Representation theory of P-adic Groups

上个世纪最著名的数学成果之一是泰勒-怀尔斯对费马大定理的证明。从本质上讲，他们证明了模形式和椭圆曲线之间的关系，这是由谷山和志村证明的，也是证明费马长期猜想的最后一块缺失的部分。这种对应关系是一个更大的图景的特殊情况，即所谓的朗兰兹纲领，它将自守形式和伽罗瓦表示联系起来。这个程序的一部分是局部朗兰兹猜想，其中GL（n）在2000年被证明。Emerton和Helm后来将这一概念扩展到GL（n）的表示族，最近Helm和Moss解决了这一问题。一个重要的一步，他们的证明是理论的伽玛因素的家庭GL（n）这是发展莫斯。在我的博士论文中，我将尝试在更一般的经典群的表示中构造这样的伽马因子。这可能有助于证明经典群族中的局部朗兰兹猜想。

One of the most celebrated mathematical results of the last century was the Taylor-Wiles proof of Fermat's Last Theorem. Essentially, they proved a relation between modular forms and elliptic curves which was conjectured by Taniyama and Shimura and was the last missing piece for a proof of Fermat's longstanding conjecture. This correspondence is a special case of a much bigger picture, the so called Langlands Program, which conjecturally relates automorphic forms and Galois representations. One part of this program is the local Langlands conjecture which for GL(n) was proven in 2000. Emerton and Helm later extended this conjectures to families of representations of GL(n) which was recently resolved by Helm and Moss. An important step of their proof was the theory of gamma factors in families for GL(n) which was developed by Moss. In my PhD thesis I will try to construct such gamma factors in families for representations of more general classical groups. This might help to prove the local Langlands conjecture in families for classical groups.