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.

BASES OF MINIMAL VECTORS IN LATTICES, III

We prove that all Euclidean lattices of dimension n ≤ 9 which are generated by their minimal vectors, also possess a basis of minimal vectors. By providing a new counterexample, we show that this is not the case for all dimensions n ≥ 10.

ON WELL-ROUNDED IDEAL LATTICES

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many real and imaginary quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.