Arakelov Theory, Motives and Special Values

Gillet教授将研究与Arakelov理论、动机以及zeta和l函数的特殊值有关的一些问题。他计划证明，在价值离散的领域中，存在一种不同品种动机的专业化地图；例如，这将描述在有理数上定义的变化的动机与在有限域上相关的可能是奇异的变化之间的关系。他计划研究是否可以在奇异变量上做算术交点理论，并提高我们对同调平凡代数环上的高度配对的理解。他还打算研究与狄利克雷级数的特殊值有关的高度和类似积分问题。这包括理解圆锥曲线的高度和Heun’s方程的特征值之间的关系，Heun’s方程是一个由物理问题引起的经典微分方程。这些研究领域的总体目标是提高我们对丢番图方程的理解，也就是说，理解具有整数系数的方程的整数解集的性质。丢番图方程特别适用于密码学和编码理论中的问题。

Professor Gillet will be studying a number of problems related to Arakelov Theory, Motives, and special values of zeta and L-functions. He plans to show that there is a specialization map for motives of varieties over discretely valued fields; this would for example describe the relationship between the motive of a variety defined over the rational numbers and the associated, possibly singular, varieties over finite fields. He plans to study whether one can do arithmetic intersection theory on singular varieties, and improve our understanding of the height pairing on algebraic cycles which are homologically trivial. He also intends to study questions relating heights and similar integrals to special values of Dirichlet series. This includes understanding the relationship between heights of conics and the eigenvalues of Heun's equation, which is a classical differential equation arising from problems in physics.The overall goal of these areas of research is to improve our understanding of Diophantine equations, that is, understanding the properties of the set of integer solutions of equations with integer coefficients. Diophantine equations have applications in particular to problems in cryptography and coding theory.