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

Abstract

Let D be an indefinite quaternion division algebra over Q. We approach the problem of bounding the sup-norms of automorphic forms φ on 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 φ ∞ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for 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 φ ∞ ≪π ∞,ǫ N 1/3+ǫ φ 2.A key application of our result is to automorphic forms φ which correspond at the ramified primes to either minimal vectors (in the sense of [11]), or p-adic microlocal lifts (in the sense of [15]). For such forms, our bound specializes to φ ∞ ≪ǫ C 1 6 +ǫ φ 2 where C is the conductor of the representation π generated by φ. This improves upon the previously known local bound φ ∞ ≪ λ,ǫ C 1 4 +ǫ φ 2 in these cases.

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

“…Proof. This is an immediate corollary of [Sah20,Proposition 2.8 and Remark 2.11]. Note that Proposition 2.8 of [Sah20] has the additional assumption 1 L N O(1) .…”

“…, which is a function of the usual hyperbolic distance on H. For the convenience of the reader, we recall a counting result from [Sah20] that will be used later.…”

“…Then, by the results of [HN18], F min, * is controlled by (1, 0; 1). Furthermore, it follows from Remark 3.2 of [Sah20] that F min, * is controlled by (γ, 1 − γ; 1) for all 0 γ 1. So, this is an example where equality is attained in both (10) and (11).…”

“…However, despite considerable recent activity on the sup-norm problem, the local bound in the depth aspect for newforms has not been improved upon so far. We refer the reader to the end of the Y. Hu and A. Saha introduction of [Sah20] for a brief discussion of why the usual methods are not sufficient to beat the local bound in this case.…”

“…Thus, it reduces the question of proving these bounds to checking these local hypotheses, and Theorem 5.4, described earlier, is essentially the statement that these local hypotheses are satisfied by the family of local newvectors of odd conductor and trivial central character. The proof of Theorem 4.6 is carried out in § 4.3 and uses as a main ingredient a lattice-point counting result proved in [Sah20].…”