Abstract: In this talk, I will introduce a family of integrals which represent the product of Rankin-Selberg L-functions of $GL(\ell)\times GL(m)$ and of $GL(\ell)\times GL(n)$ where $m+n<\ell$. When $n=0$, these integrals are those defined by Jacquet--Piatetski-Shapiro--Shalika up to a shift. As an application, we obtain a new proof of Jacquet's local converse conjecture using these new integrals and Cogdell--Shahidi--Tsai's theory on partial Bessel functions. This is joint work with Qing Zhang. If time permits, I will also briefly talk about another application, to prove an algebraicity result for the special values of certain L-functions, which is a joint work with Yubo Jin.
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