Explicit forms of the trace formula for GL(2)

There has recently been renewed interest in the trace formula–in particular, that of the initial case of GL(2)–due to counting applications in the function field case. For these applications, one needs a very precise form of the trace formula, with all terms computed explicitly. Our aim in this work is to compute the trace formula for GL(2) over a number field in as full detail as was done for the function field case and to give an accessible exposition, being motivated by these applications to counting, but also by pure curiosity as to the optimal form of this plastic formula. We also explain a correction argument in our context here of GL(2). The idea is to introduce a global summand which does not change the formula globally but changes the local weighted orbital integrals at the hyperbolic terms, so that their limit at the identity becomes a unipotent contribution to the trace formula. This gives a harmonious and pleasing form to the formula. Finally, we put the trace formula in an invariant form; thus all its terms are distributions whose value at a test function f^y(x) = f(y⁻¹xy) is independent of y in GL(2,A).