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The Density of Elliptic Dedekind Sums Over Imaginary Quadratic Fields

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The Density of Elliptic Dedekind Sums Over Imaginary Quadratic Fields THE DENSITY OF ELLIPTIC DEDEKIND SUMS OVER IMAGINARY QUADRATIC FIELDS NICOLAS BERKOPEC, JACOB BRANCH, RACHEL HEIKKINEN, CAROLINE NUNN, AND TIAN AN WONG Abstract. Elliptic Dedekind sums were introduced by Sczech as generaliza- tions of classical Dedekind sums to imaginary quadratic fields K. We show that for any such K, the values of suitably normalized elliptic Dedekind sums are dense in the real numbers. This extends an earlier result of Ito for Eu- clidean K. Our proof is an adaptation of recent work of Kohnen, which gives a new proof of density for classical Dedekind sums. 1. Introduction The classical Dedekind sum s(m; n) is defined for m; n 2 Z,(m; n) = 1, by 1 X mk k s(m; n) := cot π cot π : n n n 06≡k (mod n) The significance of this function is related to its appearance, discovered by Dedekind, in the transformation property for the logarithm of the Dedekind eta function η(τ), aτ + b 1 cτ + d πi a b log η = log(η(τ)) + (sign(c))2 log + φ ; cτ + d 2 i sign(c) 12 c d where φ is a homomorphism from SL2(Z) to Z, ( a b b if c = 0 φ = d c d a+d c − 12(sign(c))s(d; jcj)) if c 6= 0 and τ is any point in the upper half plane. It was conjectured by Grosswald and Rademacher that the values of s(m; n) are dense in R, and moreover that the graph (m=n; s(m; n)) is dense in R2 [5]. Clearly the latter implies the former. The latter was proved by Hickerson [2], and recently Kohnen gave a new proof of the former [4], using a result of Girstmair [1] on the three-term relation of Dedekind sums and Dirichlet's theorem on arithmetic progressions of primes, providing a direct approximation of any rational number. In this paper, we use Kohnen's method to provide an equivalent result for nor- malized elliptic Dedekind sums for any imaginary quadratic field. Consider an imaginary quadratic field K with discriminant D and ring of integers OK . Pick a Date: July 16, 2021. 1991 Mathematics Subject Classification. 11F20 (primary), 11A15 (secondary). Key words and phrases. Elliptic Dedekind sums, density. 1 2 N. BERKOPEC, J. BRANCH, R. HEIKKINEN, C. NUNN, AND T.A. WONG lattice L such that OK = fm 2 C : mL ⊂ Lg. For any positive integer k, we recall the Eisenstein-Kronecker series X −n −s Ek(z) = (l + z) jl + zj js=0 l2L; l+z6=0 where the value of the sum at s = 0 is evaluated by means of analytic continuation. These elliptic functions are in fact good generalizations of the cotangent function. Elliptic Dedekind sums, for a lattice L, were defined by Sczech [6] for h, k in OK , X hµ µ D(h; k) = E E : 1 k 1 k µ2L=kL As with the classical case, one defines a map Φ from SL2(OL) to C give by ( a b E (0)I a+d − D(a; c) c 6= 0 Φ := 2 c c d b E2(0)I d c = 0 where I(z) = z − z. Indeed, it was shown by Sczech [6] that Φ is a homomorphism in the additive group of complexp numbers and that it is trivial for OK = Z; Z[i]; and Z[ρ] where ρ = (−1 + −3)=2. For our purposes it is advantageous to follow Ito [3] and deal with normalized elliptic Dedekind sums p −1 D~(a; c) = (i jdjE2(0)) D(a; c): Ito [3] proved that the values of the normalized elliptic Dedekind sums are dense in R when OK is Euclidean and not equal to Z(i) or Z(ρ), in which case the Dedekind sum is trivial. In fact, Ito showed that the set (a=c; D~(a; c)) is dense in C×R, using the continued fraction method of Hickerson. As continued fraction expansions are less well-developed for non-Euclidean fields, especially for K with class number greater than 1, Ito's method does not immediately generalize. In any case, Ito's result implies that D~(a; c; ) is dense in R for the cases listed above. Using the method introduced by Kohnen [4] for the case of classical Dedekind sums, we provide an alternate proof that the normalized elliptic Dedekind sums are dense in the real line, and we show that the density result holds for any imaginary quadratic field. Theorem 1. The values of the normalized elliptic Dedekind sum D~(a; c) are dense in R. We shall first prove a lemma that provides us with an elliptic Dedekind sum identity analogous to the three term relation of Girstmair. After proving the lemma, we are prepared to show that D~(a; c) is dense in the real numbers. We prove that a simple subset of the rationals is dense in the reals, with the idea of approximating values in this subset. The latter is then follows as an application of Dirichlet's theorem and quadratic reciprocity. While our results extends Ito's in a certain sense, it remains open whether the graph (a=c; D~(a=c)) is dense in C × R for general K. Moreover, it is also possible to study the density of elliptic Dedekind sums defined for complex lattices L such that OL is not the ring of integers for some imaginary quadratic field K. THE DENSITY OF ELLIPTIC DEDEKIND SUMS OVER IMAGINARY QUADRATIC FIELDS 3 2. An application of the three-term relation Before working on density we must prove a lemma that establishes a simple formula for certain values of elliptic Dedekind sums. This lemma makes use of the homomorphism property of Φ, and follows the method of Girstmair [1] and Kohnen [4]. ai bi Lemma 1. Let Ai := 2 SL2(OL) for i 2 f1; 2; 3g. If A1 = A2A3 with ci di the conditions c1 = c2 = c 6= 0, a1a2 ≡ 1 (c) and (ai; c) = 1 satisfied, then 2 c3 D(a3; c3) = E2(0)I + 2 : c3 c Proof. Let Ai be as above. Since Ai 2 SL2(OL), the following equations must hold a2d2 − b2c = 1(1) (2) a1d1 − b1c = 1: In Section 1.2 of [6], Sczech states that Φ is a homomorphism from the multiplicative group SL2(OL) to the additive group of complex numbers. Therefore, −1 0 = Φ(I) = Φ(A1 A2A3) = −Φ(A1) + Φ(A2) + Φ(A3): After expanding and reordering terms as needed, we arrive at the three term relation a1 + d1 a2 + d2 a3 + d3 D(a1; c) = D(a2; c) + D(a3; c3) + E2(0)I − − : c c c3 Fix c1 = c2 and a1a2 ≡ 1 (c), so we can conclude D(a1; c1) = D(a2; c2). This follows from 1 X a1k k D(a ; c) = E E : 1 c 1 c 1 c k2L=cL 0 With the change of variables k = a1k, this becomes 0 0 1 X k a2k E E = D(a ; c): c 1 c 1 c 2 k02L=cL Since D(a1; c) = D(a2; c), we can conclude that a2 + d2 a3 + d3 a1 + d1 D(a3; c3) = E2(0)I + − : c c3 c −1 Now note that by assumption A3 = A2 A1, so we may write a1d2 − b2c d2b1 − d1b2 A3 = : a2c − a1c d1a2 − b1c Hence a + d a + d a + d 2 2 + 3 3 − 1 1 c c3 c (a − a )(a + d ) a d − b c + d a − b c (a − a )(a + d ) = 2 1 2 2 + 1 2 2 1 2 1 − 2 1 1 1 : (a2 − a1)c (a2 − a1)c (a2 − a1)c Which, by (1) and (2) reduces to 2 2 2 + a2 − 2a1a2 + a1 2 c3 = + 2 : (a2 − a1)c c3 c 4 N. BERKOPEC, J. BRANCH, R. HEIKKINEN, C. NUNN, AND T.A. WONG So by substitution we arrive at 2 c3 D(a3; c3) = E2(0)I + 2 c3 c as desired. 3. Proof of Theorem 1 We first explain the choice of normalization of D~(h; k) that we employ. Indeed, after normalizing, the lemma allows us to express the elliptic Dedekind sums as rational numbers. Ito [3] states that normalized elliptic Dedekind sums take values in Q(j), where j is equal to the j-invariant of the lattice OK . It is therefore sufficient to show that j is real. Recall that j is defined by g (τ)3 j(τ) = 1728 2 ∆(τ) where 3 2 ∆(τ) = g2(τ) − 27g3(τ) iπτ and g2; g3 are Eisenstein series. The q = e expansions of these series have purely real coefficients. Since τ is either of the form −1 + pjdj ipjdj or ; 2 d an integer, the nome must also be purely real. Therefore j(τ) and the normalized elliptic Dedekind Sums are also real. With this we are prepared to prove density. We note that unlike Kohnen, care must be taken when working with imaginary quadratic fields with d of different parity. Proof of Theorem 1. Since the rationals are dense in the reals it suffices to show ~ a that D(a3; c3) is dense in a dense subset of the rationals. Let x = b be a rational number written in lowest terms witha ¯ the inverse of a (mod b). We establish two cases of when d is odd and even to show that there exists infinitely many primes p such that d2e2 + 4d is a square mod p, where ap − 1 e = : b First, suppose d is odd. By Dirichlet's theorem on primes in arithmetic progres- sions and the Chinese Remainder Theorem, there exist infinitely many primes p such that p ≡ 1 (mod 4b2d + d); p ≡ a¯ (mod b); wherea ¯ is the inverse of a mod b.
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