Topic and importance:

Automorphic forms and L-functions have played prominent roles invarious branches of mathematics and physics since the 20th century,mostly at a theoretical level. However, recent advances in explicitand computational methods have made a more hands-on approach possible,and with it a host of applications in explicit and algorithmic numbertheory have arisen.

Syllabus:

This course will survey some of these methods and applications, in twoparts. The first part will focus on the computational theory ofL-functions and applications of an analytic nature, such asHelfgott's proof of the ternary Goldbach conjecture. The second partwill focus on explicit equations for modular varieties, the arithmeticof Jacobians of curves, and applications of an algebraic nature, suchas Couveignes and Edixhoven's polynomial-time algorithm for computingthe Ramanujan \tau-function.