Zeros of weakly holomorphic modular forms of levels 2 and 3
Let be the space of weakly holomorphic modular forms for that are holomorphic at all cusps except possibly at . We study a canonical basis for and and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.
2010 Mathematics Subject Classification:11F11, 11F03
In studying a complex-valued function, it is natural to attempt to locate the zeros of the function; in fact, one of the most famous unsolved problems in mathematics asks whether the nontrivial zeros of the Riemann zeta function lie on a particular line. In this paper, we study the locations of the zeros of certain modular forms, and show that most of their zeros in a fundamental domain occur on a particular circular arc.
For the Eisenstein series, perhaps the easiest examples of modular forms, a great deal is known about the locations of the zeros. In the 1960s, Wohlfahrt  showed that for even , all zeros of the Eisenstein series in the standard fundamental domain for lie on the unit circle , and R. A. Rankin  extended the range of values of for which this holds. Shortly afterward, F.K.C. Rankin and Swinnerton-Dyer  proved this result for all weights . R.A. Rankin  also obtained the result for certain Poincaré series, which generalize Eisenstein series. Similar results have been obtained for Eisenstein series for and by Miezaki, Nozaki, and Shigezumi , for Eisenstein series for and and for Poincaré series for and by Shigezumi [21, 22], and for a family of Eisenstein series for by the first author, Long, Swisher, and Treneer .
The above results which locate the zeros of Eisenstein series and Poincaré series use the same general idea of approximating the modular form by an elementary function having the required number of zeros on the arc. For example, the Eisenstein series may be written as a sum over an integer lattice. When is restricted to the unit circle, so that , the four main terms of this series combine to give . Rankin and Swinnerton-Dyer’s proof shows that the additional terms are small, so the zeros of are close to the zeros of this trigonometric function.
In 1997, Asai, Kaneko, and Ninomiya  used this idea to study the zeros of polynomials related to the modular function . The function is a Hauptmodul, or an isomorphism from the quotient of the upper half plane under the action of to the complex plane . It generates all modular functions on , and it also parameterizes the isomorphism classes of elliptic curves over . The image of under the Hecke operator is a polynomial in , which we write as . Letting be the modular discriminant, which is a weight 12 cusp form on with no zeros in , a generating function for the is given by
Using this generating function, Asai et al. approximated the polynomials by trigonometric functions well enough to prove that their zeros in the fundamental domain lie on the unit circle.
Duke and the second author  extended the results on to a two-parameter family of modular forms that form bases for spaces of weakly holomorphic modular forms of level 1. In this case, the connection to elementary functions is less direct. Cauchy’s integral formula relates the modular forms to a contour integral of a generalized version of the generating function (1.1). An application of the residue theorem produces the elementary functions, and the proof follows by bounding the integral over a range of values for and . The zeros again lie on the unit circle for many of the forms in the family, though in contrast to previous results, it is known that this property does not hold for all of the modular forms.
We mention one further result using a different technique by Hahn , who obtained general results on the zeros of Eisenstein series for genus zero Fuchsian groups; the general idea is an analogue of the classical argument that shows that the zeros of an orthogonal polynomial all lie on the real line.
This question of locating zeros of modular forms is made even more interesting by results of Rudnick  that showed that the zeros of Hecke eigenforms of weight , in a sense the orthogonal complement of the Eisenstein series, are expected to become equidistributed in the fundamental domain as ; this conjecture was proved by Holowinsky and Soundararajan  as a consequence of more general work on mass equidistribution for Hecke eigenforms. Ghosh and Sarnak  gave a lower bound for the density of zeros lying on certain arcs for such eigenforms. In a different direction, Basraoui and Sebbar  proved that the quasi-modular form has infinitely many zeros that are inequivalent under , and that none of these lie within the fundamental domain.
In this paper, we examine modular forms in a basis for certain spaces of weakly holomorphic modular forms of arbitrary integral weight and levels and . We show that for almost all of the basis elements, most of their zeros in a fundamental domain for or lie on a circular arc along the lower boundary of the fundamental domain. This is possible because we can again approximate these modular forms by elementary functions; however, the shape of the fundamental domain makes it difficult to accurately locate all of the zeros.
2. Definitions and statement of results
Let be the space of holomorphic modular forms of weight for the group , and let be the corresponding space of weakly holomorphic modular forms, or modular forms that are holomorphic on the upper half plane and meromorphic at the cusps. Let be the subspace of consisting of forms which are holomorphic away from the cusp at . This space appears, for instance, in , where it is shown that traces of negative integral weight forms in such a space appear as coefficients of certain half integral weight forms of level . Modular forms in have been studied by Ahlgren , who gave explicit formulas for the action of the -operator on forms in these spaces and obtained formulas for the exponents of their infinite product expansions, and by Andersen and the second author , who gave congruences for the coefficients of a basis for .
For the group , we use a fundamental domain in the upper half plane bounded by the lines and and the circles of radius centered at and . We include the boundary on the left half of this fundamental domain, which is equivalent to the opposite boundary under the action of the matrices and . The cusps of this fundamental domain may be taken to be at and at .
Modular forms in with real coefficients demonstrate a nice property on the lower boundary of this fundamental domain. For a Fourier series with real Fourier coefficients , note that , or . Thus, for modular forms of weight on , if we let for , so that is on the lower boundary of the (symmetric) fundamental domain, we find that . Therefore, , and the normalized modular form is real valued for between and .
It is useful to define three particular modular forms of level 2. As usual, let . Let
be the Hauptmodul for . This form has integer coefficients, has a pole at , and vanishes at . Moreover, by the above argument is real-valued on the lower boundary of the fundamental domain, taking on values in . The special value arises from the relationships between and given by
By using the fact that , it is easy to see that must be a common root of the polynomials and .
be the unique normalized holomorphic modular form of weight and level . Here is the weight 2 Eisenstein series . The form has integer coefficients and a single zero at the elliptic point . This can be seen by noting that at , so an application of the modular equation yields . Uniqueness comes from applying the valence formula for found in, for instance, . Additionally, we define the Eisenstein series as
it is easily checked that has integral Fourier coefficients and vanishes at . It does not vanish at the cusp at 0, as the valence formula shows that there are no cusp forms of weight and level .
We now use these forms to construct a basis for . For general even weight , we write , where . A basis for is given by
for all integers . We note that this is an extension of the basis given in  for ; similar sequences of modular forms for many levels appear in . The basis elements are constructed as follows. We first define and set . Next, for each we define inductively by multiplying by and subtracting off earlier basis elements. Note that since , , and have integral Fourier coefficients, each has integral Fourier coefficients and is of the form , where is a polynomial with integer coefficients of degree . (In fact, is a generalized Faber polynomial; see , .) Thus, if all of the zeros of the basis elements lie on the lower boundary of the fundamental domain, then all of the zeros of the polynomial must lie in the interval .
The main result of this paper is the following theorem.
Let be as above. If and , or if and , then at least of the nontrivial zeros of in the fundamental domain for lie on the lower boundary of the fundamental domain.
We note that the bounds and are not sharp, and that often many more of the zeros lie on the arc. In fact, for certain weights close to , all of the zeros of all of the are on the lower boundary of the fundamental domain. However, some restriction on in relation to is necessary, as there are also examples of with zeros elsewhere. We discuss this further in Section 6. Additionally, we obtain similar results for a family of modular forms in whose coefficients are dual to the Fourier coefficients of and for a basis for the space , showing that many of the zeros of these modular forms in the appropriate fundamental domain lie on the lower boundary.
The remainder of this paper proceeds as follows: in Section 3, we give a generating function for the basis elements and approximate their values on the lower boundary of the fundamental domain by a trigonometric function. In Section 4 we bound the error term to show that most of the zeros lie on the appropriate arc. Section 5 gives technical details on bounds for the error, and in Section 6 we discuss extensions of the main theorem to other modular forms for and to forms for .
3. Generating functions and integration
In this section we use Cauchy’s integral formula to relate the basis elements to a trigonometric function. Letting , a generating function for the basis elements is given in El-Guindy’s paper [8, Theorem 1.2] as
Multiply by and integrate around . Changing variables from to and noting that , we obtain
where is some real number larger than . We move the contour downward, noting that there is a pole whenever is zero, which happens when is equivalent to under the action of . There are no other poles, since has no zeros in the upper half plane, and if , the zero of is canceled by the zero of . The closed contour that is the difference between the old integral and the new integral is in the clockwise direction, so we get a factor of in Cauchy’s integral theorem, and at each pole, we obtain a term of times the residue of the integrand.
If a function has a zero of order at , its logarithmic derivative has a simple pole with residue . In calculating the residue, note that part of the integrand is the logarithmic derivative of , which has a simple zero exactly at the values we are looking at, since is a Hauptmodul for . This means that the logarithmic derivative of at points equivalent to under will just give us a factor of in the residue. Supposing that for some , this is then multiplied by
Since the denominator is a modular form of weight on , we have for , and the residue becomes .
Assume that is on the lower boundary of the fundamental domain for . The first two points where is equivalent to through which the contour moves are and . Calculating the residues, we find that
where is a contour that moves from left to right across the fundamental domain and passes below the points and and above all other points equivalent to under the action of .
We write for some , so that ; then the quantity can be simplified to
Putting all of this together and multiplying through by , we end up with
By the argument in Section 2, the left hand side is a real-valued function of . We note that the cosine function takes on alternating values of whenever is equal to for . Since this quantity moves from at to at , we know that there must be at least times that this happens. Thus, if we bound the integral term in absolute value by , then by the Intermediate Value Theorem we must have at least zeros of the modular form on this arc.
The dimension of the space of holomorphic modular forms is , and we get at most zeros not at elliptic points for these forms by the general valence formula in . (Note that has a zero at , which is already on the arc in question, so has a trivial zero there if .) Counting a pole of order at and no other poles gives us a total of zeros of the basis element whose locations are unknown. This argument proves that if the weighted modular form is close enough to the cosine function, then all of these zeros must be simple and must be on this arc on the lower boundary of the fundamental domain.
Unfortunately, it is difficult to move the contour down far enough to prove that all of the zeros are on this arc; as the weight or the order of the pole increases, the contour will need to get closer and closer to if is close to . Recall that for every fixed value of , we need to prove that the integral is bounded by 2, after moving the contour below that value of . As gets close to the real line, this becomes very difficult–either the contour is not straight and the integral is harder to estimate, as does not have a fixed imaginary part, or the contour must pass through more residues, adding additional terms to the equation.
We can still prove that the majority of the zeros do indeed lie on this arc by choosing a fixed height for the contour, estimating the value of the integral along that contour, and showing that its absolute value is bounded above by . The goal then is to choose a contour low enough to capture as many zeros as possible, yet high enough to avoid additional residues and to avoid large values inside the integral. We choose for , so that the contour has constant imaginary part .
For this choice to work, we must also limit the range of , so that our contour passes below and but above other images of under . If for , then has imaginary part , and a contour at a height of picks up residues at and but no other points equivalent to under the action of ; the maximum possible imaginary part of such a point is . In this case, the quantity inside the cosine function has the value at and the value at , and passes through at least multiples of . Bounding the integral by for the appropriate will finish the proof of Theorem 1.
4. Bounding the integral
In this section we bound
for the values with and with . We will also give some indication of how this bound might change if we allow to approach and alter the countour accordingly. Details for the computation of the numerical bounds that appear here are provided in the next section.
We seek a bound for
which is real-valued, by something less than . In absolute value, this integral is
Consider the exponential term . We have chosen , so that for , and this term has exponential decay as ; in this case . If we find an upper bound for the absolute value of the integral, then for large enough the right hand side is indeed less than 2, and we can apply the Intermediate Value Theorem as desired. It turns out that we can find a bound for the absolute value that removes the dependence on , but may be exponential in . However, if is large enough in relation to , then we will see that for a fixed weight , all but finitely many of the have zeros on the appropriate arc.
We note that the absolute value of the integral is certainly bounded above by
and already the dependence on has vanished. For the terms involving and , we find an upper bound for the maximum possible value of these terms over the appropriate ranges of and , and pull these upper bounds outside of the integral. This leaves us with the contribution from
which we consider in pieces.
Consider first the quantity
Computations, explained in more detail in the next section, yield
As a power series in , we know that has positive coefficients, as seen in (2.1). The maximum value of occurs when , when is real and positive. Heuristically, the minimum value should occur when there is maximum cancelation between terms, or when is real and negative, so that , and we confirm this computationally. Decreasing both increases the upper bound, as we are adding larger positive terms, and potentially decreases the lower bound due to cancelation.
Heuristically, the maximum value of should occur when is close to , meaning is close to , as here is real, positive, and close to 1. Similarly, the minimum value of should occur when is close to , meaning is close to ; here is real but negative, so there is extensive cancelation when adding terms. In this case, though, the size of depends on , as we have . Computationally, the minimum indeed occurs at , where . In general, we have
We compute that
Moving the lower bound on closer to increases the maximum value of , though it does not appear to affect the minimum value.
Putting this together, we have, for ,
and for ,
Next, we consider the term
If , this is , which is bounded above by 8.00067. If , this is , which is bounded above by 12.50005. Either way, the contribution is no more than 12.50005. Note that has positive coefficients, and is therefore large when is large.
Finally, we consider
The Hauptmodul is an injective mapping on the fundamental domain; it is real-valued and strictly decreasing on . We restrict ourselves to . Computation shows that on this domain. In contrast, takes on a wide range of values, including some with very large and some with very small modulus. Bounding the numerator and denominator separately yields a trivial upper bound of roughly 79000, while numerical calculations indicate that the actual maximum is a little larger than 1.
In order to achieve a sharper bound, we will instead consider a related quantity,
where we use to indicate and to indicate for ease of notation. This quantity is related to our Hauptmodul expression by the identity
We will break our path of integration into pieces, and consider in relation to on each. As we know that is real, we consider the real and imaginary parts of separately for . It is clear that , and so , while . With this in mind, we restrict our calculations to and use symmetry for .
The numerical techniques described in the next section reveal that on the interval , we have either , , or ; it follows that on this interval.
Next, we note that since is real, then if we have the bound , it follows that
If , then this is bounded above by , while if , then the maximum possible value of the right hand side for a fixed as varies is
occurring when . A lower bound for thus gives us an upper bound for .
For we have and If , then our bound is 1, and if then our bound is 1.00016. Either way, for , we have . Similarly, on we have , and we obtain a bound of 1.13192.
Finally, we consider . We have , and so
The maximum value occurs when , and we find here.
Altogether, we have . By breaking the integral into pieces, we compute more precisely that
The relationship in (4.2) allows us to conclude that
We can improve this further by noting that the fact that on implies that the integrand is bounded by 1 here, yielding
We see that if we extend closer to , and hence also decrease , this term has the most potential to blow up near , as this is where and are both small. Additionally, if , then any contour with fixed imaginary part less than , such as , will cross the arc and at least one of its images under , so restricting our values is necessary to avoid a zero in the denominator.
Putting all of these pieces together and using the fact that is decreasing on , we see that for ,
Note that if , and if ; hence, the integral is less than our desired bound if and . Similarly, for , we replace with , and find that our integral is bounded by if . We can then apply the Intermediate Value Theorem to prove that the appropriate number of zeros are on the desired arc.
5. Rigorously computing upper and lower bounds
In the previous section, while bounding our integral we used upper and lower bounds on Eisenstein series and the Hauptmodul for values on a circular arc on the boundary of the fundamental domain and on a straight line segment. In this section we justify those bounds.
It is useful for most of these calculations to truncate each series. For a modular form with Fourier series , we will choose a positive integer and let be the truncation of the Fourier series of up to and including the term, and we let be the remaining tail of the series. We bound and separately.
The calculations for the Eisenstein series are straightforward, as we have explicit formulas for the Fourier coefficients in terms of divisor functions, while calculations for require a little more finesse. We do not have a nice formula or a sharp bound for the growth rate of the Hauptmodul coefficients, and they are quite large, so there are more terms making a significant contribution to the value of the series. We begin by bounding the values of the Eisenstein series, and then use those bounds to tame .
By (2.1) we have
For , we can generously bound by