Eisenstein Series and the Trace Formula for GL(2) over a Function Field

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 $\GL(2,F)\bs\GL(2,\A)$ is studied in section 2 by means of locally free sheaves of $\OO_X$-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.

2010 Mathematics Subject Classification: Primary 11F70, 11F72; Secondary 22E35, 22E55, 11G20, 11R39, 11R52, 11R58, 14H30, 11S37

Keywords and Phrases: Eisenstein series, intertwining operators, trace formula, automorphic representations, GL(2), function fields, orbital integrals

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