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

近年来，p进伽罗瓦表示与其模p约简之间的关系一直是数论的核心。 例如，由怀尔斯开创的方法证明了定义在有理数上的每一条椭圆曲线都是从模形式产生的，从而证明了费马大定理，这一方法关键地依赖于对具有特定模p约简的p进伽罗瓦表示的某些变形空间的研究。一个猜想的Breuil和Mezard -现在一个定理在许多情况下，由于Breuil和Mezard，以PI，并Kisin-预测的结构，这些变形空间。 PI建议研究Breuil-Mezard猜想到p-adics的有限扩展的推广。 调查员进一步建议，以确定某些特定类别的p-adic表示的模p的减少。 每一个证明广义Breuil-Mezard猜想的例子都应该通过Kisin的方法得到一个模性定理。 在许多情况下，这项工作将产生其他算术感兴趣的信息，如模形式的mod p表示的形状。PI的研究是数论，数学的最古老的分支之一。 本质上，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印象。 近几十年来，数论在密码学（创建代码）和密码分析（破解代码）领域有着革命性的应用。 例如，许多蜂窝电话的通信受到基于椭圆曲线的密码系统的保护，椭圆曲线是PI领域的主要研究对象之一。 PI是加拿大/美国数学夏令营的副主任，这是一个为数学天才高中生开设的暑期项目，他认为有研究活跃的数学家参与这些努力是至关重要的。

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.