Representation theory of P-adic Groups

上个世纪最著名的数学成果之一是费马大定理的泰勒-怀尔斯证明。从本质上讲，他们证明了模形式和椭圆曲线之间的关系，这是谷山和志村猜想的，也是费马长期猜想的最后一个缺失的证明。这种对应关系是更大图景的一个特例，也就是所谓的朗兰兹纲领，它推测地将自同构形式和伽罗瓦表示联系起来。该计划的一部分是局部朗兰兹猜想，该猜想在2000年证明了GL(n)。Emerton和Helm后来将这一猜想扩展到最近由Helm和Moss解决的GL(n)表示族。他们证明的一个重要步骤是由Moss提出的GL(n)家族中的γ因子理论。在我的博士论文中，我将尝试在家庭中构建这样的伽马因子，以表示更一般的经典群体。这可能有助于证明古典群体在家庭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.