Automorphic forms and L-functions

点击翻译按钮获取中文摘要

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.