arXiv subjects

Shuichiro Takeda

Publications and source records attributed to Shuichiro Takeda.

22 records · Page 2Linked to original sources

arXiv2010

Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs

Following the approach of B. Roberts, we characterize the non-vanishing of global theta lifts for symplectic-orthogonal dual pairs in terms of its local counterpart. In particular, we replace the temperedness assumption present in Robert's work by a certain weaker assumption, and apply our results to small rank similitude groups. Among our applications is a certain instance of Langlands functorial transfer of a (non-generic) cuspidal automorphic representation of $GSp(4)$ to GL(4).

math.NT↗

arXiv2009

On the regularized Siegel-Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups

We derive a (weak) second term identity for the regularized Siegel-Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the "second term range". As an application, we show the following non-vanishing result of global theta lifts from orthogonal groups. Let $π$ be a cuspidal automorphic representation of an orthogonal group $O(V)$ with $\dim V=m$ even and $r+1\leq m\leq 2r$. Assume further that there is a place $v$ such that $π_v\congπ_v\otimes\det$. Then the global theta lift of $π$ to $Sp_{2r}$ does not vanish up to twisting by automorphic determinant characters if the (incomplete) standard $L$-function $L^S(s,π)$ does not vanish at $s=1+\frac{2r-m}{2}$. Note that we impose no further condition on $V$ or $π$. We also show analogous non-vanishing results when $m > 2r$ (the "first term range") in terms of poles of $L^S(s,π)$ and consider the "lowest occurrence" conjecture of the theta lift from the orthogonal group.

math.NT↗

arXiv2008

On Shalika Periods and a Theorem of Jacquet-Martin

Let πbe a cuspidal automorphic representation of GL_4 with central character μ^2. It is known that πhas Shalika period with respect to μif and only if the L-function L^S(s, π, \bigwedge^2\otimesμ^{-1}) has a pole at s=1. Recentlt, Jacquet and Martin considered the analogous question for cuspidal representations π_D of the inner form GL_2(D)(\A), and obtained a partial result via the relative trace formula. In this paper, we provide a complete solution to this problem via the method of theta correspondence, and give necessary and sufficient conditions for the existence of Shalika period for π_D. We also resolve the analogous question in the local setting.

math.NT↗

arXiv2008

Some local-global non-vanishing results for theta lifts from orthogonal groups

We, firstly, improve a theorem of B. Roberts which characterizes non-vanishing of a global theta lift from O(X) to Sp(n) in terms of non-vanishing of local theta lifts. In particular, we will remove all the archimedean conditions imposed upon his theorem. Secondly, following Roberts, we will apply our theorem to theta lifting of low rank similitude groups. Namely we characterize the non-vanishing condition of a global theta lift from GO(4) to GSp(2) in our improved setting. Also we consider non-vanishing conditions of a global theta lift from GO(4) to GSp(1) and explicitly compute the lift when it exists.

math.NT↗