On the dimension of the Jacquet module of a certain induced representation
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The Fourier coefficients of certain metaplectic Eisenstein series contain L-series of arithmetic interest. This fact has been repeatedly exploited by various authors in order to obtain analytic information about these L-series. Bump and Hoffstein [H] conjectured that the Hecke L-series of the n-th order residue symbol would be contained in the Fourier coefficients of an Eisenstein series on the n-cover of GL(n) induced up from a theta function on the n-cover of GL(n − 1). Some evidence for this conjecture was provided in [H], and a representation theoretic explanation for its plausibility was given in [BL]. Recently, Suzuki [S] proved the conjecture in the function field case, and he carried out many preliminary steps towards the proof of the conjecture in the number field case. Using very different techniques, the authors of this paper recently proved a generalization of the (slightly corrected version of the) Bump-Hoffstein conjecture over an arbitrary global field [BBL].