Norm bounds on Eisenstein series
Dubi Kelmer, Alex Kontorovich, Christopher Lutsko
TL;DR
The paper analyzes supremum-norm and mean-square bounds for Eisenstein series on arithmetic hyperbolic orbifolds, introducing exponents $\nu_\infty$ and $\nu_2$ that govern growth on compacta and in spectral parameter intervals. It reduces these bounds to mean-square estimates for Epstein zeta functions and to uniform lattice-point counts for indefinite quadratic forms via Eskin–Margulis–Mozes machinery, then transfers the Epstein zeta bounds to Eisenstein series on $\mathrm{SL}_2(\mathbb{Z})$ and $\mathrm{SL}_2(\mathbb{Z}[i])$. The main results include $\nu_2(\mathrm{SL}_2(\mathbb{Z}))=0$ with a mean-square bound $O(T\log^4 T)$, and $\nu_2(\mathrm{SL}_2(\mathbb{Z}[i]))=\nu_\infty(\mathrm{SL}_2(\mathbb{Z}[i]))=1/2$ with a bound $|E(z,1+iT)| \ll T^{1/2+\varepsilon}$ on compacta; together with sharp mean-square bounds for Epstein zeta functions in low dimensions ($m=2$) and higher ($m\ge 3$). A sharpness result links subconvexity of $\nu_2$ to the existence of infinitely many cusp forms, illustrating the depth of the arithmetic–spectral connection. These findings advance understanding of Eisenstein spectral bounds and their arithmetic consequences for SL$_2$ and Picard-type lattices.
Abstract
We study the sup-norm and mean-square-norm problems for Eisenstein series on certain arithmetic hyperbolic orbifolds, producing sharp exponents for the modular surface and Picard 3-fold. The methods involve bounds for Epstein zeta functions, and counting restricted values of indefinite quadratic forms at integer points.