Admissible $p$-adic measures attached to triple products of elliptic cusp forms

Abstract

We use the Siegel-Eisenstein distributions of degree three, and their higher twists with Dirichlet characters, in order to construct admissible $p$-adic measures attached to the triple products of elliptic cusp forms. We use an integral representation of Garrett's type for triple products of three cusp eigenforms. For a prime $p$ and for three primitive cusp eigenforms $f_1, f_2, f_3$ of equal weights $k_1= k_2= k_3=k$, we study the critical values of Garrett's triple product $L(f_1\times f_2\times f_3, s, \chi)$ twisted with Dirichlet characters $\chi$. The result is stated in the framework of a general program by John Coates [see Sémin. Bourbaki, Vol. 1988/89, 41e année, Exp. No. 701, Astérisque 177–178, 33–59 (1989; Zbl 0706.11064), Adv. Stud. Pure Math. 17, 23–54 (1989; Zbl 0783.11039)].