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

Let $D$ be an indefinite quaternion division algebra over $\mathbb{Q}$. We approach the problem of bounding the sup-norms of automorphic forms $\phi$ on $D^\times(\mathbb{A})$ 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 $\|\phi\|_\infty$ in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for $\|\phi\|_\infty$ 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 $\|\phi\|_\infty \ll_{\pi_\infty, \epsilon} N^{1/3 + \epsilon} \|\phi\|_2$. A key application of our result is to automorphic forms $\phi$ which correspond at the ramified primes to either minimal vectors (in the sense of Hu-Nelson-Saha), or $p$-adic microlocal lifts (in the sense of Nelson). For such forms, our bound specializes to $\| \phi\|_\infty \ll_{\epsilon} C^{\frac16 + \epsilon}\|\phi\|_2$ where $C$ is the conductor of the representation $\pi$ generated by $\phi$. This improves upon the previously known local bound $\|\phi\|_\infty \ll_{\lambda, \epsilon} C^{\frac14 + \epsilon}\|\phi\|_2$ in these cases.

1. Introduction 1.1. Background. Let φ = ⊗ v φ v be a cuspidal automorphic form on D × (A) where D is an indefinite quaternion algebra over Q. The sup-norm problem, which has seen a lot of interest recently, is concerned with bounding φ ∞ φ 2 in terms of natural parameters of φ. When the primary focus is the dependance of the bound on parameters related to the ramified primes or to the underlying level structures associated to φ (with the dependance on the archimedean parameters suppressed), this is known as the level-aspect sup-norm problem.
In the split case where D = GL 2 , this problem has been heavily studied in the special case that φ is a global newform [5,21,7,8,22,18,17]. In this case, φ transforms by a character of the subgroup 1 Γ 0 (N ) of norm 1 units in the standard Eichler order of level N , where N equals the arithmetic conductor of the representation π generated by φ. The best upper bound currently known in the case of newforms on GL 2 is due to the present author [17] and in the trivial character case this bound reads 2 for any ǫ > 0, where we write N = N 0 N 1 with N 0 the largest integer such that N 2 0 divides N . More recently, the sup-norm problem has also been considered for newforms on GL 2 over number fields, we refer the reader to [4,1] for this. 1 Technically, we need to consider the adelic counterpart of this subgroup. 2 As usual, we use the notation A ≪x,y,... B to signify that there exists a positive constant C, depending at most upon x, y, . . ., so that |A| ≤ C|B|. 1 In the compact situation where D is a division algebra, less work has been done. As in the split case, the trivial bound in the level aspect is (2) φ ∞ ≪ π∞,ǫ N 1/2+ǫ φ 2 , for any automorphic form φ that transforms by a character of the unit group of an order of level N (see below for a more precise definition of this). The first improvement in this setting was due to Templier [20,21], who proved that for φ a global newform with respect to an Eichler order of level N , one has the bound More recently, Marshall [14] proved the bound (again, only in the setting of newforms for Eichler orders of level N ): with N 1 as in (1). (It was noted in [14] that the same bound also holds in the split case of D = GL 2 , provided one restricts the domain to a fixed compact set.) Note that Marshall's bound is better than Templier's when N is squarefull, but worse when N is squarefree. This reflects the fact that Marshall's bound is the local bound, which essentially coincides with the trivial bound for newforms of squarefree level, but is stronger than the trivial bound in general. We discuss this distinction in more detail in Section 1.4; see also Remark 3.2 for a more precise formulation.

The main result.
For the rest of this paper, let D be a fixed indefinite quaternion division algebra over Q. We now describe (a simplified version of) the main result of this paper, which deals with the compact case and for the first time, improves upon the trivial bound (2)  Given an order O ⊆ O max , define the (adelic) congruence subgroup K O of D × (A f ) (where A f denotes the finite adeles) by where the product is taken over all primes, and where we denote O p = O ⊗ Z Z p . Let π be an irreducible automorphic representation of D × (A) with unitary central character. Since we are in the division algebra case, π is unitary and cuspidal. By the multiplicity one theorem, we can uniquely identify V π with a certain space of automorphic forms on D × (A). For φ ∈ V π , we define as usual Let O ⊆ O max be an order and χ be a character of K O . Note that we may write χ = p χ p where p traverses the primes, and χ p is a character of O × p with χ p trivial for almost all p. The compactness of K O and the continuity 3 of χ automatically implies that χ is of finite order. We define V π (K O , χ) ⊆ V π to be the subspace of functions φ ∈ V π that have the transformation property (5) φ(gk) = χ(k)φ(g) for all k ∈ K O , g ∈ D × (A).
Given any non-zero φ ∈ V π (K O , χ), we wish to bound the sup-norm φ ∞ φ 2 in terms of the level N O .
As is clear from the earlier discussion, previous work on this topic has been focussed on the case where φ is a newform and O is an Eichler order with level N O equal to the conductor of π. This restriction to newforms and Eichler orders is quite limiting as it does not capture the behavior of several natural families of automorphic forms. For example, there is an emerging theory of automorphic forms of minimal type [11,9,10,12]; such forms transform naturally with respect to characters of unit groups of certain non-Eichler Bass orders. The aim of this paper is to prove for the first time a non-trivial upper bound for the sup-norm in the case of general orders. Let N 1 be as in (1). Let φ ∈ V π (K O , χ) where π is an irreducible automorphic representation of D × (A) with unitary central character, and χ is a character of K O . Suppose that φ ∞ corresponds to the vector of lowest non-negative weight 4 in π ∞ . Then we have For squarefree N , our Theorem implies that φ ∞ ≪ D,π∞,ǫ N 11/24+ǫ φ 2 . However, when N is squarefull (i.e., every prime that divides N does so with exponent at least 2) Theorem A gives a much stronger "Weyl-type" exponent.
Corollary. Let the notations and assumptions be as in Theorem A and assume that N is squarefull. Then we have For an explanation why we get such different exponents for squarefree and squarefull levels, see Section 1.5 of this introduction. An interesting fact is that this squarefree/squarefull dichotomy in the sup-norm bound is also present in the case of newforms on GL 2 (see Section 1.3 of [17]), but for utterly different reasons! Remark 1.1. Our main result (Theorem 1) is significantly more general than Theorem A above, because it does not require that the space generated by φ under the action of K O is one-dimensional.

A classical reformulation.
In this subsection, we reformulate Theorem A in the language of Hecke-Laplace eigenfunctions on the upper-half plane, which may be helpful for those who are more familiar with the classical language.
We let d denote the reduced discriminant of D and we fix an isomorphism ι ∞ : D ⊗ Q R ≃ M 2 (R). This leads to an embedding D ֒→ M 2 (R) which we also denote by ι ∞ .
For any order O of D, associate a discrete subgroup Γ O of SL 2 (R) as follows: Note that Γ O \H is a compact hyperbolic surface. Now, let χ = p χ p be a unitary character of K O as in the previous subsection. We have the identity consists of the elements of D × (R) ≃ GL 2 (R) with positive reduced norm (positive determinant). We extend χ to D × (R) + K O by making it trivial on D × (R) + . Hence (6) allows us to define a character on Γ O which (by abusing notation) we also denote by χ. Let N χ be an integer such that , which is a normal subgroup of Γ O . In particular, χ is a congruence character, i.e., it is trivial on a principal congruence subgroup.
where dz = c dxdy y 2 is any SL 2 (R)-invariant measure on H. On C ∞ (Γ O \H, χ −1 ) there exist Hecke operators T n for each positive integer n defined as follows: where N χ is chosen as above, and the subset S O max (Nχ) (n) of GL 2 (Q) is defined by It can be checked that the definition of T n given above is independent of all choices, including the choice of N χ , and is well-defined on the space C ∞ (Γ O \H, χ −1 ). For all (n, dN χ ) = 1, these operators are normal and commuting. Now, let φ and π be as in Theorem A. Assume that φ is right-invariant by where g ∞ ∈ D × (R) + is any matrix such that ι ∞ (g ∞ )i = z. Then f φ has the following properties: ( (2) f φ satisfies (∆ + λ)f φ = 0 where ∆ := y −2 (∂ 2 x + ∂ 2 y ).
(3) f φ is a simultaneous eigenfunction of the Hecke operators T n for all positive integers n with (n, dN χ ) = 1.
In other words, f φ is a Maass form for Γ O with character χ −1 and Laplace eigenvalue λ that is a Hecke eigenform at the good primes. Theorem A can be reformulated as an upper-bound on the sup-norms of such f φ : This follows from the fact that sup g∈D × (A) |φ(g)| = sup z∈H |f φ (z)|. Remark 1.2. In fact, every Maass form f for Γ O with character χ −1 and Laplace eigenvalue λ that is a Hecke eigenform at the good primes, can be obtained in the above way, i.e., f = f φ for some φ as in Theorem A. This can be proved using the technique of adelization. We omit the details of this in the interest of brevity.
1.4. The local bound and application to minimal vectors. For this subsection, we assume for simplicity that π has trivial central character. We compare Theorem A with the local bound in the level aspect for automorphic forms φ inside automorphic representations π of D × (A). By the local bound for φ, we mean the immediate bound emerging from the adelic pre-trace formula where the local test function at each ramified prime is chosen to be the restriction (to a maximal compact subgroup) of the matrix coefficient of φ p . In fact, an explicit computation performed in [14,17] for the case of newforms together with the principle of formal degrees, allows us to write down this bound in terms of the conductor of π.
More precisely, let π be as in Theorem A such that π has trivial central character and π p is one-dimensional at each prime dividing d. Then, for all φ ∈ V π that satisfy a mild condition 5 , we have the following bound: where C 1 is the smallest integer such that cond(π)|C 2 1 . We call (8) the local bound (in the level aspect). A more refined local bound is given in Remark 3.2 of this paper, under slightly more restrictive assumptions on φ. Remark 1.3. The quantity C 1 is equal to cond(π ×π) 1/2 . One reason that C 1 = cond(π ×π) 1/2 shows up in (8) is that (when π is discrete series) C 1 approximately equals the formal degree of π; see the calculations in [14,17] or [10, Appendix A].
The local bound (8) is essentially due to Marshall [14]. It seems reasonable to call (8) the local bound because (to quote Marshall in [14]) it appears to be "the best bound that may be proved by only considering the behaviour of φ in one small open set at a time, without taking the global structure of the space into account". We note that the bound (8) is also true in the non-compact setting of automorphic forms on GL 2 (A), provided one restricts the domain of φ to a fixed compact set. It seems worthwhile here to comment on the relationship between the local bound (8) and the trivial bound (2). It can be shown easily that the local bound (8) is always at least as strong as the trivial bound (2). However, these two bounds have somewhat different flavours: the trivial bound applies to all forms that transform by unitary characters of compact subgroups of a particular volume (and does not depend on the conductors of the associated representations) while the local bound applies to 5 The condition is that for some This is a mild technical condition that is satisfied by several families of automorphic forms, including newforms, automorphic forms corresponding to minimal vectors, p-adic microlocal lifts, and so on. For a more down-to-earth but slightly stronger condition, see Remark 3.2. forms whose associated representations have a particular conductor (and does not depend on some choice of subgroup that transforms the form by a character).
A central problem in this field (which also generalizes to higher rank automorphic forms) is to improve upon the local bound (8) for natural families of automorphic forms φ. An obvious strategy to try to do this would be to first improve upon the trivial bound for some class of subgroups (as we do in Theorem A in wide generality), and then use this result (for some carefully chosen subgroup) to improve upon the local bound. This naive strategy works best when the local component φ p for each ramified prime p is an eigenvector of a relatively large neighbourhood of the identity.
A key class of φ p for which this is true is the family of minimal vectors. Minimal vectors may be viewed as p-adic analogues of holomorphic vectors at infinity and have several remarkable properties, which were proved in our recent work [11] (where the analytic properties of the corresponding global automorphic forms of minimal type were studied for the first time in the setting of GL 2 ). The main result of [11] proved an optimal sup-norm bound for such forms in the setting of GL 2 .
However, the techniques used in [11] relied on a very special property of the Whittaker/Fourier expansion of φ which only works in the non-compact setting. Therefore, the proof cannot carry over to the compact case, i.e., to our case of indefinite quaternion division algebras D, as no Whittaker/Fourier expansions exist here. A major impetus behind this paper was to improve upon (8) for automorphic forms of minimal type on compact arithmetic surfaces. One consequence of Theorem A is that we can now do this.
Theorem B. (see Theorem 2) Let π be an irreducible, automorphic representation of D × (A) with trivial central character whose local component at each prime dividing d is one-dimensional, and let C denote the (arithmetic) conductor of π. Assume that C is the fourth power of an odd integer and suppose, for each prime p dividing C, that π p is a supercuspidal representation. Let φ in the space of π correspond to a minimal vector at each prime dividing C, a spherical vector at all other primes, and a vector of minimal weight at infinity. Then We remark that the condition on the conductor being the fourth power of an odd integer is a convenient one that was assumed for the definition of minimal vectors in our work [11]. However, this restriction has been partially removed in more recent work of Hu and Nelson [10] where they define and study properties of minimal vectors for all supercuspidal representations of D × where D is a (split or division) quaternion algebra over a p-adic field with p = 2. Using their work, we prove a more general version of Theorem B (Theorem 2) that applies to any odd conductor C.
The quantity C 1/6 on the right side of (9) represents one-third of the progress from the local bound C 1/4 extracted from the right side of (8) (we observe that in the setting of Theorem B, C 1 = C 1/2 ) to the conjectured 6 true bound of C ǫ . Theorem B therefore gives a Weyl-type exponent, which appears to be a natural limit for the problem of improving upon the local bound with current tools, at least in cases where no Fourier expansions are available.
Theorem A also leads to a sub-local bound in certain other settings. These other settings include the case of "p-adic microlocal lifts" in the sense of [15]. The padic microlocal lifts may be naturally viewed as the principal series analogue of minimal vectors. Indeed, for the corresponding global automorphic forms, we are also able to prove a Weyl strength sub-local bound, see (40). We also obtain a general bound for newforms that generalizes and strengthens all previously known results; see Theorem 3.
Finally, we remark that the results of this paper appear to be the first time that the local bound in the conductor aspect has been improved upon for squarefull conductors, for any kind of automorphic form on a compact domain. (In the noncompact case, this had been achieved in our previous paper [11].) It seems also worth mentioning here the very recent work of Hu [9] which generalizes [11] and proves a sub-local bound in the depth aspect for automorphic forms of minimal type on GL n under the assumption that the corresponding local representations have "generic" induction datum.
1.5. Key ideas. The heart of this paper is our solution to a counting problem for the number of elements that are "close" to the identity inside a (varying) quaternion order. This counting problem arises naturally in the amplification method for the level-aspect sup-norm problem. Roughly speaking, given an order O ⊆ O max of D, we are interested in good upper bounds for the integer and u(z 1 , z 2 ) denotes the hyperbolic distance. Above, z is any point on the upper-half plane H. Note, however, that for any , for an order O ′ that is conjugate to O by an element of (O max ) × . This allows us to move z to a fixed compact set J , namely J equal to some choice of fundamental domain for the action of Γ O max on H, at the cost of changing the order O to a suitable (O max ) × -conjugate of it. Now suppose that for each m and z ∈ J , we are able to prove a bound for N O (m; z) is the same for all (O max ) × -conjugates of O. Then we have actually proved a bound that is valid for all z ∈ H. This reduction is a key piece in our strategy and can be viewed as a workaround for the situation when Γ O is not a normal subgroup of Γ O max (c.f. the comments in Section 1.3 of [21]).
In fact we prove two separate bounds for N O (m; z) for z ∈ J . The primary one among them (Proposition 2.8) is valid for general lattices O (and does not use the multiplicative structure of the order at all!). The analysis behind the proof of this proposition, carried out in Sections 2.1 -2.3, may be of independent interest. Roughly speaking, we use a workhorse lemma on small linear combinations of integers to reduce the counting problem to several elementary ones involving simple linear congruences. The reader may wonder at this point why we do not instead use standard lattice counting results such as Proposition 2.1 of [2]. The reason is that those counting results are typically in terms of the successive minima of the lattice, which is not a preserved quantity for (O max ) × -conjugates of the lattice. In contrast, our method allows us to obtain a strong upper bound (Proposition 2.8) in terms of the invariant factors of the lattice in O max (the invariant factors are the same for all (O max ) × -conjugate lattices).
However, the bound obtained in Proposition 2.8 is sufficient for our purposes only when the lattice is balanced in the sense that its largest invariant factor is not very large (relative to the level). This raises the question: how do we deal with unbalanced lattices? For this, we observe another useful fact: the sup-norm of an automorphic form φ does not change when the form is replaced by some D × (A)translate of it. Now, given φ as in Theorem A, a D × (A)-translate of φ transforms by a character of an order O ′ that is locally isomorphic to (in the same genus as) the order O that we started off with. This leads us to investigate which orders have the key property of having a locally isomorphic order whose largest invariant factor is not very large. We solve this problem by a careful case-by-case analysis (see Section 3.7) relying on the explicit classification of local Gorenstein orders due to Brzezinski. The answer (essentially) is that any order of level N is locally isomorphic to an order whose largest invariant factor divides N 1 . This result may be of independent interest.
The upshot of all this is that the only orders for which our general lattice counting result (Proposition 2.8) does not lead to a non-trivial sup-norm bound are those whose levels are close to squarefree. To deal with this remaining case, we follow Templier's method [21,20] and prove a second counting result (Proposition 2.14), which uses the ring structure of the order to prove that elements that are close to the centralizer of some point must lie in a quadratic subfield. The combination of these two counting results lead directly to Theorem A above, and explain the shape the bound therein takes. Indeed, the term max(N 1/3 , N 1/2 1 ) in Theorem A comes from our first counting result, while the term N 11/24 comes from our second counting result.
Once the counting results are in place, we feed it into the amplification machinery to prove our main Theorems, closely following the adelic language employed in our previous paper [17]. It might be worth mentioning here that we use the slightly improved amplifier introduced by Blomer-Harcos-Milićević in [3] rather than the amplifier used in works like [8,17], which saves us some technical difficulties.
1.6. Additional remarks. For simplicity, we have only proved a level aspect bound in Theorem A. It should be possible to extend the argument to prove a non-trivial hybrid bound, however we do not attempt to do so here. It may also be possible to extend some parts of this paper (with additional work) to the case of number fields, and to certain higher rank groups. This is because our counting argument for general lattices is elementary and highly flexible, and should generalise to anisotropic lattices of higher rank.
We end this introduction with a final remark. As explained in Section 1.4 of this paper, our main result leads to an improvement of the local bound (8) in the level aspect for various families of automorphic forms, particularly the ones of minimal type studied in [11,10]. This uses crucially the fact that minimal vectors in supercuspidal representations π p are eigenvectors for the action of a relatively large subgroup (having volume around cond(π p ) −1/2 ). In contrast, newvectors in π behave well only under the action of a much smaller subgroup (having volume around cond(π p ) −1 ). Therefore, the approach of this paper does not immediately lead to an improvement over the local bound for newforms in the depth aspect where the conductor varies over powers of a fixed prime. However, all hope is not lostit turns out that one can augment this approach with suitable results on decay of matrix coefficients. This is the topic of ongoing work of the author with Yueke Hu, 8 which will be published in a sequel to this paper. We expect that our method there will allow us to beat the local bound for newforms in the depth aspect for the first time.
Acknowledgements. I would like to thank Yueke Hu and Paul Nelson for useful discussions, and Raphael Steiner for his generous help with the proof of Lemma 2.3.

2.
A counting problem for lattices 2.1. A lemma on linear combinations of integers. The object of this subsection is to prove an elementary but very useful lemma on the existence of "small" linear combinations of integers coprime to another integer. It is possible that some version of this lemma has appeared elsewhere, but we were unable to find a suitable reference. The proof below is essentially due to Raphael Steiner (private correspondence, March 2018) and we are grateful to him for his help and for allowing us to include his proof here.
Remark 2.1. We encourage the reader to initially focus on the case n = 1 of the lemma below to get a feel for the statement. In this paper, we will need the lemma only in the case n = 2.
Remark 2.2. With more sophisticated sieving methods a la [13], one can replace N ǫ by log(N ) 2 in conclusion (1) of the lemma below. Also, one can replace 2 n by c n for any fixed c, if desired. Lemma 2.3. For any ǫ > 0, there is a positive constant C ǫ such that for all (n + 2)tuples of integers (a 0 , a 1 , . . . , a n , N ), with N > 0, gcd(a 0 , a 1 , . . . , a n , N ) = 1, there exists at least 2 n distinct n-tuples of integers p 1 , p 2 , . . . , p n with gcd(a 0 + a 1 p 1 + a 2 p 2 + . . . + a n p n , N ) = 1.
Proof. We may assume without loss of generality that gcd(a 0 , a 1 , . . . , a n ) = 1 and that N is squarefree. Indeed, if these conditions are not met, we can replace each a i by a i /d where d = gcd(a 0 , a 1 , . . . , a n ) and we can replace N by its largest squarefree divisor, so that this modified setup does satisfy the conditions. Any constant C ǫ that works for this modified setup will also work for the original setup. We will prove the lemma by induction on n. Let us prove the base case n = 1. The starting point for this case is the following fact: For all ǫ > 0, there is a constant D ǫ such that for all positive integers a 0 , a 1 , Q with gcd(a 1 , Q) = 1 and X > 0 we have The proof of (10) follows from the following calculation: Let us now explain how the case n = 1 of the lemma follows from (10). We may assume that D ǫ > 1. We can find a constant E ǫ such that for all Q and all ǫ > 0. Now put Q = N/ gcd(a 1 , N ) and choose C ǫ > 3D ǫ E ǫ . Then picking X = C ǫ Q ǫ in (10) we have that (11) 1≤m≤CǫQ ǫ gcd(a 0 +ma 1 ,Q)=1 So there exist at least two distinct integers m 1 , m 2 , such that for p 1 ∈ {m 1 , m 2 }, we have p 1 ≤ C ǫ N ǫ , gcd(a 0 + p 1 a 1 , N/ gcd(a 1 , N )) = 1.
The proof of the case n = 1 of the Lemma is complete. We now prove the induction step. Assume that the lemma is proved for n = k. Now suppose we have integers a 0 , a 1 , . . . , a k+1 , N , with gcd(a 0 , a 1 , . . . , a k+1 ) = 1. We need to prove the conclusion of the lemma in this case. By replacing a k+1 by its residue modulo N if necessary, we may assume that 0 ≤ a k+1 ≤ N . By the case n = k of the Lemma, we can find 2 k distinct tuples of integers (p 1 , . . . , p k ) such that each of these tuples satisfy 0 ≤ p i ≤ C ǫ N ǫ and gcd(a 0 + a 1 p 1 + . . . + a k p k , a k+1 ) = 1 (we use here that a k+1 ≤ N ). Now, given any of these 2 k -tuples, we can use the case n = 2 of the lemma, to find 2 distinct possibilities for an integer p k+1 , with 0 ≤ p k+1 ≤ C ǫ N ǫ and such that gcd(a 0 + a 1 p 1 + . . . + a k p k + a k+1 p k+1 , N ) = 1.
So we have found 2 k+1 distinct (k + 1)-tuples of integers satisfying the required conditions and thus the induction step is complete.

2.2.
Lattices in quaternion orders. Let D be an indefinite quaternion division algebra over Q. We let d denote the reduced discriminant of D, i.e., the product of all primes such that D p is a division algebra. Fix one and for all a maximal order O max of D, and an isomorphism 7 ι ∞ : For α ∈ D, let α → α be the standard involution of D and let the reduced norm nr and trace Tr be given by nr(α) = αα, Tr(α) = α + α.   (1) is the subgroup of (O max ) × with reduced norm 1. We fix, once and for all, three elements i 1 , Fix a compact subset J of H. 8 Given a subset L of D, and an element z ∈ H, δ > 0, define for each positive integer m the set The reader should think of δ ≍ 1 as fixed, since all constants will be allowed to depend on δ (in fact, for our eventual applications, we will actually fix δ = 1, however it will be useful to keep this variable δ around for now). Our goal is to bound the cardinality of L(m; z, δ) (in terms of some basic invariants of L) whenever L ⊆ O max is a lattice containing 1.
Let L ⊆ O max be a lattice containing 1. We denote and call N the level of L. By the structure theorem for finitely generated abelian groups, the finite group O max for some uniquely defined positive integers M 1 |M 2 |M 3 , which are sometimes known as invariant factors. We will refer to these integers as the shape of L.  It is easy to see that L ′ always has level N but its shape might be different in general. However, if x ∈ (O max ) × , then L ′ also has shape (M 1 , M 2 , M 3 ).
It turns out to be more convenient for us to descend to the sublattice Z ⊕ L 0 , for which the next lemma is essential. Proof. Given an element ℓ ∈ L, we have Tr(ℓ) ∈ Z and furthermore, ℓ belongs to Z ⊕ L 0 if and only if Tr(ℓ) ∈ 2Z. So if ℓ 1 and ℓ 2 are two elements in L, neither of which belong to Z ⊕ L 0 , then ℓ 1 + ℓ 2 ∈ Z ⊕ L 0 . The statement follows.
Let L be a lattice as in Lemma 2.6, and let N be the level of L, and M the level of L 0 . Using Lemma 2.6 and the fact that O max 0 + Z has index two in O max , we observe that Remark 2.9. The constants implicit in the bounds above depend only on ǫ, δ and the fixed objects D, O max , ι ∞ , J .
Remark 2.10. Note that the bounds do not depend on the elements ∆ 1 , ∆ 2 , ∆ 3 . Hence the bounds obtained are uniform over all (O max ) × -conjugates of L. This will be key for us later on.
Remark 2.11. Because of (12) one can replace N by M in the theorem above, if one wishes. Furthermore, because of (13), one can replace M 1 , M 2 in the theorem above by M ′ 1 , M ′ 2 respectively, if one wishes to. Remark 2.12. The bound obtained in Proposition 2.8 is not sufficient for our purposes when M 1 M 2 is small in relation to N . So in Section 2.4, we will prove another counting result under the additional assumption that L is an order. We now prove Proposition 2.8. Using Lemma 2.6, we may assume that L = Z⊕L 0 . Indeed, putting L ′ = Z ⊕ L 0 , we see that |L ′ (m; z, δ)| ≤ |L(m; z, δ)| ≤ |L ′ (4m; z, δ)|.
So by shrinking L if necessary, we will assume throughout the rest of this subsection that (16) L = Z ⊕ L 0 . Now, using Lemma 2.13, we see that Proposition 2.8 would follow from the following statement: Then for 1 ≤ L ≤ N O(1) we have: Let us now prove the bounds (17), (18). This will complete the proof of Proposition 2.8. For brevity, we drop T from the ≪ symbol in the rest of this subsection (so all constants henceforth are allowed to depend on T ).
Our strategy will be to associate to each such α a quadruple (a 0 , A α , B α , a 3 ) such that the function α → (a 0 , A α , B α , a 3 ) is injective. A bound for the cardinality of 1≤m≤L L(m, T ) will then follow by bounding the number of distinct tuples (a 0 , A α , B α , a 3 ) that are possible. Write α = a 0 + a 1 i 1 + a 2 i 2 + a 3 i 3 . We have that |a i | ≪ L 1/2 and furthermore we have integers u 1 , u 2 , u 3 such that for i = 1, 2, 3, we have (21) a i = u 1 M 1 δ 1,i + u 2 M 2 δ 2,i + u 3 M 3 δ 3,i .
Now that A α and B α have been chosen, we will finish the proof assuming that s 2 = r 2 . (By assumption s i = r i for some i ∈ {2, 3}; the proof for s 3 = r 3 is essentially identical).
Using (26), we see that where S[S −1 ] ≡ 1 (mod N ). Now (22) and (21) imply that So a 3 is known modulo M 3 in terms of choices that have already been made. Since From the above, we see that there are This completes the proof of (17).

A supplementary counting result for orders.
In this subsection, we give another counting result to supplement Proposition 2.8, but one that is applicable only if L = O is an order.
Proposition 2.14. Let O ⊆ O max be an order of level N . There is a constant C (depending on δ, J and ι ∞ ) such that for z ∈ J and 1 ≤ m < CN 1 3 , we have Our proof of Proposition 2.14 is broadly similar to that of Proposition 6.5 of [21] (see also [20]). The proof will follow from the following sequence of lemmas. Throughout the proof, we will use the notations introduced in Section 2.2 and we will assume (without loss of generality) that (16) holds.
Lemma 2.15. Let L be a subset of D that is closed under multiplication and contains 1. Let z ∈ H, L a positive integer, and δ > 0. Then the Q-algebra generated by all elements in 1≤m≤L L(m; z, δ) is contained in the Q-vector-space spanned by 1≤m≤L 2 L(m; z, 2δ). Proof. By basic properties of a quaternion algebra, any element of the Q-algebra generated by 1≤m≤L L(m; z, δ) is a Q-linear combination of elements of the form β = β 1 β 2 with β 1 , β 2 ∈ 1≤m≤L L(m; z, δ). So it suffices to show that any such β belongs to 1≤m≤L 2 L(m; z, 2δ). This is clear as nr(β) = nr(β 1 )nr(β 2 ) ≤ L 2 and Let A be the 3 by 3 matrix whose (i, j)th entry is a (i) j for 1 ≤ i, j ≤ 3. We need to show that det(A) = 0. Let L have shape (M 1 , M 2 , M 3 ) with M 1 M 2 M 3 ≍ N , and let the integers δ i,j be as in (19). Therefore we have integers u  where d(m) denotes the divisor function and the implied constant is independent of F .
Proof. Any element of F (m; z, δ) is a product of a unit in O × F of norm 1, and an element of O F of norm m, with the latter taken from a fixed set of cardinality ≪ d(m). So we only need to consider the action of units. By the proof of Lemma 6.4 of [21], the number of norm 1 units κ ∈ O × F satisfying u(z, ι ∞ (κ)z) ≤ δ is ≪ δ 1. This completes the proof.
3. The main result: Statement and key applications 3.1. Basic notations. We continue to use the notations established in Section 2.2, and introduce some new ones below. Let f denote the finite places of Q (which we identify with the set of primes) and ∞ the archimedean place. We let A denote the ring of adeles over Q, and A f the ring of finite adeles. Given an algebraic group H defined over Q, a place v of Q, a subset of places U of Q, and a positive integer M , we denote H v := H(Q v ), H U := v∈U H v , H M := p|M H p . Given an element g in H(Q) (resp., in H(A)), we will use g p to denote the image of g in H p (resp., the p-component of g); more generally for any set of places U , we let g U the image of g in H U .
Recall that D is an indefinite quaternion division algebra over Q with reduced discriminant d and that we have fixed a maximal order O max . We denote G = D × and G ′ = P D × = D × /Z where Z denotes the center of D × . For each prime p, let K p = (O max p ) × and let K ′ p denote the image of K p in G ′ p . Thus, for p|d, K ′ p has index 2 in the compact group G ′ p . For each place v that is not among the primes dividing d, fix an isomorphism . We assume that these isomorphisms are chosen such that for each finite prime p ∤ d, we have ι p (O p ) = M (2, Z p ). It is well known that all such choices are conjugate to each other by some matrix in GL 2 (Z p ). By abuse of notation, we also use ι v to denote the composition map We fix the Haar measure on each group G p such that vol(K p ) = 1. We fix Haar measures on Q × p such that vol(Z × p ) = 1. This gives us resulting Haar measures on each group G ′ p such that vol(K ′ p ) = 1. Fix any Haar measure on G ∞ , and take the Haar measure on R × to be equal to dx |x| where dx is the Lebesgue measure. This gives us a Haar measure on G ′ ∞ . Take the measures on G(A) and G ′ (A) to be given by the product measure.
For each continuous function φ on the space G(A), we let R(g) denote the right regular action, given by (R(g)φ)(h) = φ(hg). If a continuous function φ on G(A) satisfies that |φ| is left Z(A)G(Q) invariant, define Note above that G ′ (Q)\G ′ (A) is compact, so convergence of the integral is not an issue.

Some facts on orders and their localizations.
We recall some facts we will need. Proofs of these standard facts can be found, e.g., in [23].
We define the shape and level of an order as in Section 2.2. These quantities have obvious local analogues, and so for each order O p ⊆ O max p of D p , we can define its shape (p m 1,p , p m 2,p , p m 3,p ) and level p np . It is easy to see that vol(O × p ) ≍ p −np . If O is the global order of shape (M 1 , M 2 , M 3 ) and level N corresponding to the collection of local orders {O p } p∈f with shape and level as above, then for i = 1, 2, 3 we have: For each g ∈ G(A f ), and an order O of D, we let g · O denote the order whose localization at each prime p equals g p O p g −1 p . An order is said to be locally isomorphic to (in the same genus as) O if and only if it is equal to g · O for some g ∈ G(A f ). Note that Note also that Given g satisfying (33), the orders O and g · O need not be isomorphic or have the same shape, however they always have the same level. However, note that if k ∈ K O max , then k · O has exactly the same shape as O.

Statement of main result.
Let π = ⊗ v π v be an irreducible, unitary, cuspidal automorphic representation of G(A) where we identify V π with a (unique) subspace of functions on G(A), so that π(g) coincides with R(g) on that subspace. Given a compact open subgroup K ′ = p∈f K ′ p of K O max (where each K ′ p is a subgroup of K p , with K ′ p = K p for almost all p) and a finite dimensional representation ρ of K ′ , we say that an automorphic form φ ∈ V π is of (K ′ , ρ)-type if the right-regular action of K ′ on φ generates a representation isomorphic to ρ. Observe that the existence of a form of (K ′ , ρ)-type implies that the restrictions of ρ and ω π to the centre of K ′ must coincide.
We can now state our main theorem.
Theorem 1. Let φ be an automorphic form in the space of π such that φ 2 = 1. Let O ⊆ O max be an order of D of level N and let ρ be a finite dimensional representation of K O . Let N 1 be the smallest positive integer such that N divides N 2 1 . Let φ ∈ V π be of (K O , ρ)-type and assume that φ is of minimal weight at the archimedean place, i.e., where k = 0 if π ∞ is principal series, and k is the lowest weight if π ∞ is discrete series. Then Theorem A of the introduction is a special case of Theorem 1, where we take ρ to be a character. A key flexibility of Theorem 1 comes from the fact that given φ, one can optimise which order O to use depending on how much information one has about the dimensions of the representations ρ generated under the action of various K O . In certain cases, however, one may only know the dimension under the action of some K ′ that is not of the form K O . In such cases one can still get a bound by working with any order O containing K ′ . The following corollary makes this precise.
Now apply Theorem 1 using the fact that φ is of (K O , ρ ′ )-type.
Remark 3.2. Suppose that φ ∈ V π is an automorphic form satisfying (35) and suppose that K ′ is a compact open subgroup that acts on φ by a character. Taking O = O max in Corollary 3.1 then gives us the trivial bound: which is a mild extension of (2). On the other hand, suppose that π has trivial central character and is spherical at all p|d. Denote the conductor of π by C and let C 1 be the smallest integer such that C|C 2 1 . Suppose that φ ∈ V π has the property that some G(A f ) translate of it is fixed by the "principal congruence subgroup" K O max (C 1 ) (see (7) for the definition). Then by the results of [19, p. 96-97], the action of K O on φ generates a representation of dimension ≤ C 1 p|C (1 + p −1 ), and so by Corollary 3.1: which is a sub-local bound for sup-norms of "automorphic forms of p-adic microlocal type". Remark 3.3. Minimal vectors and p-adic microlocal lifts are analogues of each other, the only difference being that the former live in supercuspidal representations and the latter live in principal series representations. Both these classes of vectors may be viewed as special cases (in the p-adic setting) of the more general class of "localized" vectors. See also [16] for a discussion of localized vectors in the archimedean setting, where they are known as "microlocalized" vectors.
Finally, we discuss what sort of bound Theorem 1 gives us for newforms. We obtain the following general result: Theorem 3. Let π = ⊗ v π v be an irreducible, unitary, cuspidal automorphic representation of G(A) with conductor C. Let M be the conductor of the central character of π. Let φ in the space of π be a global newform, i.e., φ = ⊗ v φ v with φ p spherical if p ∤ C, φ p equal to the local newvector for p|C, and φ ∞ a vector of smallest non-negative weight. Then we have Proof. For any integer N , let O 0 (N ) denote the Eichler order of level N . We first apply Theorem 1 with O = O 0 (C). Since φ transforms by character under the action of K O 0 (C) , we obtain that Next, we apply the theorem with O = O 0 (C ′ ) where C ′ = C 2 1 /C is the squarefree integer obtained by taking the product of all primes which divide C to an odd power. Then, it was shown in [17,Sec. 2.7] that the action of K O on (a suitable right-translate of) φ generates a representation of dimension ≪ lcm(M,C 1 ) C ′ . Now applying Theorem 1 (and using the fact that right-translating does not change the sup-norm), we get that sup g∈G(A) |φ(g)| ≪ D,π∞,ǫ C ǫ (C ′ ) −1/24 lcm(M, C 1 ) 1/2 ). This completes the proof. Theorem 3 generalizes all currently known bounds for the supnorms of newforms on G(A), and its proof clearly demonstrates the flexibility of Theorem 1. Of particular interest is the fact we are able to get a good upper bound in the case of highly ramified central characters. For example, if M = C and is equal to a perfect square, then the above Theorem gives φ ∞ ≪ D,π∞,ǫ C 1/3+ǫ . As a point of comparison, the analogous bound obtained in [17] for automorphic forms on GL 2 with M = C and C a square, was φ ∞ ≪ D,π∞,ǫ C 1/2+ǫ . Note however, that when the central character of π is trivial, then Theorem 3 reduces to where C ′ = C 2 1 /C is the squarefree integer obtained by taking the product of all primes which divide C to an odd power. This bound (42) fails to improve upon the local bound (8) for any family of newforms φ ∈ V π for which C ′ C → 0. In particular, a key outstanding case concerns the problem of beating the local bound for newforms 22 of trivial central character in the depth aspect C = p n , n → ∞. This case will be treated in future work.
3.6. Preparations for the proof. We now begin the proof of Theorem 1. The main part of the proof will be completed in Section 4 (which will crucially rely on the counting results from Section 2). In this subsection, we make a few simple but key observations, which will allow us to impose additional hypotheses without any loss of generality.
First of all, the property of being of minimal weight at the archimedean place, strictly speaking, depends on the local isomorphism ι ∞ which has been fixed by us. However, a different choice of ι ∞ simply corresponds to replacing φ by a G ∞translate of it, and (by definition) the sup-norm of this translated form coincides with the sup-norm of φ. Therefore, fixing ι ∞ does not change the sup-norm. Now we fix, once and for all, a compact fundamental domain J for the action of on H. Any element of G(A) can be left-multiplied by a suitable element of Z(A)G(Q) and right-multiplied by a suitable element of K O max to get an element g ∞ ∈ G ∞ such that det(ι ∞ (g ∞ )) > 0 and ι ∞ (g ∞ )(i) lies in J . Since |φ(g)| is Z(A)G(Q) invariant, we may assume, for the purposes of proving (36), that g = v g v satisfies (43) g p ∈ K p for all p ∈ f , det(ι ∞ (g ∞ )) > 0, and ι ∞ (g ∞ )(i) ∈ J , where J is our fixed compact set. Secondly, suppose that O ′ ⊆ O max is any order in the same genus as O. So there exists h ∈ G(A f ) such that h · O = O ′ (recall the notations from Section 3.2). Clearly O ′ has the same level as O. Let the finite dimensional representation ρ ′ of K O ′ be defined via ρ ′ (k) = ρ(h −1 kh) (So ρ and ρ ′ are isomorphic). Now define the automorphic form φ ′ = π(h)φ. Then φ ′ is of (K O ′ , ρ ′ ) type and of minimal weight at the archimedean place. Further, it has the same sup-norm as φ, being a translate. So it suffices to prove the Theorem for φ ′ (which allows us to change the order from O to O ′ ). However, a very useful algebraic fact, that we will prove below in Section 3.7, is that each genus of orders contains an order with shape (M 1 , M 2 , M 3 ) and level N such that M 3 |N 1 . So, for the purpose of proving Theorem 1, we can and will assume the following: Thirdly, we may assume, for the purpose of proving Theorem 1, that (45) ρ is irreducible.
Indeed, suppose we have proved the Theorem under (45). Now for the general case, we simply write φ as an orthogonal sum of automorphic forms φ i , each of which generates an irreducible representation ρ i under the action of K O . Now apply the already proved result to each φ i , follow it by the triangle inequality and then Cauchy Schwartz, to obtain the desired result for φ (using the fact that dim(ρ) = i dim(ρ i )).

3.7.
A result on balanced representatives for orders.
Definition 3.4. Given a pair of lattices L 1 , L 2 in D such that L 1 ⊆ L 2 ,
(2) L 1 is balanced in L 2 if the invariant factors (a 1 , a 2 , a 3 , a 4 ) have the following property: If t 1 denotes the smallest integer such that a 1 a 2 a 3 a 4 divides t 2 1 , then a 4 divides t 1 .
Note that if L 1 and L 2 are orders, then the smallest invariant factor a 1 equals 1. The object of this subsection is to prove the following result, which was used in the previous subsection to show that we can always assume (44) without any loss of generality. Now let us go back to (56). We will prove two bounds. For the first, we choose Λ = 1 2 C 1/4 N 1/12 and apply Proposition 2.14 to (56). (Note here that g · O has the same level N as O). This gives us (57) |φ(g)| ≪ π∞,ǫ dim(ρ) 1/2 N 11 24 +ǫ . For the second bound, we apply Proposition 2.8 to (56). By the assumption (43), the orders O and g · O have the same shape, which we denote by (M 1 , M 2 , M 3 ). Furthermore, by (44), M 1 M 2 ≫ N/N 1 . Now, applying Proposition 2.14 to (56), we get |φ(g)| 2 ≪ π∞,ǫ dim(ρ) N 1+ǫ Λ −1 + Λ 2 N + N 1 N .