Supnorms of eigenfunctions in the level aspect for compact arithmetic surfaces
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Abstract
Let D be an indefinite quaternion division algebra over \({{\mathbb {Q}}}\). We approach the problem of bounding the supnorms 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 nontrivial upper bound for \(\Vert \phi \Vert _\infty \) in the level aspect that is valid for arbitrary orders. This generalizes and strengthens previously known upper bounds for \(\Vert \phi \Vert _\infty \) 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 \(\Vert \phi \Vert _\infty \ll _{\pi _\infty , \epsilon } N^{1/3 + \epsilon } \Vert \phi \Vert _2\). 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 et al. (Commun Math Helv, to appear) or padic microlocal lifts, in the sense of Nelson in “Microlocal lifts and and quantum unique ergodicity on \(\mathrm{GL}_2({{\mathbb {Q}}}_{p})\)” (Algebra Number Theory 12(9):2033–2064, 2018). For such forms, our bound specializes to \(\Vert \phi \Vert _\infty \ll _{ \epsilon } C^{\frac{1}{6} + \epsilon }\Vert \phi \Vert _2\) where C is the conductor of the representation \(\pi \) generated by \(\phi \). This improves upon the previously known local bound\(\Vert \phi \Vert _\infty \ll _{\lambda , \epsilon } C^{\frac{1}{4} + \epsilon }\Vert \phi \Vert _2\) in these cases.
1 Introduction
1.1 Background
Let \(\phi =\otimes _v \phi _v\) be a cuspidal automorphic form on \(D^\times ({{\mathbb {A}}})\) where D is an indefinite quaternion algebra over \({{\mathbb {Q}}}\). The supnorm problem, which has seen a lot of interest recently, is concerned with bounding \(\frac{\Vert \phi \Vert _\infty }{\Vert \phi \Vert _2}\) in terms of natural parameters of \(\phi \). 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 \(\phi \) (with the dependance on the archimedean parameters suppressed), this is known as the levelaspect supnorm problem.
1.2 The main result
As is clear from the earlier discussion, previous work on this topic has been focussed on the case where \(\phi \) is a newform and \({{\mathcal {O}}}\) is an Eichler order with level \(N_{{\mathcal {O}}}\) equal to the conductor of \(\pi \). 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 [9, 10, 11, 12]; such forms transform naturally with respect to characters of unit groups of certain nonEichler Bass orders. The aim of this paper is to prove for the first time a nontrivial upper bound for the supnorm in the case of general orders.
Theorem A
For squarefree N, our Theorem implies that \(\Vert \phi \Vert _\infty \ll _{D, \pi _\infty , \epsilon } N^{11/24 + \epsilon }\Vert \phi \Vert _2\). 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 “Weyltype” exponent.
Corollary
For an explanation why we get such different exponents for squarefree and squarefull levels, see Sect. 1.5 of this introduction. An interesting fact is that this squarefree/squarefull dichotomy in the supnorm bound is also present in the case of newforms on \({\text {GL}}_2\) (see Sect. 1.3 of [17]), but for quite 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 \(\phi \) under the action of \(K_{{\mathcal {O}}}\) is onedimensional.
1.3 A classical reformulation
In this subsection, we reformulate Theorem A in the language of Hecke–Laplace eigenfunctions on the upperhalf 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 \(\iota _\infty : D \otimes _{{\mathbb {Q}}}{{\mathbb {R}}}\simeq M_2({{\mathbb {R}}})\). This leads to an embedding \(D \hookrightarrow M_2({{\mathbb {R}}})\) which we also denote by \(\iota _\infty \).
 (1)
\(f_\phi \in C^\infty (\Gamma _{{\mathcal {O}}}\backslash {\mathbb {H}}, \chi ^{1})\).
 (2)
\(f_\phi \) satisfies \((\Delta + \lambda ) f_\phi = 0\) where \(\Delta := y^{2} (\partial _x^2 + \partial _y^2)\).
 (3)
\(f_\phi \) is a simultaneous eigenfunction of the Hecke operators \(T_n\) for all positive integers n with \((n,dN_\chi )=1\).
Remark 1.2
In fact, every Maass form f for \(\Gamma _{{\mathcal {O}}}\) with character \(\chi ^{1}\) and Laplace eigenvalue \(\lambda \) that is a Hecke eigenform at the good primes, can be obtained in the above way, i.e., \(f=f_\phi \) for some \(\phi \) 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 \(\pi \) has trivial central character. We compare Theorem A with the local bound in the level aspect for automorphic forms \(\phi \) inside automorphic representations \(\pi \) of \(D^\times ({{\mathbb {A}}})\). By the local bound for \(\phi \), we mean the immediate bound emerging from the adelic pretrace 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 \(\phi _p\). 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 \(\pi \).
Remark 1.3
The quantity \(C_1\) is equal to \(\mathrm{cond}(\pi \times {\tilde{\pi }})^{1/2}\). One reason that \(C_1=\mathrm{cond}(\pi \times {\tilde{\pi }})^{1/2}\) shows up in (8) is that (when \(\pi \) is discrete series) \(C_1\) approximately equals the formal degree of \(\pi \); 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 \(\phi \) 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 noncompact setting of automorphic forms on \({\text {GL}}_2({{\mathbb {A}}})\), provided one restricts the domain of \(\phi \) 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 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 \(\phi \). 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 \(\phi _p\) for each ramified prime p is an eigenvector of a relatively large neighbourhood of the identity. A key class of \(\phi _p\) for which this is true is the family of minimal vectors. Minimal vectors may be viewed as padic 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 \({\text {GL}}_2\)). The main result of [11] proved an optimal supnorm bound for such forms in the setting of \({\text {GL}}_2\).
However, the techniques used in [11] relied on a very special property of the Whittaker/Fourier expansion of \(\phi \) which only works in the noncompact 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
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 \(D^\times \) where D is a (split or division) quaternion algebra over a padic field with \(p\ne 2\). 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 onethird 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 \(C^\epsilon \). Theorem B therefore gives a Weyltype 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 sublocal bound in certain other settings. These other settings include the case of “padic 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 sublocal bound, see (43). We also obtain a bound for newforms that generalizes and strengthens previously known results; see Theorem 3.^{7}
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 sublocal bound in the depth aspect for automorphic forms of minimal type on \({\text {GL}}_n\) under the assumption that the corresponding local representations have “generic” induction datum.
1.5 Key ideas
Above, z is any point on the upperhalf plane \({\mathbb {H}}\). Note, however, that for any \(\gamma \in \Gamma _{{{\mathcal {O}}}^\mathrm{max}}\), we have \(N_{{\mathcal {O}}}(m;z) = N_{{{\mathcal {O}}}'}(m;\gamma z)\), for an order \({{\mathcal {O}}}'\) that is conjugate to \({{\mathcal {O}}}\) by an element of \(({{\mathcal {O}}}^\mathrm{max})^\times \). This allows us to move z to a fixed compact set\({{\mathcal {J}}}\), namely \({{\mathcal {J}}}\) equal to some choice of fundamental domain for the action of \(\Gamma _{{{\mathcal {O}}}^\mathrm{max}}\) on \({\mathbb {H}}\), at the cost of changing the order \({{\mathcal {O}}}\) to a suitable \(({{\mathcal {O}}}^\mathrm{max})^\times \)conjugate of it. Now suppose that for each m and \(z \in {{\mathcal {J}}}\), we are able to prove a bound for \(N_{{\mathcal {O}}}(m;z)\) that depends only on m, \({{\mathcal {J}}}\), and the \(({{\mathcal {O}}}^\mathrm{max})^\times \)conjugacy class of \({{\mathcal {O}}}\). Then we have actually proved a bound that is valid for all \(z\in {\mathbb {H}}\). This reduction is a key piece in our strategy and can be viewed as a workaround for the situation when \(\Gamma _{{\mathcal {O}}}\) is not a normal subgroup of \(\Gamma _{{{\mathcal {O}}}^\mathrm{max}}\) (c.f. the comments in Sect. 1.3 of [21]).
In fact we prove two separate bounds for \(N_{{\mathcal {O}}}(m;z)\) for \(z \in {{\mathcal {J}}}\). The primary one among them (Proposition 2.8) is valid for general lattices\({{\mathcal {O}}}\) (and does not use the multiplicative structure of the order at all!). The analysis behind the proof of this proposition, carried out in Sects. 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 \(({{\mathcal {O}}}^\mathrm{max})^\times \)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 \({{\mathcal {O}}}^\mathrm{max}\) (the invariant factors are the same for all \(({{\mathcal {O}}}^\mathrm{max})^\times \)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 supnorm of an automorphic form \(\phi \) does not change when the form is replaced by some \(D^\times ({{\mathbb {A}}})\)translate of it. Now, given \(\phi \) as in Theorem A, a \(D^\times ({{\mathbb {A}}})\)translate of \(\phi \) transforms by a character of an order \({{\mathcal {O}}}'\) that is locally isomorphic to (in the same genus as) the order \({{\mathcal {O}}}\) 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 casebycase analysis (see Sect. 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 nontrivial supnorm bound are those whose levels are close to squarefree. To deal with this remaining case, we follow Templier’s method [20, 21] 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 \(\max (N^{1/3}, N_1^{1/2})\) 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 nontrivial 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 Sect. 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 [10, 11]. This uses crucially the fact that minimal vectors in supercuspidal representations \(\pi _p\) are eigenvectors for the action of a relatively large subgroup (having volume around \(\mathrm{cond}(\pi _p)^{1/2}\)). In contrast, newvectors in \(\pi \) behave well only under the action of a much smaller subgroup (having volume around \(\mathrm{cond}(\pi _p)^{1}\)). 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 lost—it 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, which will be published in a sequel to this paper. Our method there will allow us to beat the local bound for newforms in the depth aspect for the first time.
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.
Lemma 2.1
 (1)
\(0 \le p_i \le C_{c,\epsilon } N^\epsilon \) for \(1 \le i \le n\),
 (2)
\(\gcd (a_0 + a_1p_1 + a_2p_2 + \cdots +a_np_n, N)=1\).
Remark 2.2
With more sophisticated sieving methods a la [13], one can replace \(N^{\epsilon }\) by \(\log (N)^2\) in the lemma above.
Remark 2.3
We encourage the reader to initially focus on the case \(n=1\) of the lemma above to get a feel for the statement. In this paper, we will need the lemma only in the case \(c=2\), \(n=2\).
Proof
We may assume without loss of generality that \(\gcd (a_0, a_1, \ldots , a_{n})=1\) and that N is squarefree. Indeed, if these conditions are not met, we can replace each \(a_i\) by \(a_i/d\) where \(d = \gcd (a_0, a_1, \ldots , a_{n})\) and we can replace N by its largest squarefree divisor, so that this modified setup does satisfy the conditions. Any constant \(C_{c,\epsilon }\) that works for this modified setup will also work for the original setup.
We now prove the induction step. Assume that the lemma is proved for \(n=k\). We will prove the lemma for \(n=k+1\). Suppose we have integers \(a_0, a_1, \ldots , a_{k+1}, N\), with \(\gcd (a_0, a_1, \ldots , a_{k+1})=1\). By replacing \(a_{k+1}\) by its residue modulo N if necessary, we may assume that \(0 \le a_{k+1} \le N\).

For each \((p_1, \ldots , p_k) \in S_k\), we have \(0 \le p_i \le C_{c,\epsilon } N^\epsilon \) and \(\gcd (a_0 + a_1p_1 + \cdots +a_kp_k, a_{k+1})=1\).

For each nonempty subset I of \(\{1, \ldots , k\}\), \(P_I(S) \ge c^{I}\).
2.2 Lattices in quaternion orders
Let D be an indefinite quaternion division algebra over \({{\mathbb {Q}}}\). 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 once and for all a maximal order \({{\mathcal {O}}}^\mathrm{max}\) of D, and an isomorphism^{8}\(\iota _\infty : D_\infty \rightarrow M(2, {{\mathbb {R}}})\).
Definition 2.4
 (1)
\({{\mathcal {O}}}^\mathrm{max}_0 = {{\mathbb {Z}}}\Delta _1 \oplus {{\mathbb {Z}}}\Delta _2 \oplus {{\mathbb {Z}}}\Delta _3\),
 (2)
\({{\mathcal {L}}}_0 = M_1{{\mathbb {Z}}}\Delta _1 \oplus M_2{{\mathbb {Z}}}\Delta _2\oplus M_3{{\mathbb {Z}}}\Delta _3\).
Remark 2.5
Let \({{\mathcal {L}}}\subseteq {{\mathcal {O}}}^\mathrm{max}\) be a lattice of shape \((M_1,M_2, M_3)\) and level N. If \(x\in D\) satisfies \(x {{\mathcal {L}}}x^{1} \subseteq {{\mathcal {O}}}^\mathrm{max}\), then one may ask about the shape and level of \({{\mathcal {L}}}':= x {{\mathcal {L}}}x^{1}\).
It is easy to see that \({{\mathcal {L}}}'\) always has level N but its shape might be different in general. However, if \(x \in ({{\mathcal {O}}}^\mathrm{max})^\times \), then \({{\mathcal {L}}}'\) also has shape \((M_1,M_2, M_3)\).
It turns out to be more convenient for us to descend to the sublattice \({{\mathbb {Z}}}\oplus {{\mathcal {L}}}_0\), for which the next lemma is essential.
Lemma 2.6
Proof
Given an element \(\ell \in {{\mathcal {L}}}\), we have \(\mathrm{Tr}(\ell ) \in {{\mathbb {Z}}}\) and furthermore, \(\ell \) belongs to \({{\mathbb {Z}}}\oplus {{\mathcal {L}}}_0\) if and only if \(\mathrm{Tr}(\ell ) \in 2{{\mathbb {Z}}}\). So if \(\ell _1\) and \(\ell _2\) are two elements in \({{\mathcal {L}}}\), neither of which belong to \({{\mathbb {Z}}}\oplus {{\mathcal {L}}}_0\), then \(\ell _1 + \ell _2 \in {{\mathbb {Z}}}\oplus {{\mathcal {L}}}_0\). The statement follows. \(\square \)
Remark 2.7
We now state our main counting result.
Proposition 2.8
Remark 2.9
The constants implicit in the bounds above depend only on \(\epsilon , \delta \) and the fixed objects \(D, {{\mathcal {O}}}^\mathrm{max}, \iota _\infty , {{\mathcal {J}}}\).
Remark 2.10
Note that the bounds do not depend on the elements \(\Delta _1\), \(\Delta _2\), \(\Delta _3\). Hence the bounds obtained are uniform over all \(({{\mathcal {O}}}^\mathrm{max})^\times \)conjugates of \({{\mathcal {L}}}\). This will be key for us later on.
2.3 Proof of Proposition 2.8
Lemma 2.13
Proof
It remains to bound the number of distinct tuples of integers \((a_0, A_\alpha , B_\alpha , a_3)\) that are possible. To achieve that, we will prove certain bounds and congruences satisfied by such tuples. First of all, using (26) and the fact that \(r_2\), \(r_3\) are both \(\ll _\epsilon N^\epsilon \), it follows that \(A_\alpha  \ll _\epsilon L^{1/2}N^\epsilon \) and \(A_\alpha \equiv 0 \pmod {M_1}\). So there are \(\ll _\epsilon N^\epsilon \left( 1 + \frac{L^{1/2}}{M_1}\right) \) choices for \(A_\alpha \). Henceforth, assume such a choice has been made.
Finally, as \(a_0 \ll L^{1/2}\), there are \(\ll L^{1/2}\) possible choices for \(a_0\).
2.4 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 \({{\mathcal {L}}}={{\mathcal {O}}}\) is an order.
Proposition 2.14
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 Sect. 2.2 and we will assume (without loss of generality) that (16) holds.
Lemma 2.15
Let \({{\mathcal {L}}}\) be a subset of D that is closed under multiplication and contains 1. Let \(z\in {\mathbb {H}}\), L a positive integer, and \(\delta >0\). Then the \({{\mathbb {Q}}}\)algebra generated by all elements in \(\bigcup _{1 \le m \le L}{{\mathcal {L}}}(m;z, \delta )\) is contained in the \({{\mathbb {Q}}}\)vectorspace spanned by \(\bigcup _{1 \le m \le L^2}{{\mathcal {L}}}(m;z, 2\delta ).\)
Proof
Lemma 2.16
Let \({{\mathcal {L}}}\) be a lattice in D of level N, \(z\in {{\mathcal {J}}}\), L a positive integer, and \(\delta >0\). Then there is a constant C (depending on \(\delta \), \({{\mathcal {J}}}\) and \(\iota _\infty \)) such that the \({{\mathbb {Q}}}\)vectorspace spanned by \(\bigcup _{1 \le m \le L^2}{{\mathcal {L}}}(m;z, 2\delta )\) is proper whenever \(L < C N^{1/3}\).
Proof
Now let \({{\mathcal {O}}}\subseteq {{\mathcal {O}}}^\mathrm{max}\) be an order of level N. The above two lemmas imply that if \(1\le m <CN^{1/3}\), then all elements of \({{\mathcal {O}}}(m;z, \delta )\) lie in a quadratic field (since the only proper \(\mathbb {Q}\)algebras in a given quaternion algebra are \({{\mathbb {Q}}}\) and various embedded quadratic fields). Now the proof of Proposition 2.14 follows from the following lemma and the fact that \(d(m) \ll _\epsilon m^\epsilon \).
Lemma 2.17
Proof
Any element of \(F(m;z, \delta )\) is a product of a unit in \({{\mathcal {O}}}_F^\times \) of norm 1, and an element of \({{\mathcal {O}}}_F\) of norm m, with the latter taken from a fixed set of cardinality \(\ll d(m)\). 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 \(\kappa \in {{\mathcal {O}}}_F^\times \) satisfying \(u(z, \iota _\infty (\kappa )z) \le \delta \) is \(\ll _\delta 1\). This completes the proof. \(\square \)
3 The main result: statement and key applications
3.1 Basic notations
We continue to use the notations established in Sect. 2.2, and introduce some new ones below. Let \({{\mathbf {f}}}\) denote the finite places of \({{\mathbb {Q}}}\) (which we identify with the set of primes) and \(\infty \) the archimedean place. We let \({{\mathbb {A}}}\) denote the ring of adeles over \({{\mathbb {Q}}}\), and \({{\mathbb {A}}}_{{\mathbf {f}}}\) the ring of finite adeles. Given an algebraic group H defined over \({{\mathbb {Q}}}\), a place v of \({{\mathbb {Q}}}\), a subset of places U of \({{\mathbb {Q}}}\), and a positive integer M, we denote \(H_v:= H({{\mathbb {Q}}}_v)\), \(H_U:=\prod _{v \in U} H_v\), \(H_M :=\prod _{pM} H_p\). Given an element g in \(H({{\mathbb {Q}}})\) (resp., in \(H({{\mathbb {A}}})\)), we will use \(g_p\) to denote the image of g in \(H_p\) (resp., the pcomponent 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 \({{\mathbb {Q}}}\) with reduced discriminant d and that we have fixed a maximal order \(O^\mathrm{max}\). We denote \(G=D^\times \) and \(G' =PD^\times = D^\times /Z\) where Z denotes the center of \(D^\times \). For each prime p, let \(K_p=({{\mathcal {O}}}^\mathrm{max}_p)^\times \) and let \(K_p'\) denote the image of \(K_p\) in \(G'_p\). Thus, for pd, \(K'_p\) has index 2 in the compact group \(G'_p\).
For each place v that is not among the primes dividing d, fix an isomorphism \(\iota _v: D_v \xrightarrow {\cong } M(2,{{\mathbb {Q}}}_v)\). We assume that these isomorphisms are chosen such that for each finite prime \(p\not \mid d\), we have \(\iota _p({{\mathcal {O}}}_p) = M(2,{{\mathbb {Z}}}_p)\). It is well known that all such choices are conjugate to each other by some matrix in \({\text {GL}}_2({{\mathbb {Z}}}_p)\). By abuse of notation, we also use \(\iota _v\) to denote the composition map \(D({{\mathbb {Q}}}) \rightarrow D_v \rightarrow M(2,{{\mathbb {Q}}}_v)\).
We fix the Haar measure on each group \(G_p\) such that \({\text {vol}}(K_p)=1\). We fix Haar measures on \({{\mathbb {Q}}}_p^\times \) such that \({\text {vol}}({{\mathbb {Z}}}_p^\times )=1\). This gives us resulting Haar measures on each group \(G'_p\) such that \({\text {vol}}(K'_p)=1\). Fix any Haar measure on \(G_\infty \), and take the Haar measure on \({{\mathbb {R}}}^\times \) to be equal to \(\frac{dx}{x}\) where dx is the Lebesgue measure. This gives us a Haar measure on \(G'_\infty \). Take the measures on \(G({{\mathbb {A}}})\) and \(G'({{\mathbb {A}}})\) to be given by the product measure.
3.2 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].
3.3 Statement of main result
Let \(\pi = \otimes _v \pi _v\) be an irreducible, unitary, cuspidal automorphic representation of \(G({{\mathbb {A}}})\) where we identify \(V_\pi \) with a (unique) subspace of functions on \(G({{\mathbb {A}}})\), so that \(\pi (g)\) coincides with R(g) on that subspace. Given a compact open subgroup \(K' = \prod _{p\in {{\mathbf {f}}}} K'_p\) of \(K_{{{\mathcal {O}}}^\mathrm{max}}\) (where each \(K'_p\) is a subgroup of \(K_p\), with \(K'_p = K_p\) for almost all p) and a finite dimensional representation \(\rho \) of \(K'\), we say that an automorphic form \(\phi \in V_\pi \) is of \((K', \rho )\)type if the rightregular action of \(K'\) on \(\phi \) generates a representation isomorphic to \(\rho \). Observe that the existence of a form of \((K', \rho )\)type implies that the restrictions of \(\rho \) and \(\omega _\pi \) to the centre of \(K'\) must coincide.
We can now state our main theorem.
Theorem 1
Theorem A of the introduction is a special case of Theorem 1, where we take \(\rho \) to be a character. A key flexibility of Theorem 1 comes from the fact that given \(\phi \), one can optimise which order \({{\mathcal {O}}}\) to use depending on how much information one has about the dimensions of the representations \(\rho \) generated under the action of various \(K_{{\mathcal {O}}}\).
In certain cases, however, one may only know the dimension under the action of some \(K'\) that is not of the form \(K_{{\mathcal {O}}}\). In such cases one can still get a bound by working with any order \({{\mathcal {O}}}\) containing \(K'\). The following corollary makes this precise.
Corollary 3.1
Proof
Remark 3.2
3.4 The case of automorphic forms of minimal type
 (1)
If \(c_p \equiv 0 \pmod {4}\), then \(n_p = \frac{c_p}{2}\), \(d_p=0\).
 (2)
If \(c_p \equiv 2 \pmod {4}\), then \(n_p = \frac{c_p}{2}  1\), and \(d_p=1.\)
 (3)
If \(c_p \equiv 1 \pmod {2}\), then \(n_p = \frac{c_p}{2} + \frac{1}{2}\), \(d_p=0\).
In a recent work [10], Hu and Nelson extended the concept of a minimal vector to cases (2) and (3) above, so that now there is a welldefined notion of a minimal vector for all twistminimal supercuspidal representations of \({\text {GL}}_2({{\mathbb {Q}}}_p)\) for p odd. We remark here that the twistminimality is merely for convenience since the minimal vector in the general case is defined in terms of the twistminimal case. In principle, the case of \(p=2\) can also be dealt with similarly but in this case the computations get more technical and these have not been performed so far. The analogous vectors for the case of principal series representations has also been dealt with in separate work of Nelson [15]; in this case the relevant vectors are known as padic microlocal lifts.
Going back to the case of a twistminimal supercuspidal representation \(\pi _p\) of \({\text {GL}}_2({{\mathbb {Q}}}_p)\) for p odd, we define a “Type 2 minimal vector” as in [10]. If \(c_p \not \equiv 2 \pmod {4}\), then the relevant space is one dimensional and so any minimal vector is automatically of Type 2. In the case \(c_p \equiv 2 \pmod {4}\), the space of minimal vectors is pdimensional (except for the case \(c_p=2\), when the space is \(p1\) dimensional) and has a basis consisting of Type 2 minimal vectors.
A Type 2 minimal vector \(\phi _p\) in the space of \(\pi _p\) has the property that there exists an order \({{\mathcal {O}}}_p \in {{\mathcal {O}}}^\mathrm{max}_p\) of level \(p^{n_p}\) such that the action of \({{\mathcal {O}}}_p^\times \) on \(\phi _p\) generates an irreducible representation \(\rho _p\) of \({{\mathcal {O}}}_p^\times \) with dimension \(p^{d_p}\), except in the special case \(c_p=2\), in which the representation has dimension \(p1\). Now Theorem 1 leads to the following theorem.
Theorem 2

If pd, then \(\pi _p\) has a (nonzero) vector fixed by \(K_p\). (This implies that \(\pi _p\) is onedimensional for each pd.)

If \(p \not \mid d\), and the representation \(\pi _p\) of \(G_p \cong {\text {GL}}_2({{\mathbb {Q}}}_p)\) is ramified, then p is odd and \(\pi _p\) is a twistminimal supercuspidal representation with conductor \(p^{c_p}\).
Proof
Theorem 2 improves upon the local bound (8) except when \(\sqrt{C}\) is a squarefree integer (in which case we recover the local bound).
3.5 Bounds for padic microlocal lifts and for newforms
In fact, Theorem 1 implies sublocal bounds in the level aspect for certain families of automorphic forms in addition to the ones of minimal type described above. For example, consider the case where \(\pi \) has trivial central character and whose (awayfromd) conductor C equals \(N^4\) where \(N=\prod _p p^{n_p}\) is an odd integer. For two characters \(\chi _1\), \(\chi _2\) on \({{\mathbb {Q}}}_p^\times \), let \(\chi _1 \boxplus \chi _2\) denote the principal series representation on \({\text {GL}}_2({{\mathbb {Q}}}_p)\) that is unitarily induced from the corresponding representation of its Borel subgroup. Now assume that for each pN, \(\pi _p\) is of the form \(\chi _p \boxplus \chi _p^{1}\) with \(a(\chi _p)=2n_p\). Let \(\phi _p \in V_{\pi _p}\) at these primes p correspond to padic microlocal lifts in the sense of [15].
Remark 3.3
Minimal vectors and padic 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 padic 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 termed “microlocalized” vectors.
Finally, we discuss what sort of bound Theorem 1 gives us for newforms. We obtain the following general result:
Theorem 3
Proof
In particular, a key outstanding case concerns the problem of beating the local bound for newforms of trivial central character in the depth aspect \(C=p^n, \ n \rightarrow \infty \) where p is a fixed prime. This case will be treated in a sequel to this paper written with Y. Hu, where we will introduce a new technique in the setting described above (under an additional assumption that p is odd) which will enable us to replace the exponent 1 / 2 in (8) by the exponent 5 / 24 in this particular aspect.
3.6 Preparations for the proof
We now begin the proof of Theorem 1. The main part of the proof will be completed in Sect. 4 (which will crucially rely on the counting results from Sect. 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.
3.7 A result on balanced representatives for orders
Definition 3.4
 (1)The invariant factors of \({{\mathcal {L}}}_1\) in \({{\mathcal {L}}}_2\) are the unique quadruple of positive integers \((a_1,a_2,a_3,a_4)\) such that \(a_1a_2a_3a_4\) and$$\begin{aligned} {{\mathcal {L}}}_2 /{{\mathcal {L}}}_1 \simeq ({{\mathbb {Z}}}/ a_1{{\mathbb {Z}}}) \times ({{\mathbb {Z}}}/ a_2{{\mathbb {Z}}}) \times ({{\mathbb {Z}}}/ a_3{{\mathbb {Z}}}) \times ({{\mathbb {Z}}}/ a_4{{\mathbb {Z}}}).\end{aligned}$$
 (2)
\({{\mathcal {L}}}_1\) is balanced in \({{\mathcal {L}}}_2\) 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 \(a_1a_2a_3a_4\) divides \(t_1^2\), then \(a_4\) divides \(t_1\).
Note that if \({{\mathcal {L}}}_1\) and \({{\mathcal {L}}}_2\) are orders, then the smallest invariant factor \(a_1\) equals 1.
Remark 3.5
Let \({{\mathcal {O}}}\subseteq {{\mathcal {O}}}^\mathrm{max}\) be an order with shape \((M_1, M_2, M_3)\) and level N, and let \(N_1\) be the smallest integer such that \(NN_1^2\). Now, suppose that \({{\mathcal {O}}}\) is balanced in \({{\mathcal {O}}}^\mathrm{max}\). Then by Remark 2.7, we see that \(M_3N_1\). In particular assumption (47) holds.
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 (47) without any loss of generality.
Proposition 3.6
Let \({{\mathcal {O}}}\) be an order in D. Then there exists \(g \in G({{\mathbb {A}}}_{{\mathbf {f}}})\) such that \(g\cdot {{\mathcal {O}}}\) is balanced in \({{\mathcal {O}}}^\mathrm{max}\).
To prove the above Proposition, we first of all recall (see, e.g., [23, Chapter 24]) that the order \({{\mathcal {O}}}\) can be written as \({{\mathcal {O}}}= {{\mathbb {Z}}}+ f {{\mathcal {O}}}^{\mathrm{gor}}\) where \(f \in {{\mathbb {Z}}}\) and \({{\mathcal {O}}}^{\mathrm{gor}}\) is a Gorenstein order. If for some \(g \in G({{\mathbb {A}}}_{{\mathbf {f}}})\) we know that \(g \cdot {{\mathcal {O}}}^{\mathrm{gor}} \subseteq {{\mathcal {O}}}^\mathrm{max}\) and that the invariant factors of \(g \cdot {{\mathcal {O}}}^{\mathrm{gor}}\) in \({{\mathcal {O}}}^\mathrm{max}\) are \((1, a_2, a_3, a_4)\) then it easily follows that the invariant factors of \(g \cdot {{\mathcal {O}}}\) in \({{\mathcal {O}}}^\mathrm{max}\) are \((1, fa_2, fa_3, fa_4)\). In particular, \(g \cdot {{\mathcal {O}}}\) is balanced in \({{\mathcal {O}}}^\mathrm{max}\) whenever \(g \cdot {{\mathcal {O}}}^{\mathrm{gor}}\) is. So it suffices to prove Proposition 3.6 for Gorenstein orders.
Being Gorenstein is a local property. Now from the localglobal principle for orders (see Sect. 3.2), Proposition 3.6 follows from the next statement.
Proposition 3.7
 (1)
\({{\mathcal {O}}}'_p \simeq {{\mathcal {O}}}_p\),
 (2)
\({{\mathcal {O}}}'_p \subseteq {{\mathcal {O}}}^\mathrm{max}_p\),
 (3)If \((m_1, m_2, m_3)\) are the unique triple of nonnegative integers such that \(m_1 \le m_2 \le m_3\) and there is an isomorphism as \({{\mathbb {Z}}}_p\)modules$$\begin{aligned} {{\mathcal {O}}}^\mathrm{max}_p/{{\mathcal {O}}}'_p \simeq ({{\mathbb {Z}}}_p / p^{m_1}{{\mathbb {Z}}}_p) \times ({{\mathbb {Z}}}_p / p^{m_2}{{\mathbb {Z}}}_p) \times ({{\mathbb {Z}}}_p / p^{m_3}{{\mathbb {Z}}}_p), \text { then} \end{aligned}$$$$\begin{aligned} m_3 \le \left\lceil \frac{m_1+m_2+m_3}{2} \right\rceil \text { holds.} \end{aligned}$$(49)
We now prove Proposition 3.7. We rely heavily on the work of Brzezinski [6] who gives a complete list of Gorenstein orders (up to isomorphism) and their resolutions in terms of explicit linear combinations of generators of \({{\mathcal {O}}}^\mathrm{max}\). It is therefore easy (albeit tedious) to compute the triple \((m_1, m_2, m_3)\) for each order in his list (by bringing the corresponding matrices to Smith normal form). We do this and observe that most orders in his list already satisfy (49); the ones that aren’t can be conjugated by a simple element and made to satisfy it. We give the key details below, omitting some of the routine calculations.
First consider the case when \(D_p \simeq M_2({{\mathbb {Q}}}_p)\). Put Open image in new window , Open image in new window , Open image in new window . Note that \({{\mathcal {O}}}_p^\mathrm{max}= \langle 1, x_1, x_2, x_3 \rangle \). According to Prop. 5.4 of [6], \({{\mathcal {O}}}_p\) is isomorphic to one of the cases \((a)  (d_3')\) described there. We denote Open image in new window . We write down the required order \({{\mathcal {O}}}_p'\) in each case, using the notation from Proposition 5.4 of [6].
Case (a). In this case we take \({{\mathcal {O}}}_p' = r_n E_n^{(1)}r_n^{1}\).
Case (b). In this case we take \({{\mathcal {O}}}_p' = E_n^{(1)}\).
Case (c). In this case we take \({{\mathcal {O}}}_p' = E_n^{(0)}\).
Case (\(d_1\)). In this case we take \({{\mathcal {O}}}_p' = r_n E_{n,s}^{(1)}r_n^{1}\).
Case (\(d_2\)). In this case we take \({{\mathcal {O}}}_p' = E_{n,s}^{(1)}\).
Case (\(d_3\)). In this case we take \({{\mathcal {O}}}_p' = E_{n,s}^{(0)}\).
Case (\(d_3'\)). In this case we take \({{\mathcal {O}}}_p' = E_{2,s^+}^{(0)}\).
Next, we consider the case when pd, i.e, \(D_p\) is a division algebra. Let \(x_1\), \(x_2\), \(x_3\) be as in [6, (5.5)]. Then \({{\mathcal {O}}}_p^\mathrm{max}= \langle 1, x_1, x_2, x_3 \rangle \). According to Prop. 5.6 of [6], \({{\mathcal {O}}}_p\) is isomorphic to one of the cases \((a)  (c_2)\) described there.
Case (a). In this case we take \({{\mathcal {O}}}_p' = \Gamma _n^{(1)}\).
Case (b). In this case we take \({{\mathcal {O}}}_p' = \Gamma _n^{(0)}\).
Case (\(c_1\)). In this case we take \({{\mathcal {O}}}_p' =\Gamma _{n,s}^{(1)}\).
Case (\(c_2\)). In this case we take \({{\mathcal {O}}}_p' = \Gamma _{n,s}^{(0)}\).
In all cases above, the description of \({{\mathcal {O}}}_p'\) given in [6] provides an explicit \({{\mathbb {Z}}}_p\)basis for \({{\mathcal {O}}}_p'\) in terms of a \({{\mathbb {Z}}}_p\)basis for \({{\mathcal {O}}}^\mathrm{max}_p.\) We reduce the resulting matrix into Smith normal form via elementary operations, and observe that the invariant factors \((1, p^{m_1}, p^{m_2}, p^{m_3})\) of the resulting matrix always satisfies (49). (We remark here that in the case \((d_2)\) for \(p \not \mid d\), and the cases (b), \((c_1)\) for pd, the reference [6] gives five generators for \({{\mathcal {O}}}_p'\). However, once the generator matrix is brought into Smith normal form, we get exactly four nonzero rows; these correspond to a \({{\mathbb {Z}}}_p\)basis of the form required.)
This completes the proof of Proposition 3.7, and therefore of Proposition 3.6.
4 Amplification
4.1 Test functions
Our main tool is the amplification method. From the adelic point of view, amplification corresponds to an appropriate choice of test function \(\kappa \) on \(G({{\mathbb {A}}})\) which increases the contribution of the particular automorphic form \(\phi \) in the resulting pretrace formula. In this subsection, we describe this test function \(\kappa \) (which will depend on \(\phi \) and \({{\mathcal {O}}}\)) and note its key properties.
 (1)
S contains all primes dividing dN,
 (2)
If \(p \notin S\), then \(\rho _p\) is trivial.
Let \(\mathbf{ur}= {{\mathbf {f}}}{\setminus } S\) be the set of primes not in S. We will choose \(\kappa \) of the form \(\kappa = \kappa _S \kappa _\mathbf{ur}\ \kappa _\infty \).
Next, we consider the primes \(p \in \mathbf{ur}\). Note that \(\pi _p\) is unramified for each such prime (indeed for such p, \(\rho _p\) is trivial and hence \(\phi \) is \(K_p\)fixed). Let \({\mathcal {H}}_{\mathbf{ur}}\) be the set of all compactly supported functions on \(\prod _{p \in \mathbf{ur}}{\text {GL}}_2({{\mathbb {Q}}}_p)\) that are bi\({\text {GL}}_2({{\mathbb {Z}}}_p)\) invariant for each \(p \in \mathbf{ur}\) and transform under the action of the centre by \(\omega _{\pi }^{1}\). For each positive integer \(\ell \) satisfying \((\ell , S)=1\), define the functions \(\kappa _\ell \) in \({\mathcal {H}}_{\mathbf{ur}}\) as in Section 3.5 of [17]; these correspond to the usual Hecke operators \(T_\ell \).
For each positive integer m such that \((m,S)=1\), we let \(\lambda _\pi (m)\) be the coefficient of \(m^{s}\) in the Dirichlet series corresponding to \(L(s, \pi )\), where we normalize the Lfunction to have functional equation \(s \mapsto 1s\).
4.2 The automorphic kernel and spectral expansion
Definition 4.1
 (1)
\(\gamma _p \in {{\mathbb {Q}}}_p^\times {{\mathcal {L}}}_p(\ell )\) for all \(p \in {{\mathbf {f}}}\), where \({{\mathcal {L}}}_p(\ell ) = \{ \alpha \in {{\mathcal {L}}}_p: \mathrm{nr}(\alpha ) \in \ell {{\mathbb {Z}}}_p^\times \}\).
 (2)
\(\det (\iota _\infty (\gamma _\infty ))>0\), \(u(z, \iota _\infty (\gamma _\infty ) z) \le 1\).
4.3 The endgame
We can now wrap up the proof, beginning with a simple proposition that links it all back to Sect. 2.
Proposition 4.2
Proof
It is clear that any element of \({{\mathcal {L}}}(\ell ;z,1)\) satisfies the two conditions defining \(S(\ell , {{\mathcal {L}}}; z)\). Furthermore, if two elements \(\gamma _1\), \(\gamma _2\) in \({{\mathcal {L}}}(\ell ;z,1)\) represent the same class in \(G'({{\mathbb {Q}}})\), then putting \(t \gamma _1 = \gamma _2\), we obtain (taking norms) that \(t^2=1\) which means that \(\gamma _1 = \pm \gamma _2\). Therefore we get an injective map \(\pm 1 \backslash {{\mathcal {L}}}(\ell ;z,1) \rightarrow S(\ell , {{\mathcal {L}}}; z)\). To complete the proof, we need to show that this map is surjective. Let \(\gamma \in G({{\mathbb {Q}}})\) be an element whose image in \(G'({{\mathbb {Q}}})\) lies in \(S(\ell , {{\mathcal {L}}}; z)\). We need to prove that there exists \(t_0 \in {{\mathbb {Q}}}^\times \) such that \(t_0 \gamma \in {{\mathcal {L}}}(\ell ;z,1)\). By Definition 4.1, we can find for each prime p, an element \(t_p \in {{\mathbb {Q}}}_p^\times \) such that \(\gamma _p \in t_p {{\mathcal {L}}}_p(\ell )\) for all primes p. By strong approximation for \({{\mathbb {Q}}}\), we can choose \(t_0\in {{\mathbb {Q}}}\) such that \(t_0 t_p \in {{\mathbb {Z}}}_p^\times \) for all primes p. Now consider the element \(t_0 \gamma \). For each prime p, we have that the pcomponent of \(t_0\gamma \) lies in \({{\mathcal {L}}}_p\). So by (34), we have \(t_0 \gamma \in {{\mathcal {L}}}\). Furthermore, we have that the pcomponent of \(\mathrm{nr}(t_0 \gamma )\) lies in \(\ell {{\mathbb {Z}}}_p^\times \) for all primes p, and hence \(\mathrm{nr}(t_0 \gamma ) = \pm \ell \). But by assumption \(\mathrm{nr}(\gamma )>0\). Hence \(\mathrm{nr}(t_0 \gamma ) = \ell \). It follows that \(t_0 \gamma \in {{\mathcal {L}}}(\ell )\). Since \(u(z, \iota _\infty (\gamma ) z) \le 1\), it is now immediate that \(t_0 \gamma \in {{\mathcal {L}}}(\ell ;z,1)\). \(\square \)
Footnotes
 1.
Technically, we need to consider the adelic counterpart of this subgroup.
 2.
As usual, we use the notation \(A \ll _{x,y,\ldots } B\) to signify that there exists a positive constant C, depending at most upon \(x,y,\ldots \), so that \(A \le C B\).
 3.
All our characters are assumed to be continuous.
 4.
This assumption on \(\phi _\infty \) is merely for convenience; our result can be stated without this assumption but then the implied constant will depend on \(\phi _\infty \).
 5.
The condition is that for some \(g \in D^\times ({{\mathbb {A}}}_{{\mathbf {f}}})\) we have \(\int _{gK_{{{\mathcal {O}}}^\mathrm{max}}g^{1}} \langle \pi (h)\phi , \phi \rangle ^2 dh \gg _\epsilon C_1^{1\epsilon } \langle \phi , \phi \rangle ^2\). This is a mild technical condition that is satisfied by several families of automorphic forms, including newforms, automorphic forms corresponding to minimal vectors, padic microlocal lifts, and so on. For a more downtoearth but slightly stronger condition, see Remark 3.2.
 6.
To be fair, not a lot of evidence exists for this conjecture beyond the fact that it the best possible bound one can hope to prove, and no theoretical obstructions to achieving it have been found.
 7.
A much stronger bound in the setting of newforms of trivial character in the depth aspect will appear in a sequel to this paper.
 8.
Such an isomorphism \(\iota _\infty \) is unique up to conjugation by \({\text {GL}}_2({{\mathbb {R}}})\).
 9.
Later in this paper, we will take \({{\mathcal {J}}}\) to be a fundamental domain for the action of \(\iota _\infty ({{\mathcal {O}}}^\mathrm{max}(1))\) on \({\mathbb {H}}\).
Notes
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 2.1. I would also like to thank the anonymous referee for his suggestion to rephrase the argument of Sect. 2.3 in terms of matrix manipulations and for other suggestions which have improved the readability of this paper.
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