### 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>