MATH 7370 - Automorphic Forms

Nicolas Templier, fall 2016.

We will explore different approaches to automorphic forms through examples. Topics include arithmetic and combinatorial functions, such as the partition function; exponential sums over finite fields, such as Kloosterman sums; linear ordinary differential equations with singularities, such as Bessel functions. In the traditional approach we study holomorphic functions on the Poincare upper-half plane and their Fourier expansion and in the geometric approach we study vector bundles on the Riemann sphere. The lectures will emphasize connections between these constructions and their unity in the context of automorphic forms.

Prerequisites

Concepts in algebra, complex analysis and differential geometry at the level of the core courses. No prior knowledge of modular forms is necessary.

References

• Serre, A course in arithmetic. Graduate Texts in Mathematics, No. 7. (available through Cornell library)
• Serre, Lectures on $N_X(p)$. Chapman & Hall/CRC Research Notes in Mathematics, 11.
• Katz, Exponential sums and differential equations. Annals of Mathematics Studies, 124.
• Katz, Gauss sums, Kloosterman sums, and monodromy groups. Annals of Mathematics Studies, 116.