Sup-norms of Eigenfunctions on Arithmetic Ellipsoids

Abstract

is shown, where t = (1 + λ) 1/2 and V = vol(X(O)).

Hybrid bounds for automorphic forms on ellipsoids over number fields

“…The present paper extends [BM11] in two further directions: on the one hand, by a new treatment of the amplifier we improve significantly the main result in [BM11]; on the other hand, we extend the argument to varieties attached to quadratic lattices L ⊂ V for (V, q) a totally definite ternary quadratic space defined over some fixed, totally real number field F of degree d over Q; the corresponding variety X(L, V ) is then a finite union of d-fold products of 2-spheres. We stress that this extension to number fields is not solely for the sake of generality: in the next subsection, we use these results to study similar problems for varieties associated to quaternary quadratic spaces.…”

“…Bounds on 2-dimensional ellipsoids. Our previous work [BM11] dealt with a family of varieties X = X(L, q) associated to pairs (L, q) for q a definite quadratic form on a three dimensional Q-vector space V and L a suitable lattice in V ; X was then a finite union of (quotients) of 2-spheres indexed by a set of representatives of genus classes of L.…”

“…Application to 3-dimensional ellipsoids. We illustrate the extension of [BM11] to general totally real number fields by providing non-trivial sup-norm bounds for Hecke-Laplace eigenfunctions on manifolds X that are finite unions of d-fold products of 3-spheres S 3 = SO 4 /SO 3 (i.e. bounds for automorphic forms of orthogonal groups in 4 variables).…”

“…The second bound in (1.7) is more complicated and requires Lemmata 3 -5. There are at least two sources of improvement compared to the analysis in [BM11]: in the present paper we use explicitly the fact that the considered quadratic forms are associated to an order in a quaternion algebra and in particular represent 1. Moreover, we exploit the average over the amplifier and treat the quadratic part of variable in the amplifier essentially as a new variable of the quadratic form.…”

“…Similar non-adelic treatments can be found in [IS95,Van97]. The (only) new input compared to [BM11] is that we choose an amplifier that has support on "balanced" algebraic integers (i.e. algebraic integers whose various conjugates have roughly the same size) that generate principal prime ideals.…”