Relative completed cohomologies and modular symbols

Abstract

Generalizing Emerton's completed cohomologies, we define relative completed cohomologies of arithmetic manifolds. We also define modular symbols for them, and show that the relative completed cohomology spaces interpolate the ``nearly ordinary part" of the classical automorphic cohomologies, and the modular symbols defined for them interpolate the classical modular symbols. As applications, we use these modular symbols to construct three families of nearly ordinary $p$-adic L-functions: (i) Rankin-Selberg $p$-adic L-functions for $\mathrm{GL}_n\times \mathrm{GL}_{n-1}$, (ii) Rankin-Selberg $p$-adic L-functions for $\mathrm{U}_n\times \mathrm{U}_{n-1}$, and (iii) Standard $p$-adic L-functions of symplectic type for $\mathrm{GL}_{2n}$. We define and calculate explicitly the modifying factors at $\infty$ and at $p$, and determine the exceptional zeros of the $p$-adic L-functions for these examples. The modifying factors at $\infty$ are consistent with the conjectures given by Deligne and Blasius, and the modifying factors at $p$ are consistent with the conjecture given by Coates and Perrin-Riou.