Borcherds products approximating Gersten complex
Shouhei Ma
TL;DR
The paper builds a modular-analytic bridge to higher Chow groups of orthogonal modular varieties by constructing a Borcherds-Gersten complex ${oldsymbol{ frac{M}{oldsymbol{\Gamma}}}^{ullet}_{p}}$ from lattices $L$ and spaces of input modular forms $M(L)$. It defines a Borcherds lift $oldsymbol{\\psi}_{L}$ sending modular data into Milnor $K$-groups and introduces residue maps that imitate tame symbols, yielding a boundary structure and a complex with $oldsymbol{\\partial}oldsymbol{\\partial}=0$. The main result shows that for $p\,\leq\,n$, this complex maps to the Gersten complex ${oldsymbol{\\mathsf{K}}_{p,X}^{ullet}}$, inducing connections to ${ extrm{CH}}^{p}(X,m)_{oldsymbol{\ frac{oldsymbol{\oldmath{Q}}}{}}}$ for $m\le 2$ and providing a framework to study special higher cycles via modular forms. The work also develops detailed functorial and transfer properties, presents explicit constructions of special $(p,1)$- and $(p,2)$-cycles, and discusses regulators and non-compact cases, highlighting potential links to Beilinson conjectures and Kudla's program. Overall, the paper offers a principled, computable path to access higher Chow groups of modular varieties through Borcherds products and CM values of theta lifts.
Abstract
For an orthogonal modular variety, we construct a complex which is defined in terms of lattices and elliptic modular forms, which resembles the Gersten complex in Milnor K-theory, and which has a morphism to the Gersten complex of the modular variety by the Borcherds lifting. This provides a formalism for approaching the higher Chow groups of the modular variety by special cycles and Borcherds products. The construction is an incorporation of the theory of Borcherds products and ideas from Milnor K-theory.