Notes on Analytic Properties of Residual Eisenstein Series, I

Details Notes on Analytic Properties of Residual Eisenstein Series, I Notes on Analytic Properties of Residual Eisenstein Series, I

2009-08-25

arxiv.org/abs/0908.3667v1

Mathematics

Number Theory

This is the 61 page version of 40 page paper submitted for publication

Scientific paper

We partially generalize the results of Kudla and Rallis on the poles of degenerate, Siegel-parabolic Eisenstein series to residual-data Eisenstein series. In particular, for $a,b$ integers greater than 1, we show that poles of the Eisenstein series induced from the Speh representation $\Delta(\tau,b)$ on the Levi $\mathrm{GL}_{ab}$ of $\mathrm{Sp}_{2ab}$ are located in the "segment" of half integers $X_{b}$ between a "right endpoint" and its negative, inclusive of endpoints. The right endpoint is $\pm b/2$, or $(b-1)/2$, depending on the analytic properties of the automorphic $L$-functions attached to $\tau$. We study the automorphic forms $\Phi_{i}^{(b)}$ obtained as residues at the points $s_i^{(b)}$ (defined precisely in the paper) by calculating their cuspidal exponents in certain cases. In the case of the "endpoint" $s_0^{(b)}$ and `first interior point' $s_1^{(b)}$ in the segment of singularity points, we are able to determine a set containing \textit{all possible} cuspidal exponents of $\Phi_0^{(b)}$ and $\Phi_1^{(b)}$ precisely for all $a$ and $b$. In these cases, we use the result of the calculation to deduce that the residual automorphic forms lie in $L^2(G(k)\backslash G(\mathbf{A}))$. In a more precise sense, our result establishes a relationship between, on the one hand, the actually occurring cuspidal exponents of $\Phi_i^{(b)}$, residues at interior points which lie to the right of the origin, and, on the other hand, the "analytic properties" of the original residual-data Eisenstein series at the origin. This preprint is a longer version of the paper "Analytic Properties of Residual Eisenstein Series, I", with the details of some proofs added and some additional examples adduced in support of the main conjecture.

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