The Tamagawa Number Conjecture

本课题的目的是探索Taylor-Wiles系统在Hasse-Weil l -函数的特殊值上建立Tamagawa数猜想的有用性。在Taylor和Wiles的初步突破之后，这已经在与Diamond和Guo的联合工作中完成了椭圆模形式的伴随l函数。下一个我们要看的情况是那些泰勒-怀尔斯方法已经解决了，但与l函数值的关系仍然缺失的情况，特别是格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，甚至在所有纯数学中都占有重要地位，如果把七个克莱千年问题作为基准的话，其中两个问题直接与l函数有关。在高中代数中，我们在两个变量x和y的代数方程平面上画解集。我们也可以看到有理数、整数或整数模质数的解。l函数是由模素数的解（任意变量的任意一组方程的解）的个数建立起来的，并且期望给出有理数的解的信息。众所周知，用有理数求解方程是非常困难的，而l函数的值通常是非常可计算的，因此这种关系非常深刻。最原始的例子当然是伯奇和斯温纳顿-戴尔的猜想，这是千年问题之一。正如在数学中经常出现的那样，为了更好地理解一个问题，人们将其一般化，但代价是一个越来越抽象的框架（在这种情况下是“上同调”）。从积极的方面来看，广义框架的其他实例实际上可以用当前的方法来证明，这正是项目的目的所在。目的是证明l函数的期望特殊值公式的新情况，使用相对较新的泰勒-怀尔斯系统方法。

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.