On p-local Topological Automorphic Forms for $U(1,1;\mathbb{Z}[i])$

TL;DR

This paper introduces a new p-local topological automorphic forms (TAF) cohomology theory of height two, constructed via hyperelliptic curves and automorphic forms, with proven integrality and Landweber's criterion verification.

Contribution

It develops a novel p-local TAF-type cohomology theory based on isometry groups of hermitian lattices over Gaussian integers, with explicit genus construction and algebraic properties.

Findings

Construction of a genus to automorphic forms

Proof of integrality of the constructed forms

Verification of Landweber's criterion for the theory

Abstract

We present a new flavor of TAF-type (co)homology theories, which are p-local of height two and based on the isometry group of the odd unimodular hermitian lattice of signature (1,1) over the Gaussian integers. Using a suitable family of hyperelliptic curves, we explicitly construct a genus to automorphic forms, prove an integrality statement and verify Landweber's criterion.