“解析延续的过度收敛希尔伯特模特征形”在惰性希尔伯特的情况下，推广的主要定理Buzzard-Taylor希尔伯特的情况。模仿泰勒的“阿廷II”论文的方法，人们应该得到某些模p伽罗瓦表示的模性，这些模p伽罗瓦表示服从某些局部条件，然后阿廷猜想的许多情况应该变得容易理解。其次，我建议表明，有另一个例子的对应关系预测朗兰兹之间的n维表示的绝对伽罗瓦群的有理数Q和（尖）自守表示GL（n）在Q。根据Kisin-Lai对希尔伯特模形式的几何论证，应该可以通过对GL（n）在Q上的非正则代数自守表示的“专门化”来构造自守伽罗瓦表示和伴随维伽罗瓦表示的p进解析族（其Hodge-Tate权重"在无穷远处“不一定是不同的）。也可以证明GL（n）的非正则自守表示出现在Chenevier的特征簇上，并通过专门化Chenevier构造的伽罗瓦表示族来关联伽罗瓦表示。另一种方法，看看之间的自守形式的“正规重量”和“非正规重量”，类似于德利涅-塞尔的同余关系，也将被考虑。