# A bridge built using differential equations in automorphic forms

NC2W02 - Crossing the bridge: New connections in number theory and physics

In this talk we will attempt to build our metaphircal bridge by making a connection between zeros and special values of L-functions and scattering amplitudes. The connection is best seen through solutions to differential equations of the form $$(\Delta – \lambda ) f = S$$ on $SL_2(\mathbb{Z})\backslash\mathfrak{H}$ for $\Delta=y2(\partial_x2+\partial_y2)$ and $S \in H{-infty}(SL_2(\mathbb{Z})\backslash\mathfrak{H})\cup M$ where $M$ is the space of moderate growth functions. Recently, Bombieri and Garrett (following work of Hass, Hejhal, and Colin de Verdiere) laid out the possibly connection with eigenvalue solutions to equations of this form with zeros of L-functions.  On the other hand, physicists such as Green, Kwan, Russo, Vanhove found that eigenfunction solutions to equations of this form give coefficients of the 4-graviton scattering amlitude. We will elaborate on these connections and discuss some recent work on finiding solutions for such equations. This work is in collaboration with Ksenia Fedosova, Stephen D. Miller, and Danylo Radchenko.

This talk is part of the Isaac Newton Institute Seminar Series series.