The Poincaré upper half-plane (with a boundary consisting of the real line together with the point at infinity ) carries an action of the projective special linear group
Here and in the rest of the post we will abuse notation by identifying elements of the special linear group with their equivalence class in ; this will occasionally create or remove a factor of two in our formulae, but otherwise has very little effect, though one has to check that various definitions and expressions (such as (1)) are unaffected if one replaces a matrix by its negation . In particular, we recommend that the reader ignore the signs that appear from time to time in the discussion below.
As the action of on is transitive, and any given point in (e.g. ) has a stabiliser isomorphic to the projective rotation group , we can view the Poincaré upper half-plane as a homogeneous space for , and more specifically the quotient space of of a maximal compact subgroup . In fact, we can make the half-plane a symmetric space for , by endowing with the Riemannian metric
(using Cartesian coordinates ), which is invariant with respect to the action. Like any other Riemannian metric, the metric on generates a number of other important geometric objects on , such as the distance function which can be computed to be given by the formula
the volume measure , which can be computed to be
and the Laplace-Beltrami operator, which can be computed to be (here we use the negative definite sign convention for ). As the metric was -invariant, all of these quantities arising from the metric are similarly -invariant in the appropriate sense.
The Gauss curvature of the Poincaré half-plane can be computed to be the constant , thus is a model for two-dimensional hyperbolic geometry, in much the same way that the unit sphere in is a model for two-dimensional spherical geometry (or is a model for two-dimensional Euclidean geometry). (Indeed, is isomorphic (via projection to a null hyperplane) to the upper unit hyperboloid in the Minkowski spacetime , which is the direct analogue of the unit sphere in Euclidean spacetime or the plane in Galilean spacetime .)
One can inject arithmetic into this geometric structure by passing from the Lie group to the full modular group
or congruence subgroups such as
These are discrete subgroups of , nested by the subgroup inclusions
There are many further discrete subgroups of (known collectively as Fuchsian groups) that one could consider, but we will focus attention on these three groups in this post.
Any discrete subgroup of generates a quotient space , which in general will be a non-compact two-dimensional orbifold. One can understand such a quotient space by working with a fundamental domain – a set consisting of a single representative of each of the orbits of in . This fundamental domain is by no means uniquely defined, but if the fundamental domain is chosen with some reasonable amount of regularity, one can view as the fundamental domain with the boundaries glued together in an appropriate sense. Among other things, fundamental domains can be used to induce a volume measure on from the volume measure on (restricted to a fundamental domain). By abuse of notation we will refer to both measures simply as when there is no chance of confusion.
For instance, a fundamental domain for is given (up to null sets) by the strip , with identifiable with the cylinder formed by gluing together the two sides of the strip. A fundamental domain for is famously given (again up to null sets) by an upper portion , with the left and right sides again glued to each other, and the left and right halves of the circular boundary glued to itself. A fundamental domain for can be formed by gluing together
copies of a fundamental domain for in a rather complicated but interesting fashion.
While fundamental domains can be a convenient choice of coordinates to work with for some computations (as well as for drawing appropriate pictures), it is geometrically more natural to avoid working explicitly on such domains, and instead work directly on the quotient spaces . In order to analyse functions on such orbifolds, it is convenient to lift such functions back up to and identify them with functions which are -automorphic in the sense that for all and . Such functions will be referred to as -automorphic forms, or automorphic forms for short (we always implicitly assume all such functions to be measurable). (Strictly speaking, these are the automorphic forms with trivial factor of automorphy; one can certainly consider other factors of automorphy, particularly when working with holomorphic modular forms, which corresponds to sections of a more non-trivial line bundle over than the trivial bundle that is implicitly present when analysing scalar functions . However, we will not discuss this (important) more general situation here.)
An important way to create a -automorphic form is to start with a non-automorphic function obeying suitable decay conditions (e.g. bounded with compact support will suffice) and form the Poincaré series defined by
which is clearly -automorphic. (One could equivalently write in place of here; there are good argument for both conventions, but I have ultimately decided to use the convention, which makes explicit computations a little neater at the cost of making the group actions work in the opposite order.) Thus we naturally see sums over associated with -automorphic forms. A little more generally, given a subgroup of and a -automorphic function of suitable decay, we can form a relative Poincaré series by
where is any fundamental domain for , that is to say a subset of consisting of exactly one representative for each right coset of . As is -automorphic, we see (if has suitable decay) that does not depend on the precise choice of fundamental domain, and is -automorphic. These operations are all compatible with each other, for instance . A key example of Poincaré series are the Eisenstein series, although there are of course many other Poincaré series one can consider by varying the test function .
for any function with sufficient decay, and any -automorphic function of reasonable growth (e.g. bounded and compact support, and bounded, will suffice). Note that is viewed as a function on on the left-hand side, and as a -automorphic function on on the right-hand side. More generally, one has
whenever are discrete subgroups of , is a -automorphic function with sufficient decay on , and is a -automorphic (and thus also -automorphic) function of reasonable growth. These identities will allow us to move fairly freely between the three domains , , and in our analysis.
When computing various statistics of a Poincaré series , such as its values at special points , or the quantity , expressions of interest to analytic number theory naturally emerge. We list three basic examples of this below, discussed somewhat informally in order to highlight the main ideas rather than the technical details.
where is the divisor function. This can be rewritten (by factoring and ) as
This sum is not exactly the same as (8), but will be a little easier to handle, and it is plausible that the methods used to handle this sum can be modified to handle (8). Observe from (2) and some calculation that the distance between and is given by the formula
and so one can express the above sum as
(the factor of coming from the quotient by in the projective special linear group); one can express this as , where and is the indicator function of the ball . Thus we see that expressions such as (7) are related to evaluations of Poincaré series. (In practice, it is much better to use smoothed out versions of indicator functions in order to obtain good control on sums such as (7) or (9), but we gloss over this technical detail here.)
At first glance this does not look like a sum over a modular group, but one can manipulate this expression into such a form in one of two (closely related) ways. First, observe that any factorisation of into Gaussian integers gives rise (upon taking norms) to an identity of the form , where and . Conversely, by using the unique factorisation of the Gaussian integers, every identity of the form gives rise to a factorisation of the form , essentially uniquely up to units. Now note that is of the form if and only if , in which case . Thus we can essentially write the above sum as something like
and one the modular group is now manifest. An equivalent way to see these manipulations is as follows. A triple of natural numbers with gives rise to a positive quadratic form of normalised discriminant equal to with integer coefficients (it is natural here to allow to take integer values rather than just natural number values by essentially doubling the sum). The group acts on the space of such quadratic forms in a natural fashion (by composing the quadratic form with the inverse of an element of ). Because the discriminant has class number one (this fact is equivalent to the unique factorisation of the gaussian integers, as discussed in this previous post), every form in this space is equivalent (under the action of some element of ) with the standard quadratic form . In other words, one has
which (up to a harmless sign) is exactly the representation , , introduced earlier, and leads to the same reformulation of the sum (10) in terms of expressions like (11). Similar considerations also apply if the quadratic polynomial is replaced by another quadratic, although one has to account for the fact that the class number may now exceed one (so that unique factorisation in the associated quadratic ring of integers breaks down), and in the positive discriminant case the fact that the group of units might be infinite presents another significant technical problem.
Note that has real part and imaginary part . Thus (11) is (up to a factor of two) the Poincaré series as in the preceding example, except that is now the indicator of the sector .
Sums involving subgroups of the full modular group, such as , often arise when imposing congruence conditions on sums such as (10), for instance when trying to estimate the expression when and are large. As before, one then soon arrives at the problem of evaluating a Poincaré series at one or more special points, where the series is now over rather than .
where is a natural number and is a -automorphic form that is of the form
To compute this, we use the double coset decomposition
where for each , are arbitrarily chosen integers such that . To see this decomposition, observe that every element in outside of can be assumed to have by applying a sign , and then using the row and column operations coming from left and right multiplication by (that is, shifting the top row by an integer multiple of the bottom row, and shifting the right column by an integer multiple of the left column) one can place in the interval and to be any specified integer pair with . From this we see that
The first integral is just . The second expression is more interesting. We have
so we can write
which on shifting by simplifies a little to
and then on scaling by simplifies a little further to
is a certain integral involving and a parameter , but which does not depend explicitly on parameters such as . Thus we have indeed expressed the expression (13) in terms of Kloosterman sums. It is possible to invert this analysis and express varius weighted sums of Kloosterman sums in terms of expressions (possibly involving inner products instead of norms) of Poincaré series, but we will not do so here; see Chapter 16 of Iwaniec and Kowalski for further details.
Traditionally, automorphic forms have been analysed using the spectral theory of the Laplace-Beltrami operator on spaces such as or , so that a Poincaré series such as might be expanded out using inner products of (or, by the unfolding identities, ) with various generalised eigenfunctions of (such as cuspidal eigenforms, or Eisenstein series). With this approach, special functions, and specifically the modified Bessel functions of the second kind, play a prominent role, basically because the -automorphic functions
for and non-zero are generalised eigenfunctions of (with eigenvalue ), and are almost square-integrable on (the norm diverges only logarithmically at one end of the cylinder , while decaying exponentially fast at the other end ).
However, as discussed in this previous post, the spectral theory of an essentially self-adjoint operator such as is basically equivalent to the theory of various solution operators associated to partial differential equations involving that operator, such as the Helmholtz equation , the heat equation , the Schrödinger equation , or the wave equation . Thus, one can hope to rephrase many arguments that involve spectral data of into arguments that instead involve resolvents , heat kernels , Schrödinger propagators , or wave propagators , or involve the PDE more directly (e.g. applying integration by parts and energy methods to solutions of such PDE). This is certainly done to some extent in the existing literature; resolvents and heat kernels, for instance, are often utilised. In this post, I would like to explore the possibility of reformulating spectral arguments instead using the inhomogeneous wave equation
This equation somewhat resembles a “Klein-Gordon” type equation, except that the mass is imaginary! This would lead to pathological behaviour were it not for the negative curvature, which in principle creates a spectral gap of that cancels out this factor.
The point is that the wave equation approach gives access to some nice PDE techniques, such as energy methods, Sobolev inequalities and finite speed of propagation, which are somewhat submerged in the spectral framework. The wave equation also interacts well with Poincaré series; if for instance and are -automorphic solutions to (15) obeying suitable decay conditions, then their Poincaré series and will be -automorphic solutions to the same equation (15), basically because the Laplace-Beltrami operator commutes with translations. Because of these facts, it is possible to replicate several standard spectral theory arguments in the wave equation framework, without having to deal directly with things like the asymptotics of modified Bessel functions. The wave equation approach to automorphic theory was introduced by Faddeev and Pavlov (using the Lax-Phillips scattering theory), and developed further by by Lax and Phillips, to recover many spectral facts about the Laplacian on modular curves, such as the Weyl law and the Selberg trace formula. Here, I will illustrate this by deriving three basic applications of automorphic methods in a wave equation framework, namely
- Using the Weil bound on Kloosterman sums to derive Selberg’s 3/16 theorem on the least non-trivial eigenvalue for on (discussed previously here);
- Conversely, showing that Selberg’s eigenvalue conjecture (improving Selberg’s bound to the optimal ) implies an optimal bound on (smoothed) sums of Kloosterman sums; and
- Using the same bound to obtain pointwise bounds on Poincaré series similar to the ones discussed above. (Actually, the argument here does not use the wave equation, instead it just uses the Sobolev inequality.)
This post originated from an attempt to finally learn this part of analytic number theory properly, and to see if I could use a PDE-based perspective to understand it better. Ultimately, this is not that dramatic a depature from the standard approach to this subject, but I found it useful to think of things in this fashion, probably due to my existing background in PDE.
I thank Bill Duke and Ben Green for helpful discussions. My primary reference for this theory was Chapters 15, 16, and 21 of Iwaniec and Kowalski.
— 1. Selberg’s theorem —
We begin with a proof of the following celebrated result of Selberg:
Theorem 1 Let be a natural number. Then every eigenvalue of on (the mean zero functions on ) is at least .
One can show that has only pure point spectrum below on (see this previous blog post for more discussion). Thus, this theorem shows that the spectrum of on is contained in .
We now prove this theorem. Suppose this were not the case, then we have a non-zero eigenfunction of in with eigenvalue for some ; we may assume to be real-valued, and by elliptic regularity it is smooth (on ). If it is constant in the horizontal variable, thus , then by the -automorphic nature of it is easy to see that is globally constant, contradicting the fact that it is mean zero but not identically zero. Thus it is not identically constant in the horizontal variable. By Fourier analysis on the cylinder , one can then find a -automorphic function of the form for some non-zero integer which has a non-zero inner product with on , where is a smooth compactly supported function.
to obtain a smooth function such that and ; the existence (and uniqueness) of such a solution to this initial value problem can be established by standard wave equation methods (e.g. parametrices and energy estimates), or by the spectral theory of the Laplacian. (One can also solve for explicitly in terms of modified Bessel functions, but we will not need to do so here, which is one of the main points of using the wave equation method.) Since the initial data obeyed the translation symmetry for all and , we see (from the uniqueness theory and translation invariance of the wave equation) that also obeys this symmetry; in particular is -automorphic for all times . By finite speed of propagation, remains compactly supported in for all time , in fact for positive time it will lie in the strip , where we allow the implied constants to depend on the initial data .
Taking the inner product of with the eigenfunction on , differentiating under the integral sign, and integrating by parts, we see that
Since is initially non-zero with zero velocity, we conclude from solving the ODE that is a non-zero multiple of . In particular, it grows like as . Using the unfolding identity (6) to write
as , where we allow implied constants to depend on .
We complement this lower bound with slightly crude upper bound in which we are willing to lose some powers of . We have already seen that is supported in the strip . Compactly supported solutions to (16) on the cylinder conserve the energy
In particular, this quantity is for all time (recall we allow implied constants to depend on ). From Hardy’s inequality, the quantity
is non-negative. Discarding this term and using , and using the fact that is non-zero, we arrive at the bounds
(We allow implied constants to depend on , but not on .) From the fundamental theorem of calculus and Minkowski’s inequality in , the latter inequality implies that
for , which on combining with the former inequality gives
The function also obeys the wave equation (16), so a similar argument gives
Applying a Sobolev inequality on unit squares (for ) or on squares of length comparable to (for ) we conclude the pointwise estimates
for . In particular, we write , we have the somewhat crude estimates
for all and . (One can do better than this, particularly for large , but this bound will suffice for us.)
By repeating the analysis of (13) at the start of this post, we see that the quantity
can be expressed as
Since is supported on and is bounded by , the integral is for . We also see that vanishes unless (otherwise and cannot simultaneously be , and for such values of , we have the triangle inequality bound
Evaluating the integral and then the integral, we arrive at
and so we can bound (18) (ignoring any potential cancellation in ) by
Now we use the Weil bound for Kloosterman sums, which gives
(see e.g. this previous post for a discussion of this bound) to arrive at the bound
as . Comparing this with (17) we obtain a contradiction as since we have , and the claim follows.
as for any fixed ; this, when combined with a more refined analysis of the above type, implies the Selberg eigenvalue conjecture that all eigenvalues of on are at least .
— 2. Consequences of Selberg’s conjecture —
In the previous section we saw how bounds on Kloosterman sums gave rise to lower bounds on eigenvalues of the Laplacian. It turns out that this implication is reversible. The simplest case (at least from the perspective of wave equation methods) is when Selberg’s eigenvalue conjecture is true, so that the Laplacian on has spectrum in . Equivalently, one has the inequality
for all , where the gradient and its magnitude are computed using the Riemannian metric in .
Now suppose one has a smooth, compactly supported in space solution to the inhomogeneous wave equation
for some forcing term which is also smooth and compactly supported in space. We assume that has mean zero for all . Introducing the energy
which is non-negative thanks to (19) and integrating by parts, we obtain the energy identity
and hence by Cauchy-Schwarz
(in a distributional sense at least), giving rise to the energy inequality
for some forcing term which is also smooth and compactly supported in space, with mean zero for all time, we have the energy inequality
One can use this inequality to analyse the norm of Poincaré series by testing on various functions (and working out using (20)). Suppose for instance that is a fixed natural number, and is a smooth compactly supported function. We consider the traveling wave given by the formula
where is the primitive of ; the point is that this is an approximate solution to the homogeneous wave equation, particularly at small values of . Clearly is compactly supported with mean zero for , in the region (we allow implied constants to depend on but not on ). In the region , and its first derivatives are , giving a contribution of to the energy (note that the shifts of the region by have bounded overlap). In particular we have
and thus by the energy inequality (using only the portion of the energy)
for , where
Clearly is supported on the region . For , one can compute that , giving a contribution of to the right-hand side. When is much less than but much larger than , we have , which after some calculation yields . As this decays so quickly as , one can compute (using for instance the expansion (14) of (13) and crude estimates, ignoring all cancellation) that this contributes a total of to the right-hand side also. Finally one has to deal with the region , but is much less than . Here, is equal to , and is equal to , which after some computation makes equal to . Again, one can compute the contribution of this term to the energy inequality to be . We conclude that
The expression is only non-zero when , and the integrand is only non-zero when and , which makes the phase of size . For much smaller than , the phase is thus largely irrelevant and the quantity is roughly comparable to for . As such, the bound (21) can be viewed as a smoothed out version of the estimate
which is basically Linnik’s conjecture, mentioned in Remark 2. One can make this connection between Selberg’s eigenvalue conjecture and Linnik’s conjecture more precise: see Section 16.6 of Iwaniec and Kowalski, which goes through modified Bessel functions rather than through wave equation methods.
— 3. Pointwise bounds on Poincaré series —
The formula (14) for (13) allows one to compute norms of Poincaré series. By using Sobolev embedding, one can then obtain pointwise control on such Poincaré series, as long as one stays away from the cusps. For instance, suppose we are interested in evaluating a Poincaré series at a point of the form for some . From the Sobolev inequality we have
for any smooth function , and thus by translation
The ball meets only boundedly many translates of the standard fundamental domain of , and hence does too. Since is a subgroup of , we conclude that meets only boundedly many translates of a fundamental domain for . In particular, we obtain the Sobolev inequality
for any smooth -automorphic function . This estimate is unfortunately a little inefficient when is large, since the ball has area comparable to one, whereas the quotient space has area roughly comparable to , so that one is conceding quite a bit by replacing the ball by the quotient space. Nevertheless this estimate is still useful enough to give some good results. We illustrate this by proving the estimate
for with coprime to , where is a fixed smooth function supported in, say, (and implied constants are allowed to depend on ), and the asymptotic notation is with regard to the limit . This type of estimate (appearing for instance (in a stronger form) in this paper of Duke, Friedlander, and Iwaniec; see also Proposition 21.10 of Iwaniec and Kowalski.) establishes some equidistribution of the square roots as varies (while staying comparable to ). For comparison, crude estimates (ignoring the cancellation in the phase ) give a bound of , so the bound (23) is non-trivial whenever is significantly smaller than . Estimates such as (23) are also useful for getting good error terms in the asymptotics for the expression (10), as was first done by Hooley.
One can write (23) in terms of Poincaré series much as was done for (10). Using the fact that the discriminant has class number one as before, we see that for every positive and with , we can find an element of such that has imaginary part and real part modulo one (thus, and ); this element is unique up to left translation by . We can thus write the left-hand side of (23) as
and are the bottom two entries of the matrix (determined up to sign). The condition implies (since must be coprime) that are coprime to with for some with ; conversely, if obey such a condition then . The number of such is at most . Thus it suffices to show that
for each such .
The constraint constrains to a single right coset of . Thus the left-hand side can be written as
which is just . Applying (22) (and interchanging the Poincaré series and the Laplacian), it thus suffices to show that
We can compute
By hypothesis, the coefficient is bounded, and so has all derivatives bounded while remaining supported in . Because of this, the arguments used to establish (24) can be adapted without difficulty to establish (25).
The quantity is vanishing unless . In that case, the integrand vanishes unless and , so by the triangle inequality we have . So the left-hand side of (26) is bounded by
By the Weil bound for Kloosterman sums, we have , so on factoring out from we can bound the previous expression by
and the claim follows.