Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces

Let D be an indefinite quaternion division algebra over $${{\mathbb {Q}}}$$Q. We approach the problem of bounding the sup-norms of automorphic forms $$\phi $$ϕ on $$D^\times ({{\mathbb {A}}})$$D×(A) that belong to irreducible automorphic representations and transform via characters of unit groups of orders of D. We obtain a non-trivial upper bound for $$\Vert \phi \Vert _\infty $$‖ϕ‖∞ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for $$\Vert \phi \Vert _\infty $$‖ϕ‖∞ in the setting of newforms for Eichler orders. In the special case when the index of the order in a maximal order is a squarefull integer N, our result specializes to $$\Vert \phi \Vert _\infty \ll _{\pi _\infty , \epsilon } N^{1/3 + \epsilon } \Vert \phi \Vert _2$$‖ϕ‖∞≪π∞,ϵN1/3+ϵ‖ϕ‖2. A key application of our result is to automorphic forms $$\phi $$ϕ which correspond at the ramified primes to either minimal vectors, in the sense of Hu et al. (Commun Math Helv, to appear) or p-adic microlocal lifts, in the sense of Nelson in “Microlocal lifts and and quantum unique ergodicity on $$\mathrm{GL}_2({{\mathbb {Q}}}_{p})$$GL2(Qp)” (Algebra Number Theory 12(9):2033–2064, 2018). For such forms, our bound specializes to $$\Vert \phi \Vert _\infty \ll _{ \epsilon } C^{\frac{1}{6} + \epsilon }\Vert \phi \Vert _2$$‖ϕ‖∞≪ϵC16+ϵ‖ϕ‖2 where C is the conductor of the representation $$\pi $$π generated by $$\phi $$ϕ. This improves upon the previously known local bound$$\Vert \phi \Vert _\infty \ll _{\lambda , \epsilon } C^{\frac{1}{4} + \epsilon }\Vert \phi \Vert _2$$‖ϕ‖∞≪λ,ϵC14+ϵ‖ϕ‖2 in these cases.

Computations of Eigenvalues and Resonances on Perturbed Hyperbolic Surfaces with Cusps

In this paper we describe a simple method that allows for a fast direct computation of the scattering matrix for a surface with hyperbolic cusps from the Neumann-to-Dirichlet map on the compact manifold with boundary obtained by removing the cusps. We illustrate that even if the Neumann-to-Dirichlet map is obtained by a finite element method (FEM) one can achieve good accuracy for the scattering matrix. We give various interesting examples of how this can be used to investigate the behaviour of resonances under conformal perturbations or when moving in Teichmüller space. For example, based on numerical experiments we rediscover the four arithmetic surfaces of genus one with one cusp. This demonstrates that it is possible to identify arithmetic objects using FEM. All the videos accompanying this paper are available with its online version, or externally either at michaellevitin.net/hyperbolic.html or as a dedicated YouTubeplaylist.

Grothendieck-Serre duality and theta-invariants on arithmetic surfaces

In the paper, a description of the Grothendieck-Serre duality on an arithmetic surface by means of fixing a horizontal divisor is given and this description is applied to the generalization of theta-invariants.