The Tamagawa Number Conjecture

摘要奖DMS-0401403的Flach该项目的目的是探索泰勒-怀尔斯系统的有用性，建立的情况下，玉川数猜想的特殊值的Hasse-Weil L-函数。继最初的突破泰勒和怀尔斯这已经完成的情况下，伴随L-函数的椭圆模形式的联合工作与钻石和郭。接下来我们要研究的是泰勒-怀尔斯方法已经被解决，但L-函数的值之间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，甚至在所有的纯数学中，如果把七个克莱千禧年问题作为基准，其中两个直接与L-函数有关。在高中代数中，我们要在平面上画出一个含有两个变量x和y的代数方程的解集。人们也可以看看有理数，整数或整数模素数的解决方案。L-函数是从（任何变量的任何方程组的）解模素数的数量构建的，并且期望给出关于有理数的解的信息。用有理数求解方程是出了名的困难，而L函数的值通常是非常可计算的，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。在数学中，为了更好地理解问题，人们通常会概括一个问题，但代价是越来越抽象的框架（在这种情况下是“上同调”）。从积极的方面来看，广义框架的其他实例实际上可以用当前的方法证明，这就是该项目的目的。目的是证明新的情况下，期望的特殊值公式的L-函数，使用相对较新的方法泰勒-怀尔斯系统。

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.