Periods of automorphic forms

Abstract

Let E / F E/F be a quadratic extension of number fields and G = Res E / F ⁡ H G= \operatorname {Res}_{E/F}H , where H H is a reductive group over F F . We define the integral (in general, non-convergent) of an automorphic form on G G over H ( F ) ∖ H ( A ) 1 H(F)\backslash H(\mathbb A)^1 via regularization. This regularized integral is used to derive a formula for the integral over H ( F ) ∖ H ( A ) 1 H(F)\backslash H(\mathbb A)^1 of a truncated Eisenstein series on G G . More explicit results are obtained in the case H = G L ( n ) H=GL(n) . These results will find applications in the expansion of the spectral side of the relative trace formula.