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 generalizations 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 ∈ 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, |c|)) 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 normalized 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 = {m ∈ C : mL ⊂ L}. For any positive integer k, we recall the Eisenstein-Kronecker series

X −n −s Ek(z) = (l + z) |l + z| |s=0 l∈L, 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 µ∈L/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 complex√ 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 |d|E2(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 := ∈ SL2(OL) for i ∈ {1, 2, 3}. 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 ∈ 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 k∈L/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 k0∈L/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 + p|d| ip|d| 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 progressions 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. Thus, d2e2 + 4d d2 + 4b2d = p p p = d2 + 4b2d 1 = d2 + 4b2d = 1. THE DENSITY OF ELLIPTIC DEDEKIND SUMS OVER IMAGINARY QUADRATIC FIELDS 5

Here we have used quadratic reciprocity, noting that since d is odd, d2 + 4b2d ≡ 1 (mod 4). Then since 1 is a square mod all the prime factors of d2 + 4b2d, the final Jacobi symbol is 1. Therefore, d2e2 + 4d is a square (mod p). Now suppose d is even. Let d = 2d0. Suppose b is even and (d0, b) = 1. Then (d02 + db2, b) = (d02, b) because d02 ≡ db2 (mod b)= 1. Then again by the Dirichlet theorem and Chinese Remainder theorem there exist infinitely many primes p such that p ≡ 1 (mod d2 + d0b2), p ≡ a¯ (mod b) Once again we exploit quadratic reciprocity to find that d2e2 + 4d is a square mod p. d2e2 + 4d 4d02e2 + 4d = p p Substituting e, we have d02(ap − 1)2 + db2 d02 + db2 p = = = 1 p p d02 + db2 where again we use quadratic reciprocity. Thus, in both cases, we have that d2e2 + 4d is a square mod p. Therefore there exists an `, with (`, p) = 1, such that d2e2 + 4d = (2` − de)2 (1 + è)d = `2. √ Let k be the inverse√ of ` (mod p). Then k(k + e)d ≡ 1 (mod p). We set a1 := k d and a2 := (k + e) d. Then (ai, c) = 1 and

a1a2 = k(k + e)d ≡ 1 (mod c).

Since (p, k) = (p, d) = 1, we have (p, kd) = 1. Therefore, there exist integers x1, y1 such that px1 + kdy1 = 1, and thus √ px1 + a1(y1 d) = 1.

Similarly, there exist integers x2, y2 such that px2 + a2y2 = 1. Let a1 −x1 a2 −x2 A1 = √ A2 = √ p y1 d p y2 d and a3 b3 −1 A3 = = A2 A1. c3 d3

By construction, each of these matrices are in SL2(OL), and this Theorem holds for any OL. Now applying Lemma 1, we have that √ ! 2 e d D(a3, c3) = E2(0)I √ + pe d p √ d 2 e = E (0)2 + . 2 d ep p Thus 2 e D˜(a , c ) = + . 3 3 ep p 6 N. BERKOPEC, J. BRANCH, R. HEIKKINEN, C. NUNN, AND T.A. WONG

ap−1 and substituting e = b , we have 2b ap − 1 + . p(ap − 1) bp Taking the limit as p → ∞ and using the fact that √ √ E2(0)I(x d) = 2E2(0) dx, we have thus √ lim D˜(1 − kd, pe d) = 2x. p→∞ Therefore, since 2x is a rational number, we can conclude that the Dedekind sums are dense in the rational numbers. This concludes the proof. Acknowledgments. This research was completed at the REU Site: Mathematical Analysis and Applications at the University of Michigan-Dearborn. We would like to thank the National Science Foundation (DMS-1950102), the National Security Agency (H98230-21), the College of Arts, Sciences, and Letters, and the Depart- ment of Mathematics and Statistics for their support.

References [1] Kurt Girstmair. Dedekind sums with predictable signs. Acta Arith., 83(3):283–294, 1998. [2] Dean Hickerson. Continued fractions and density results for Dedekind sums. J. Reine Angew. Math., 290:113–116, 1977. [3] Hiroshi Ito. A density result for elliptic Dedekind sums. Acta Arith., 112(2):199–208, 2004. [4] Winfried Kohnen. A short note on Dedekind sums. Ramanujan J., 45(2):491–495, 2018. [5] Hans Rademacher and Emil Grosswald. Dedekind sums. The Carus Mathematical Monographs, No. 16. The Mathematical Association of America, Washington, D.C., 1972. [6] Robert Sczech. Dedekindsummen mit elliptischen Funktionen. Invent. Math., 76(3):523–551, 1984. Email address: [email protected]

Uiversity of New Mexico, Albuquerque NM, USA

Email address: [email protected]

Fairmont State University, Fairmont WV, USA

Email address: [email protected]

Augustana College, Rock Island Il, USA

Email address: [email protected]

University of Maryland, College Park MD, USA

Email address: [email protected]

University of Michigan, Dearborn MI, USA