p-adic and mod p Galois Representations, and Generalized Breuil-Mezard Conjectures

模p的p进Galois表示与它们的约化之间的关系是近年来数论的核心问题。例如，Wiles首创的证明定义在有理数上的每条椭圆曲线都来自模形式的方法--从而证明Fermat最后定理--关键依赖于对具有指定约化模p的p-adyGalois表示的某些变形空间的研究。对于p-ady数的绝对Galois群的表示，Breuil和Mezard的猜想--现在在许多情况下是一个定理，由于Breuil和Mezard、Pi和Kisin--预测了这些变形空间的结构。PI建议研究Breuil-Mezard猜想对p-adics的有限扩张的推广。研究者进一步建议确定某些特定类型的p-进表示的模p约简。用Kisin方法证明广义Breuil-Mezard猜想的每一种情形都应该产生一个模性定理。在许多情况下，这项工作将产生其他算术感兴趣的信息，如附加到模形式的mod p表示的形状。PI的研究是在数论中进行的，数论是数学中最古老的分支之一。从本质上讲，数论是研究方程的整数解，尽管复杂的现代技术有时会给人一种远离这一目标的感觉。近几十年来，数论在密码学(创码)和密码分析(破译密码)领域有了革命性的应用。例如，许多蜂窝电话的通信都受到基于椭圆曲线的密码系统的保护，这是PI领域的主要研究对象之一。PI是加拿大/美国数学营的副主任，这是一个针对数学有天赋的高中生的暑期项目，他认为有积极研究的数学家参与这样的努力是至关重要的。

The relationship between p-adic Galois representations and their reductions modulo p has been central to number theory in recent years. For instance, the method pioneered by Wiles to prove that every elliptic curve defined over the rational numbers arises from a modular form -- and hence to prove Fermat's Last Theorem -- relies crucially on the study of certain deformation spaces of p-adic Galois representations with specified reduction modulo p. For representations of the absolute Galois group of the p-adic numbers, a conjecture of Breuil and Mezard -- now a theorem in many cases, due to Breuil and Mezard, to the PI, and to Kisin -- predicts the structure of these deformation spaces. The PI proposes to study generalizations of the Breuil-Mezard conjecture to finite extensions of the p-adics. The investigator further proposes to determine the reduction modulo p of certain specific classes of p-adic representations. Each case in which a generalized Breuil-Mezard conjecture is proved should yield a modularity theorem, by a method of Kisin. In many cases this work will yield other information of arithmetic interest, such as the shape of the mod p representation attached to a modular form.The PI's research is in number theory, one of the oldest branches of mathematics. At heart, number theory is the study of whole number solutions to equations, although sophisticated modern techniques can sometimes give the appearance of being far-removed from this goal. In recent decades, number theory has had revolutionary applications to the fields of cryptography (creating codes) and cryptanalysis (breaking codes). For instance, the communications of many cellular telephones are protected by a cryptosystem based on elliptic curves, one of the primary objects of study in the PI's field. The PI is deputy director of Canada/USA Mathcamp, a summer program for mathematically talented high school students, and believes it is essential that there be research-active mathematicians participating in such endeavors.