Session S09 - Number Theory in the Americas

Thursday, July 22, 20:30 ~ 21:00 UTC-3

Integral theta correspondence between two $\lambda$-resolvent Green functions

Hugo Chapdelaine

U. Laval, Canada   -   Hugo.Chapdelaine@mat.ulaval.ca

Let $F$ be a real quadratic field and let $\{\infty_1,\infty_2\}$ be its two real places. Let $B_1/F$ and $B_2/F$ be two quaternion algebras defined over $F$. We shall assume that $B_1$ is everywhere unramified (so that $B_1\simeq M_2(F)$) and that $B_2$ is ramified exactly in the two places $\{\infty_1,w\}$ where $w$ is a finite place of $F$. Let $O_i\subseteq B_i$ ($i=1,2$) be two orders which have been suitably chosen. One may associate to $O_i$ a couple $(V_i,\Delta_i)$ where $V_i$ is a Hilbert vector space of automorphic functions and where $\Delta_i$ is a Laplacian-like linear operator. The resolvent of $\Delta_i$, namely $(\Delta_i-\lambda)^{-1}$, can be written as an integral against a kernel which is given by some explicit automorphic Green function $G_{\lambda}^{i}$ ($i=1,2$). In this talk which shall present an equality between two integrals where $G_{\lambda}^{1}$ appears on the left hand side while $G_{\lambda}^{2}$ appears on the right hand side. Afterwards, we shall sketch how one can develop the integrals which appear on both side of this equality in order to obtain, a priori, non-trivial \,\lq\lq automorphic identities\rq\rq. A main motivation for this project was the famous Jacquet-Langlands correspondence which was published in 1970.

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