arXiv subjects

Richard M. Hill

Publications and source records attributed to Richard M. Hill.

4 recordsLinked to original sources

arXiv2026

Residual finiteness and cuspidal cohomology of Picard modular surfaces

We prove that, for every non-uniform arithmetic lattice in $\mathrm{SU}(2,1)$, its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice $Γ$ for which $$H^1_{\mathrm{cusp}}(Γ\backslash\mathbb B^2,\mathbb C)\ne 0.$$ In particular, the first inner cohomology of this ball quotient is non-zero. The proof uses Rogawski's endoscopic classification for $\mathrm{U}(3)$. A cohomological criterion proved previously by the author then gives the residual-finiteness result. Residual finiteness also yields multiplier systems of arbitrary denominator on suitable finite-index subgroups.

math.NT↗

arXiv2022

Residual finiteness of extensions of arithmetic subgroups of SU(d,1) with cusps

Let $Γ$ be an arithmetic subgroup of $SU(d,1)$ with cusps, and let $X_Γ$ be the associated locally symmetric space. We prove that if the first inner cohomology group $H^1_!(X_Γ,\mathbb{C})$ is non-zero then the pre-image of $Γ$ in each connected cover of $SU(d,1)$ is residually finite. We also give an example of a such a group $Γ$ for which $H^1_!(X_Γ,\mathbb{C})$ is non-zero.

math.NT↗

arXiv2022

Modular forms on SU(2,1) with weight $\frac{1}{3}$

In this note, we describe several new examples of holomorphic modular forms on the group SU(2,1). These forms are distinguished by having weight $\frac{1}{3}$. We also describe a method for determining the levels at which one should expect to find such fractional weight forms.

math.NT↗

arXiv2021

Fractional weight multiplier systems on SU(d,1)

For a class of arithmetic subgroups $Γ$ in SU(d,1) we prove that for every positive integer $n$ there exists a subgroup $Γ_n$ of finite index in $Γ$, which lifts to the $n$-fold connected cover of of SU(d,1). Consequently $Γ_n$ has a multiplier system of weight $\frac{1}{n}$.

math.GR↗