ZEROS OF SOME LEVEL 2 EISENSTEIN SERIES. Accepted for publication by the Proceedings of the American Mathematical Society

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S XX ZEROS OF SOME LEVEL EISENSTEIN SERIES SHARON GARTHWAITE, LING LONG, HOLLY SWISHER, STEPHANIE TRENEER Abstract The zeros of classical Eisenstein series satisfy many intriguing properties Work of F Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property In this paper we extend these distribution properties to a particular family of Eisenstein series on Γ because of its elegant connection to a classical Jacobi elliptic function cnu which satisfies a differential equation see formula As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allow us to calculate zeros of these Eisenstein series We end with a result linking the zeros of these Eisenstein series to an L-series Accepted for publication by the Proceedings of the American Mathematical Society Introduction Modular forms are functions defined on the complex upper half-plane, H, that transform in a nice way with respect to subgroups of SL Z These functions have proved to be intricately related to deep results in many areas of number theory, combinatorics, algebraic geometry, and other areas of mathematics cf [Ono04] For example, modular forms are deeply involved in the proof of Fermat s last theorem, results in partition theory, and denominator formulas for the monster group Eisenstein series are primary components of modular forms That is, the weighted algebra of all integral weight holomorphic modular forms for SL Z is generated by E 4 z and E 6 z, the unique normalized Eisenstein series of weight 4 and 6 for SL Z, respectively A nice generating function of the classical Eisenstein series E k z is the Weierstrass function, which is an elliptic function satisfying u = 4 u 3 a 4 u a 6, where a 4, a 6 are scalar multiples of E 4 and E 6, respectively Location of the zeros of Eisenstein series is of fundamental importance Provided with such information, Li, Long, and Yang construct noncongruence cuspforms from congruence Eisenstein series [LLY05] Additionally, in a paper of Ono and Papanikolas [OP04], an interesting formula is highlighted which relates the location of the zeros of E k z to special values of the Riemann Zeta function ζs: ζ k = 60 k e τ ord τ E k τjτ, where e i = /, e + 3 τ H/SL Z = /3, and e τ = for non-elliptic points, τ Received by the editors August 4, 009 The second author was supported in part by the NSA grant #H c XXXX American Mathematical Society

2 GARTHWAITE, LONG, SWISHER, TRENEER In [RSD70], FKC Rankin and HPF Swinnerton-Dyer study the location of the zeros of E k z By using an elementary and elegant argument, they demonstrate that all zeros of E k z inside the fundamental domain for SL Z are located on the arc {z = e iθ, π/ θ π/3} Equivalently, the j-values of these zeros are all real and are within the interval [0, 78], where jz is the well-known classical modular function which parametrizes isomorphism classes of elliptic curves In [Koh04], Kohnen gives an explicit formula for the zeros of E k z Recently, there are several results generalizing [RSD70] to Eisenstein series of other groups [Hah07, MNS07, Shi07, etc], other modular forms [Get04, Gun06], and certain weakly holomorphic modular forms [DJ08] Moreover, Nozaki proves that the zeros of the classical Eisenstein series E k z interlace with the zeros of E k+ z [Noz08] Despite these results, many mysteries remain about Eisenstein series, and their zeros For example, Gekeler studies polynomials ϕ k x which encode the j-values of the non-elliptic zeros of E k z [Gek0] He conjectures that the ϕ k x are irreducible with Galois group S d where d is the degree of ϕ k x In Section of this paper, we define a series of odd weight Eisenstein series denoted by G k+ z for Γ, the principal level congruence subgroup In [LY05], Long and Yang use G k+ z to give a second proof to some beautiful formulae of Milne [Mil0] on representing natural numbers in terms of sums of squares or triangular numbers Our attention was drawn to this family of Eisenstein series G k+ z largely because it has an elegant generating function cnu, a classical Jacobi elliptic function [Han58] satisfying d cnu = cn u λ + λcn u d u Here λ is the classical lambda function which parameterizes all isomorphism classes of elliptic curves with full -torsion structure The equation gives rise to recursions satisfied by G k+ z Consequently, one can compute the λ-values of the zeros of G k+ z with great convenience and ease Data from the first few dozen k values indicates that all λ-values of the zeros of G k+ z are real and are within the interval, 0] Moreover, this data indicates that the zeros of G k z interlace with the zeros of G k+ z In Section 3 we investigate the location of the zeros of G k+ z within the following fixed fundamental domain of Γ cf Figure in Section : D = {z H : Rez, z / /, z + / /} A direct application of the argument of Rankin and Swinnerton-Dyer shows that at least one third of the zeros of G k+ z lie in {z H : z + / = /}, or equivalently, have λ-values in, 0] By refining this argument, we will show the following improvement Theorem At least 90% of the zeros of G k+ z have real λ-values in the range, 0] In addition, by restricting our domain slightly we obtain the following result about the separation property of the zeros of G k z and G k+ z following an approach of Nozaki [Noz08] We state this result in terms of the related function

3 ZEROS OF SOME LEVEL EISENSTEIN SERIES 3 Figure A fundamental domain for Γ F k+ z θ defined in 3 where z θ := eiθ Before stating the result, we define for an integer k > 5 the intervals I j,k = α j,k π k + k, α j,k + π k + k for each j = 0,, k, where α j,k := πj/k Furthermore, let I k := j I j,k Theorem Let k > 5 be an integer, and I j,k, I k as defined above Then the zeros of F k z θ and F k+ z θ in [π/0, 9π/0] are restricted to I k and I k+ respectively in fact each I j,k and I j,k+ in [π/0, 9π/0] contains an odd number of zeros Moreover, the intervals are pairwise disjoint, and the I j,k are separated by the I j,k+ In Section 4, we obtain a formula Corollary 4 similar to equation which relates the sum of the λ-values of the zeros of G k+ z to the special values of an explicit L-series We note that the properties of our Eisenstein series G k+ z are similar to other families of level Eisenstein series studied by Tim Huber to which we expect our results can be extended Furthermore, the behavior of our Eisenstein series is similar to all other known cases for congruence subgroups of genus zero with relatively simple fundamental domains It would be very interesting to know what to expect for Eisenstein series or cuspforms that are invariant under congruence subgroups with more complex fundamental domains or higher genus The authors would like to acknowledge that this project and collaboration was initiated at the Women in Numbers Conference at BIRS The authors thank organizers Kristen Lauter, Rachel Pries and Renate Scheidler, along with BIRS, the Fields Institute, Microsoft Research, the NSA, PIMS, and the University of Calgary for their generous support of this conference Γ and its modular forms Let Γ be the principal level congruence subgroup consisting of matrices which become the identity under the natural modulo homomorphism This group is of genus 0 and has a fundamental domain shown in Figure Let q = e πiz ; the

4 4 GARTHWAITE, LONG, SWISHER, TRENEER Jacobi theta functions are defined by Θ z := q /8 q nn+/, Θ 3 z := q n /, and Θ 4 z := n Z n Z n Z n q n / The square of each of the above series is a weight modular form for Γ The classical lambda function is defined as λz := Θ4 z Θ 4 and it generates the field of 3 z meromorphic modular functions for Γ Some odd weight level Eisenstein series We now focus on a particular family of Eisenstein series of odd weight on Γ investigated in [LLY05] To begin, we define the character χγ := χd = where γ = a b c d and E k+,χ z := d denotes a Legendre symbol We now define c,d 0, c,d= χd cz + d k+, and recall some results about E k+,χ z see [LLY05, Lemmas 5-55] e n Lemma Define the nth Euler number e n by sect = n! tn The function n=0 E k+,χ z is a modular form of weight k + on Γ with Fourier expansion at given by For any integer k let E k+,χ z = + 4 k e k S k α, β = S k α, βz := By the explicit description of χ, r= c,d α,β 4 c,d= χr rk q r/ q r/ cz + d k E k+,χ z = S k+0, + S k+, S k+ 0, 3 S k+, 3 Recall that if f is a meromorphic function on H, γ = a b c d SL Z, and k is an integer then the slash operator in weight k is fz k γ := cz + d k fγz We use f γ to denote fz k γ The following lemma for odd k follows analogously from Lemma 33 in [LLY05] Lemma If γ SL Z, and S k is defined as above for odd k, then S k α, β γ = S k α, β γ mod 4 where α, β γ mod 4 denotes matrix multiplication modulo 4

5 ZEROS OF SOME LEVEL EISENSTEIN SERIES 5 A generating function Recall that cnu is a classical Jacobi elliptic function which satisfies cf [Han58, pp 56] κk Ku q cos u π cn q3 cos 3u q5 cos 5u = + π + q + q q 5 +, where κ = Θ z Θ 3 z cf [Han58, pp 4] and hence λz = κ and K = π Θ 3z For the remainder of this section we simply write λ for λz To see how the family of Eisenstein series E k+,χ relates to cnu, we consider the expansion of E k+,χ at the cusp, given by E k+,χ γ0, where γ 0 = 0 The previous lemma allows us to easily calculate E k+,χ γ0 in terms of the S k, which brings us to the following conclusion E k+,χ γ0 z = S k+, 0 + S k+ 3, S k+ 3, 0 S k+, It is this function G k+ z := E k+,χ γ0 z whose zeros we study It has been shown [LY05] Lemma 55 that G k+ z = 4 ik+ r k q r /4 e k + q r / In particular, G z = iθ z cf [LY05] Lemma 54 Using the series expansion cosx = n xn, we have n! n=0 κk Ku π cn k r k q r / = π k! + q r k=0 r= ie k = 4k! G k+zu k k=0 As an immediate consequence, we have the following proposition Proposition 3 Let κ = Θ z Θ 3 z and K = π Θ 3z We have that ie k π k+ G k+ z cnu = k+ k! K k+ κ uk k=0 r= From cnu satisfying cf [Han58, pp 47], we have that 3 cnu = u + + 4λu4! 4! + 44λ + 6λ u6 6! + Note that the choice K = π Θ 3z aligns the constant terms of equations and 3 We have that k iπ k+ e k k+ Gk+z K k+ κ = k ie k G k+ z Θ 4k 3 zθ z = k e kg k+z Θ 4k 3 zg z is a modular function for Γ which is holomorphic everywhere except possibly a pole of finite order at the cusp infinity coming from Θ 4k 3 z Consequently, for k it is a degree k polynomial in terms of λ, denoted p k+ λ, having the same zeros as G k+ in the fundamental domain We list the first six below u k

6 6 GARTHWAITE, LONG, SWISHER, TRENEER p λ =, p 7 λ = + 44λ + 6λ, p 3 λ =, p 9 λ = + 408λ + 9λ + 64λ 3, p 5 λ = + 4λ, p λ = λ λ λ λ 4 There are k nontrivial zeros counting multiplicity, in addition to the trivial zeros of G k+ at the cusps Calculations for k + 5 reveal some striking numerical trends The λ-values of the zeros of G k+ all lie in, 0 We list the numerical zeros of the first few p k+ λ below k= { 05} k=4 { 0005, 04598, 3788} k=3 { 09, 77} k=5 { 0007, 80, 879, 59745} Moreover, the λ-zeros of p k λ interlace with the λ-zeros of p k+ λ In addition, the polynomials p k+ λ are irreducible with Galois group S k for k 9 3 Locating the zeros of G k+ z Based on the numerical evidence above, we now turn our focus to the λ-values of the zeros of G k+ z Proving that the λ-values lie in, 0] is equivalent to proving that the zeros of G k+ z lie on the line Rez =, a boundary arc of the fundamental domain for Γ This, in turn, is equivalent to proving that the zeros of G k+ γ z, where γ = 0 0, lie on the arc z + =, as we mentioned in Section Our main strategy is to generalize the approach of Rankin and Swinnerton-Dyer In particular, we will need to refine estimates of the terms arising in this approximation We first find, using Lemma, that G k+ γ z = S k+0, 3 + S k+, S k+ 0, S k+, 3 We now wish to count the number of zeros of G k+ γ z that must lie on the arc z + = Lemma 3 Let α be even and z θ = eiθ Then α S k α, βz θ = S k, β α α e iθ + S k +, β α e iθ Proof The lemma follows by separating the terms in S k α, βz θ for pairs c, d according to the two possible values of c modulo 4, and by the fact that c, d = implies c, d = c, d c By Lemma 3 and the observation that S k a, b = S k a, b, we have G k+ γ z θ = S k+ 0, 3 + S k+, 0 + S k+, + S k+ 3, e iθ We next factor out e iθ k following [RSD70] and define 3 F k+ z θ := e iθ k G k+ γ z θ = Sk+ 0, 3 + S k+, 0 + S k+, + S k+ 3, θ, where S k α, βθ := c,d α,β 4 c,d= ce iθ/ + de iθ/ k

7 ZEROS OF SOME LEVEL EISENSTEIN SERIES 7 From equation 3 we see that F k+ z θ is purely imaginary by noting that F k+ z θ = F k+ z θ, as conjugation interchanges each pair of sums Moreover, in each sum c and d have opposite parity A direct application of the analysis in [RSD70] shows there are exactly the right number of zeros when θ ranges through [ π 3, π 3 ] In other words, one third of the zeros are expected to be on the above arc with θ in [ π 3, π 3 ] The real difficulty comes from the analysis when θ is close to 0 or π 3 Extraction of the main term To begin our analysis of G k+ γ z θ on the arc z + =, we first extract the two terms for which c + d = one occurring in each of S k+ 0, 3 and S k+, 0 to create our main term, e iθ k e iθ θk + k = i sin Thus, we have θk + 3 F k+ z θ = i sin + R k+ z θ, where R k+ z θ or simply Rz θ is purely imaginary and denotes the error term obtained by summing over all remaining terms 3 Bounding the error term We now turn to bounding Rz θ Here we will consider the contribution of terms satisfying c + d = N, for N > We take advantage of the symmetry in our remaining terms and define the following partial sum For each ordered pair of nonnegative integers a, b chosen such that a is odd and b is even to avoid double-counting, define 33 P a, bθ = ce iθ/ + de iθ/ k, c, d =a,b or b,a where the sum contains only terms occurring in F k+ z θ For example, P 3, 0 = 3e iθ/ k + 3e iθ/ k = i θk + sin 3k+ Due to symmetry, each partial sum P a, b is purely imaginary as before 33 The case when N 00 We now give upper bounds on the terms for which < N 00 We assume that k + > 5 as we have numerically verified the location of zeros for low weight cases When b = 0 we have the following cases: a, b {3, 0, 5, 0, 7, 0, 9, 0} Here, P a, bθ = k + θ sin ak+ a 5 The contribution from these terms to the error term is smaller than 4 E = n + 5 < 0 4 n= When b 0, then for each term in P a, b we have ce iθ/ + de iθ/ = ae iθ/ ± be iθ/ or be iθ/ ± ae iθ/

8 8 GARTHWAITE, LONG, SWISHER, TRENEER In the first case, 34 ae iθ/ ± be iθ/ k = a ± b cos θ/ + ia b sin θ/ k { a b + 4ab cos θ/ k / if +, = a b + 4ab sin θ/ k / if By symmetry, the second case yields the same result When a and b are nonzero, P a, b contains four terms, two of each type in 34 Therefore, P a, b a b + 4ab sin θ/ + k a b + 4ab cos θ/ k Note that maxcos θ/, sin θ/ / Moreover, if we limit our choice of θ to the interval 005π, 095π then mincos θ/, sin θ/ > 079 Hence we conclude that when b is nonzero, 35 P a, b Now consider the cases a b + 4ab079 k + a b + ab k a, b {,, 3,, 3, 4, 5, 4, 5, 6, 7, 6}, with a b = Summing the bounds in 35 for the six pairs a, b listed above yields that the contribution from these terms is smaller than The remaining cases with N 00 are E = 0656 a, b {, 4,, 6,, 8, 3, 6, 3, 8, 5,, 5, 8, 7,, 7, 4, 9,, 9, 4} For these we note that a b 3 Reasoning as above yields P a, b < 4 9 k Hence, the total contribution to the error term is bounded by In total, we have a +b 00 E 3 = 4 9 k < 0 0 P a, b E + E + E 3 < 0657

9 ZEROS OF SOME LEVEL EISENSTEIN SERIES 9 34 The case when N > 00 We now consider terms c +d = N with N 0, again assuming that k + > 5 The number of terms satisfying c + d = N is at most N / + 5N / Note that ce iθ/ + de iθ/ = c + cd cos θ + d If we restrict ourselves to θ values with cos θ α, for some α 0,, then Thus, we have c + cd cos θ + d αc + d Rz θ < N=0 5N / αn k / By bounding this latter sum with an appropriate integral, we have Rz θ < α k 5 k 00 k+ We pick α = 9877, which corresponds to θ 005π, 095π Then Rz θ < 35 Proof of Theorem Following [RSD70], we consider the values θ for which sin = ± and then apply the Intermediate Value Theorem to the θk+ function if k+ z θ In doing so we conclude that this function, and hence also G k+ γ z θ, must have at least zero in each interval of the form [ ] Aπ A + π, 005π, 095π, k + k + where A is an odd positive integer We also recall that G k+ has altogether k nontrivial zeros Thus, having previously considered k + 5 computationally, we can now verify that at least 90% of the zeros of G k+ z do indeed lie on the boundary Rez =, and hence have real λ-values in, 0] This concludes the proof of Theorem 36 Proof of Theorem Now that we have pinpointed the location of the zeros of G k+ z, we turn our attention to the relationship between the zeros of G k z and G k+ z Numerical evidence suggests that the zeros of F k z θ and F k+ z θ, the shifted Eisenstein series, interlace in the same manner as demonstrated by Nozaki for the classical case with E k z and E k+ z We now follow Nozaki s strategy by relating the zeros of F k+ z θ to the zeros of the main term of this series, a well-understood trigonometric function with regularly spaced zeros We then show that the additional error terms for the Eisenstein series will not cause the zeros of F k+ z θ to stray far from the zeros of the main term We must first strengthen the bounds on the error estimates given in the previous section In order to do this we will need to change our lower bound on sin θ/ and cos θ/ near the cusps θ = 0, π Thus, we narrow our focus to theta values in the range [π/0, 9π/0] Recall that the main term of F k+ z θ given in 3 is θk + f k+ θ := i sin

10 0 GARTHWAITE, LONG, SWISHER, TRENEER Let α j,k denote the zeros of f k on 0, π, that is, α j,k = πj, j =,, k k Note that α j,k+ < α j,k < α j+,k+ for j =,, k, and min {α j β j, β j+ α j } = j=,,k π k k + Our aim is to show that the zeros of F k z θ and F k+ z θ are within this distance of α j,k and α j,k+, respectively We now revisit the error estimates for 33 specifically for the range θ [π/0, 9π/0] We start by choosing a parameter γ = 056 in order that + 8γ > 95 Note that for θ [π/0, 9π/0] we have both sinθ/, cosθ/ γ Next note that in terms of k, the contribution of P a, b from the terms 4 {3, 0, 5, 0, 7, 0, 9, 0} is smaller than e = n + k+ 8 We 3k+ n= now use our new θ bounds to strengthen the bound on the second error term arising from the set of pairs of the form a, a ± with N 00 Replacing 0079 with γ in 35 yields P, 95 k + 5 k The contribution from 3,, 3, 4, 5, 4, 5, 6, 7, 6 is smaller than k k Thus the total contribution from these terms is less than e = 95 k + 5 k k k We again note that the contribution from the remaining cases with N 00 is e 3 44 Similarly, the contribution from terms with N > 00, is 9 k e 4 = 5N / CN k / < 5 C k / k 00 k+ < N=0 500 k C C k < k 48 k, where C = 95, so that cosθ < C on π/0, 9π/0 Putting the error terms altogether, we have R k+ z θ < gk := 8 3 k k + 5 k k k k + 83 k 48 k Note that the value of gk goes to 0 rapidly when k gets large We now return our focus to the relationship between the zeros of F k+ and the zeros of f k+ Let hx = π x + π 3 gx 6x + 3 π Note that 3 6x+ + gx decays more rapidly than π 3 x+ ; hence, we see that for k 5 we have hk > 0 Similarly, note that as sinx x x3 6 for any 0 x <,

11 ZEROS OF SOME LEVEL EISENSTEIN SERIES we can conclude that π sin π k + k + π 3 = hx + gx 6k + 3 Consider an interval I j,k = α j,k π k + k, α j,k + π [π/0, 9π/0] k + k about α j,k, a zero of f k We want to show that a zero of F k lies in this interval By construction, I j,k I i,k+ = for any integers i, j Note that π if k α j,k ± = k + k sin Hence for even j, π if k α j,k + k + k jπ ± π = sin sin π = j sin ± π k + k + k + π k + which is positive Similarly, if k α j π k+k + ir k z αj+ π k+k gk hk, hk is negative Likewise when j is odd, if k has opposite signs at the endpoints of I j,k It follows that F k has an odd number of zeros within I i,k as long as it is contained in [π/0, 9π/0] Moreover, since the error term ir k z θ is strictly less than the value of sinθk + / outside of the intervals I j,k, we can conclude that F k z θ has no zero in [π/0, 9π/0] \ j I j,k The analogous result holds for the zeros of F k+ with respect to the intervals I j,k+ As in the classical case, we can conclude that within [π/0, 9π/0], the zero intervals I j,k for F k z θ are separated by the zero intervals I j,k+ for F k+ z θ As mentioned in Section, our method can be generalized to some other families of Eisenstein series, in particular if the zeros of the family under consideration are expected to mainly locate on some fixed hyperbolic line However, careful analysis is needed to handle the error terms in a more general setting 4 Zeros of Eisenstein series and special values of L-series We now relate the Eisenstein series E k+,χ z defined by to special values of a particular L-series by defining C := 4 k k π k+ 4 = e k k!lk +, χ We have the following result in the style of Theorem 4 Let k 0 be an integer Then k π k+ = 4k + 6 k!lk +, χ τ H\Γ ord τ E k+,χ λτ

12 GARTHWAITE, LONG, SWISHER, TRENEER Proof We start by building the function Ẽ k+,χ z := E k+,χz Θ 4k zθ 3 z By [LY05, Lemma ], Θ 3 has the same character as E k+,χ, thus Ẽk+,χ is a modular function on Γ By examining the poles of Ẽk+,χ we conclude that it is a monic polynomial in /λ of degree k We write 4 Ẽ k+,χ z = λz λα λz λα λz λα k Since Θ and Θ 3 are holomorphic on H, the α j are exactly the zeros of E k+,χ in H, with multiplicity Expanding the product in 4, we have k k k Ẽ k+,χ z = + O λz λα j λz λz k j= Comparing the coefficients of the q k /, we have 6 k C 4 8k = 6 k k j= λα j = 6 k Solving for C and using 4 yields the desired result τ H\Γ ord τ E k+,χ λτ We note that the set of zeros of G k+ z = E k+,χ z γ 0 is given by {τ = γ 0 τ 0 : τ 0 H\Γ, E k+ τ 0 = 0} Therefore, we have the following corollary Corollary 4 Let k 0 be an integer Then k π k+ = 4k + 6 k!lk +, χ τ H\Γ ord τ G k+,χ λγ 0 τ Proof This follows directly from Theorem 4 and the comment above We now consider Eisenstein series of even weight for Γ With S k x, y defined as in and k an integer, we define E ± k z := S k0, ± S k, 0 These are modular forms of weight k for Γ with trivial character χ 0 In [LY05], Long and Yang show that E + k and E k have the Fourier expansions E ± k z = ± πi k 4 k ΓkLk, χ 0 Theorem 43 Let k be an integer Then r= πi k 4 k ΓkLk, χ 0 = ±4k + 6 r k q r/ + r q r/ τ H\Γ ord τ E ± k λτ

13 ZEROS OF SOME LEVEL EISENSTEIN SERIES 3 Proof To prove the first result, we set Ẽ + k z := E+ k z Θ 4k Then Ẽ+ k z is a modular function for Γ and is a monic polynomial in /λ of degree k Arguing as in the proof of Theorem 4, we have k Ẽ + k z = λz τ H\Γ ord τ E + k k + O λz λz k Equating the coefficients of q k / on either side, we find that 6 k πi k 43 4 k ΓkLk, χ 0 8k = 6 k ord τ E + k Solving 43 yields the formula for E + k The one for E k τ H\Γ Theorem 43 leads directly to the following surprising identity Corollary 44 Let k be an integer Then k = ord τ E + k + ord τ E k λτ = τ H\Γ References τ H\Γ is similarly obtained ord τ E + k E k λτ [DJ08] W Duke and Paul Jenkins, On the zeros and coefficients of certain weakly holomorphic modular forms, Pure Appl Math Q 4 008, no 4, part, [Gek0] E U Gekeler, Some observations on the arithmetic of Eisenstein series for the modular group SL, Z, Arch Math Basel 77 00, no, 5 [Get04] J Getz, A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms, Proc Amer Math Soc 3 004, no 8, 3 electronic [Gun06] S Gun, On the zeros of certain cusp forms, Math Proc Cambridge Philos Soc 4 006, no, 9 95 [Hah07] H Hahn, On zeros of Eisenstein series for genus zero Fuchsian groups, Proc Amer Math Soc , no 8, electronic [Han58] H Hancock, Lectures on the theory of elliptic functions: Analysis, Dover Publications Inc, New York, 958 [Koh04] W Kohnen, Zeros of Eisenstein series, Kyushu J Math , no, 5 56 [LLY05] W C Li, L Long, and Z Yang, On Atkin and Swinnerton-Dyer congruence relations, J of Number Theory 3 005, no, 7 48 [LY05] L Long and Y Yang, A short proof of Milne s formulas for sums of integer squares, Int J Number Theory 005, no 4, [Mil0] S C Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions, Ramanujan J 6 00, no, 7 49 [MNS07] T Miezaki, H Nozaki, and J Shigezumi, On the zeros of Eisenstein series for Γ 0 and Γ 0 3, J Math Soc Japan , no 3, [Noz08] H Nozaki, A separation property of the zeros of Eisenstein series for SL, Z, Bull Lond Math Soc , no, 6 36 [Ono04] K Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics, vol 0, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 004 [OP04] K Ono and M A Papanikolas, p-adic properties of values of the modular j-function, Galois theory and modular forms, Dev Math, vol, Kluwer Acad Publ, Boston, MA, 004, pp

14 4 GARTHWAITE, LONG, SWISHER, TRENEER [RSD70] F K C Rankin and H P F Swinnerton-Dyer, On the zeros of Eisenstein series, Bull London Math Soc 970, [Shi07] J Shigezumi, On the zeros of the Eisenstein series for Γ 0 5 and Γ 0 7, Kyushu J Math 6 007, no, Bucknell University, Lewisburg, PA 7837 sharongarthwaite@bucknelledu Department of Mathematics, Iowa State University, Ames, IA 500 linglong@iastateedu Oregon State University, Corvallis, OR 9730 swisherh@mathoregonstateedu Western Washington University, Bellingham, WA 985 stephanietreneer@wwuedu