Eisenstein series and the trace formula for $\mathrm{GL}(2)$ over a function field

Abstract

We write out and prove the trace formula for a convolution operator on the space of cusp forms on $GL(2)$ over the function field $F$ of a smooth projective absolutely irreducible curve over a finite field. The proof – which follows Drinfeld – is complete and all terms in the formula are explicitly computed. The structure of the homogeneous space $\mathrm{GL}(2,F)\backslash \mathrm{GL}(2,\mathbb{A})$ is studied in section 2 by means of locally free sheaves of ${\mathcal{O}}_X$-modules. Section 3 deals with the regularization and computation of the geometric terms, over conjugacy classes. Section 4 develops the theory of intertwining operators and Eisenstein Series, and the trace formula is proven in section 5.