Relative completed cohomologies and modular symbols
Dongwen Liu, Binyong Sun
TL;DR
The article develops a unified framework of relative completed cohomologies and modular symbols to interpolate the nearly ordinary parts of automorphic cohomology. It constructs stable and relative modular symbols compatible with cup products and pull-backs, yielding $p$-adic L-functions across several families. Three main Rankin–Selberg and standard L-function settings are treated: GL$_n$×GL$_{n-1}$, U$_n$×U$_{n-1}$, and GL$_{2n}$ of symplectic type, with explicit modifying factors at infinity and at $p$, and an analysis of exceptional zeros. The approach integrates rationality of complex special values with $p$-adic interpolation via relative completed cohomology and modular symbols, aligning with Deligne–Blasius and Coates–Perrin-Riou conjectures and suggesting broad applicability to Langlands-related L-functions.
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.