Automorphic forms and L-functions

19世纪后半叶最重要的数学思想之一是，解析公式通常对离散信息进行编码。例如，一个人可能想计算一个特定方程的解的数量，但发现很难找到解。另一方面，人们可能想计算一个方程序列的解的数量，并将答案作为一个序列。数学家称这类问题为“离散”问题。出现在微积分中的函数，以及微积分如此有效的函数，不是“离散的”，而是“分析的”。“令人惊讶的是，正确的解析函数往往能为离散问题提供答案。L函数是一种生成函数，由与算术几何对象（如伽罗瓦群和椭圆曲线）或模形式相关联的局部数据形成。这些L-函数提供了正确的解析函数。例如，黎曼zeta函数的零点提供了计数素数问题的答案。另一个典型的例子是，椭圆曲线的L-函数的零点提供了椭圆曲线是否有无穷多个有理解的答案。

One of the most important mathematical ideas of the second half of the 19th century is that an analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, one might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kinds of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kinds of analytic functions often provide the answers to the discrete problems. An L-function is a type of a generating function formed out of local data associated with either an arithmetic-geometric object (such as Galois groups and elliptic curves) or a modular form. These L-functions provide the right kinds of analytic functions. For example, zeros of the Riemann zeta functions provide the answer to the problem of counting prime numbers. Another conjectural example is that the zero of the L-functions of an elliptic curve provide the answer to whether the elliptic curve has an infinitely many rational solutions.