The Tamagawa Number Conjecture

本项目的目的是探索泰勒-威尔斯系统在建立关于哈斯-韦尔L函数的特定值的玉马川数猜想方面的用处。继Taylor和Wiles最初的突破之后，在与Diamond和Guo合作的椭圆模形式的伴随L函数的情况下，已经做到了这一点。我们接下来要看的是那些泰勒-怀尔斯方法已经得到解决，但与L函数的值的关系仍然缺失的情况，特别是亏格2的西格尔模形式，希尔伯特模形式和可能的酉群。L-函数在现代数论中占有重要地位，如果你以七个克莱千年问题为基准，甚至在所有纯数学中，其中两个问题与L函数直接相关。在高中代数学中，你可以画出代数方程平面上的两个变量x和y的解集。你也可以看有理数、整数或以素数为模的整数的解。L函数是由模素数的解的个数(任何变量中的任何一组方程)构成的，并且被期望以有理数的形式给出关于解的信息。用有理数解方程是出了名的难，而L函数的值往往是非常可计算的，所以这种关系非常深。最原始的例子当然是Birch和Swinnerton-Dyer的猜想，这是千禧年问题之一。就像在数学中经常发生的那样，人们为了更好地理解一个问题而概括了它，但权衡的是一个越来越抽象的框架(在这种情况下是“上同调”)。从积极的方面来说，通用框架的其他实例实际上可能可以用当前的方法来证明，这也是该项目旨在探索的。目的是用相对较新的Taylor-Wiles系统方法证明L函数的期望特值公式的新情形。

Abstract for award DMS-0401403 of FlachThe aim of the project is to explore the usefulness of Taylor-Wiles systems for establishing cases of the Tamagawa number conjecture on special values of Hasse-Weil L-functions. Following the initial breakthrough by Taylor and Wiles this has been done in the case of the adjoint L-function of elliptic modular forms in joint work with Diamond and Guo. The next cases we intend to look at are those where the Taylor-Wiles method has been worked out but the relationship to values of L-functions is still missing, notably Siegel modular forms of genus 2, Hilbert modular forms and possibly unitary groups.L-functions figure prominently in modern number theory, or even in all of pure mathematics if one takes as a benchmark the seven Clay millenium problems, two of which are directly concerned with L-functions. In high school algebra one draws the solution set in the plane of an algebraic equation in two variables x and y. One may also look at the solutions in rational numbers, integers or integers modulo a prime number. L-functions are built from the number of solutions modulo primes (of any set of equations in any number of variables) and are expected to give information about solutions in rational numbers. Solving equations in rational numbers is notoriously hard, whereas values of L-functions are often very computable, so such a relationship lies very deep. The primordial example is of course the conjecture of Birch and Swinnerton-Dyer, one of the millenium problems. As often in mathematics, one generalizes a problem in order to understand it better but the tradeoff is an increasingly abstract framework ("cohomology" in this case). On the positive side, other instances of the generalized framework may actually be provable with current methods, and this is what the project aims to explore. The aim is to prove new cases of the expected special value formula for L-functions, using the relatively recent method of Taylor-Wiles systems.