On the Calegari-Venkatesh conjecture connecting modular forms spaces and algebraic K-Theory
Alexander D. Rahm, Torti Emiliano
TL;DR
This work develops a concrete bridge between algebraic K-theory and the (co)homology of arithmetic groups by constructing a lift map $\varphi$ from $K_N(\mathcal{O})$ to a direct sum of Steinberg-related homology groups $H_{N+1-j}(GL_j(\mathcal{O}),St_j(F))$, using the Hurewicz map and Quillen’s $Q$-construction. Under odd class number and suitable real-embedding conditions, the construction yields lifts that, via Bieri–Eckmann duality, land in high-degree cohomology such as $H^{\nu(2)-1}(GL_2(\mathcal{O});R)$ and $H^{\nu(1)-2}(GL_1(\mathcal{O});R)$, connecting to CV’s surjection in the $N=2$ case and potentially to the CV isomorphism. The paper provides explicit real-quadratic and cubic-ring examples where nontrivial torsion in $K_2$ lifts to nonvanishing cohomology classes, illustrating the mechanism in concrete terms. A vanishing-theorem extension to large torsion supports these lifts by ensuring the relevant top-degree Steinberg coinvariants vanish after inverting orbifold primes. Finally, the authors give a practical method to compute $|K_2(O)|$ for real quadratic rings via $24\zeta_F(-1)$ (with explicit exceptions) and outline how to obtain these values computationally using Pari/GP.
Abstract
Calegari and Venkatesh did construct, modulo small torsion, a surjection from the degree 2 homology of the rank 2 projective general linear group over a ring of algebraic integers (of odd class number, and with enough embeddings) to the 2nd algebraic K-group of that ring. They asked whether this surjection becomes an isomorphism when passing to the quotient modulo the Eisenstein ideal on the left hand side. We provide a new method (together with numerical examples) to lift elements in the opposite direction, enabled by a theorem in a more general setting, where we exploit a connection between the algebraic K-groups and the Steinberg homology groups.