p-adic measures and square roots of triple product L-functions, by Michael Harris and Jacques Tilouine

Let p be a prime number, and let f, g, and h be three modular forms of weights $\kappa$, $\lambda$, and $\mu$ for $SL(2,\Bbb{Z})$. We suppose $\kappa \geq \lambda + \mu$. In joint work with Kudla, one of the authors obtained a formula for the normalized {\it square root} of the value at $s = \frac{1}{2}(\kappa + \lambda + \mu - 2)$ (the {\it central critical value}) of the triple product $L(s,f,g,h)$. We apply this formula, letting $f$ (and thus $\kappa$) vary in a $p$-adic analytic family ${\bold f}$ of ordinary modular forms (a Hida family). By modifying Hida's construction of the $p$-adic Rankin-Selberg convolution, we obtain a generalized $p$-adic measure whose associated analytic function gives a $p$-adic interpolation of the square roots of the central critical values of $L(s,f,g,h)$, normalized by certain universal correction factors. The archimedean correction factor is not determined explicitly. This is an example of what appears to be a very general phenomenon of $p$-adic interpolation of normalized square roots of $L$-functions along the so-called "anti-cyclotomic hyperplane." We note that the $p$-adic triple product itself has not been constructed in the half-space $\kappa \geq \lambda + \mu$.

Michael Harris and Jacques Tilouine <harris@mathp7.jussieu.fr,tilouine@math.univ-paris13.fr>