where ((x)) := {x} -1 2 for x ∈ R -Z, 0 for x ∈ Z and {x} is the fractional part of x. Following conjectures of Grosswald and Rademacher [9], Hickerson showed that the image of s(c, d) is dense in R [6], and Vardi later showed that classical Dedekind sums are furthermore equidistributed mod 1 [12]. Viewing s(c, d) as a function on SL 2 (Z), a generalization of this result to non-cocompact lattices in SL 2 (R) was also recently proven by Burrin [2]. Elliptic Dedekind sums are analogues of classical Dedekind sums for imaginary quadratic fields. The goal of this paper is to study the value distribution of elliptic Dedekind sums. To further describe our results and motivation, we introduce some basic notation. Let L = ω 1 Z + ω 2 Z be a lattice in C with Im(ω 1 /ω 2 ) > 0 and let O L := {m ∈ C | mL ⊂ L} be its ring of multipliers. For x ∈ C and k ∈ Z ≥0 , define the Eisenstein-Kronecker series

© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.

where the value at s = 0 is understood in the sense of analytic continuation. Using these, Sczech [11] Let K be an imaginary quadratic number field with discriminant d K . We take L to be the ring of integers O K , so O L = O K . Define also the normalizations, for c = 0, D(c, d) = D(c, d)/(2E 2 (0) √ -1) and = /(2E 2 (0) √ -1).

In this paper weassume that K is a fixed imaginary quadratic field different from

and Q( √ -3) (in which case D is identically zero). In [1], Berkopec, Branch, Heikkinen, Nunn, and Wong showed that the values of D are real and dense in R, extending work of Ito [7] from Euclidean K to the general case. Following Vardi's work in the classical case, it is then natural to ask how the values of D are distributed.

In this paper, we show that the values of the normalized elliptic Dedekind sum are equidistributed mod 1.

Theorem 1 For all positive r ∈ R, the sequence

To prove Theorem 1, by the Weyl criterion for equidistribution, it suffices to obtain a nontrivial bound on the following sum for any positive integer n and real X > 0,

where r ∈ R and e(x) = e 2π ix (the trivial bound being O(X 4 )). If we let = PSL 2 (O K ) and ∞ be the stabilizer of the cusp at infinity, then the indexing set corresponds to a complete set of representatives of the quotient ∞ \ . Using the homomorphism (1.1), the inner sum transforms into a generalized Selberg-Kloosterman sum associated to and χ at the cusp ∞,

for c = 0, where we set α = -r + -r with x the greatest integer less than or equal to x and

A more general definition is given in (1.7). Note that is integral by [11,Satz 4] and moreover real-valued. The function S ∞ in defined in (1.7) and can also be viewed as a Kloosterman sum twisted by a unitary character related to the 'phase' of the elliptic Dedekind sum.

In the classical case, the required bounds for Selberg-Kloosterman sums were proved by Goldfeld and Sarnak [5]. The equidistribution of classical Dedekind sums was proved by Vardi using a similar relation between Dedekind sums and Selberg-Kloosterman sums on SL 2 (Z) [12,Theorem 1.4], which take the form for c = 0 and where S is defined as on [12, p.191]. We will use a similar strategy to prove Theorem 1 in a more general setting.

1.2 Sums of generalized Selberg-Kloosterman sums Let be a cofinite and non-cocompact discrete subgroup of PSL 2 (C) with an associated cusp ζ ∈ P 1 (C) and A ∈ PSL 2 (C) so that Aζ = ∞. Let ζ be the stabilizer of ζ in , and ζ its maximal unipotent subgroup. Let R be a complete set of representatives of double cosets in the quotient

which we also write as (c, d) for short. Given ζ , there is an associated complex lattice described in (2.1), with dual lattice . Note that is non-standard and is defined with respect to the inner product in (1.5). For this reason, the Selberg-Kloosterman sums we define are also non-standard.

Define a bilinear form on C 2 as

We note that this choice of bilinear form generally differs from convention, such as in [3]; this difference is due to the fact that (1.1) involves the function I(z). For any fixed α ∈ C, let χ α : → S 1 be a homomorphism such that for each lattice associated to a cusp ζ of ,

This includes the special case of = PSL 2 (O K ) and α ∈ R with χ α as in (1.3), which follows from the fact that for any a ∈ O K , we have ( 1 a 0 1 ) = Im(a) and also (M) = -(M) by [11].

For any m, n ∈ , c ∈ , and α ∈ C, we define the generalized Selberg-Kloosterman sum on associated to ζ and χ α to be

where e(z) = e 2π iIm (z) . Note that in the original Selberg-Kloosterman sums over Q, the character χ is obtained from the multiplier system associated to a given factor of automorphy. For this reason, the estimate given in Theorem 2 is of independent interest.

The proof of Theorem 1 follows from bounds on both the generalized Selberg-Kloosterman sums and the Kloosterman zeta function given by

The main ingredients in the proof of Theorem 1 are the associated Eisenstein and Poincaré series. These Poincaré series give bounds for our generalized Selberg-Kloosterman sums and hence generalize a classical result of Goldfeld and Sarnak [5] to the imaginary quadratic case. For simplicity, we state the theorem for the case ζ = ∞.

Theorem 2 Let m, n ∈ , c ∈ and α ∈ C. Let χ α be a unitary character of satisfying (1.6). Then for any > 0, we have

for certain constants τ j and real 0 < β j < 2. The implied constant depends only on , χ α , k, m and n.

Theorem 2 can be adapted to hold for other and ζ by applying the argument in Section 3 to the Poincaré series defined in (2.3). In Section 4.2, together with a routine application of the analytic theory of Eisenstein series on , we deduce the main result by specializing to

). The general nature of these results draws, in part, on the work of Elstrodt, Grunewald, and Mennicke [3]. In particular, similar to the work of Burrin [2] in the classical case, our methods should apply to future generalizations of elliptic Dedekind sums to more general subgroups of PSL 2 (C).

We first recall some basic definitions and fix the notation that we require, referring to [3] and [10] for further details.

The upper half-space in Euclidean 3-space is a model for 3-dimensional hyperbolic space

We can view H 3 as a subset of Hamilton's quaternions, and a point P ∈ H 3 is written

where z = x + iy and j = (0, 0, 1). The action of SL 2 (C) on P ∈ H 3 is given by the isometry

where the inverse is taken in the skew field of the quaternions. More explicitly, we write

We are interested in discrete subgroups of PSL 2 (C) and functions on the quotient \H 3 . Define D to be a fundamental domain of . Define also the Laplace-Beltrami operator

and the associated volume form dv = dx dy dr/r 3 and length form ds 2 = (dx 2 + dy 2 + dr 2 )/r 2 . We write ∞ for the stabilizer of the cusp at infinity in . In the case that = PSL 2 (O K ), we have that

and a coset representation of ∞ \ is given by

More generally, given a cusp ζ of a general such that Aζ = ∞, the set

forms a lattice in C. We shall denote by ζ the stabilizer of ζ in and ζ the maximal unipotent subgroup of ζ . We have then that

The Eisenstein series on associated to the cusp ζ is defined by

which converges absolutely for Re(s) > 2. The map z + rj → r s satisfies the differential equation r s = s(s -2)r s , so that E A (P, s) is a -invariant eigenfunction of with

but is not square-integrable on \H 3 . By Theorem 6.1.2 of [3], E A (P, s) has meromorphic continuation to the whole s-plane. Furthermore, Theorem 6.1.11 of [3] shows E A (P, s) is holomorphic for Re(s) > 0, except for a finite number of simple poles on the real line (0, 2], and the simple pole of E A (P, s) at s = 2 has residue equal to | |/vol( ). Note also that the series

agrees with E A (P, s) term by term by the change of variable M → AMA -1 whenever the series converges absolutely. The analytic properties of E A (P, s) are controlled by the behaviour of the zeroth coefficient of its Fourier expansion at the cusp. The Fourier expansion is given by ω ∈ a ω (r, s)q ω , where q ω = e(ω • z), with e(x) = e 2π ix and the operation • denotes the Euclidean scalar product in C, and is the dual lattice of with respect to the latter inner product. The Fourier coefficients can be expressed as

and

where d 0 is a constant given in [3, Theorem 2.1], K s denotes the modified Bessel function of the second kind, and φ ω (s) is the Dirichlet series

with the summation running over R as defined in (1.4).

where h is the class number of K and L j (s) = a ψ j (a)N (a) -s , the sum running over nonzero ideals of O K and ψ j an ideal class character [10, p.210]. For general , the functional equation for φ 0 follows from the functional equation of the associated Eisenstein series (see [3, p. 232-233]).

For s ∈ C and 0 = m ∈ , the dual lattice of , we define the Poincaré series on associated to χ α as defined in 1.6 and ζ as

where the inner product is as defined in (1.5). The series converges absolutely for s ∈ C and Re(s) > 2 following the same argument as [10, §3]. For the purpose of clarity, we will set ζ = ∞ in the future proofs and calculations and we more simply define

These computations can be applied to U A,m (P, s, χ α , , ζ ) with proper scaling.

The function U m is ∞ -invariant, and U m (MP, s, χ α ) = χ α (M) U m (P, s, χ α ).

(2.5)

Proof For any M ∈ ∞ , we have that M acts trivially on r M and by translation on z M . We then have, up to conjugaton,

where we note that m, ω ∈ Z so that e( m, ω ) = 1. Secondly, by direct computation we have

and by the homomorphism property of χ α , the identity (2.5) follows.

Letbe the self-adjoint extension of the Laplace operator on L 2 ( \H 3 ). Let ρ(-) be the resolvent set ofand

for λ ∈ ρ(-) the corresponding resolvent operator. Note that R s(2-s) is holomorphic for Re(s) > 2 and meromorphic for Re(s) > 1 with at most finitely many poles of order 1 in the interval (1,2).

Similarly to [3, p.355] (see also [10, (2.16)]), we also have

In particular, the region of convergence for U m (P, s, χ α ) extends meromorphically to Re(s) > 1 with at most a finite number of poles in (1, 2).

In this section, we take to be a general cofinite but not cocompact discrete subgroup of PSL 2 (C), ζ = ∞ a cusp of , and χ α a homomorphism on as in Section 2.3. Let D denote a fundamental domain for in H 3 and P a fundamental polytope.

We shall first prove an estimate for the Kloosterman zeta function in (1.8). We continue to assume m, n are in the dual lattice of .

The function Z ∞ (m, n, s, χ α ) is meromorphic in Re(s) > 1 with at most a finite number of simple poles in (1, 2) and satisfies the growth condition

The implied constant depends only on , χ α , m, n.

For the proof of Theorem 5, we rely on two lemmas and Stirling's formula. The first result we need is an upper bound for the L 2 -norm of our Poincaré series. The proof and formulation of the following lemma is similar to that of [5,Theorem 1].

where D is a fundamental domain for \H 3 .

Proof For 2 ≤ σ ≤ 3, we have U m (P, s, χ α ) = O(1) uniformly in P since the series converges absolutely. From (2.6) and the property that

The next lemma computes the inner product of two Poincaré series.

Let be a discrete subgroup of PSL 2 (O K ). For m, n = 0, we have

Proof First note that from (2.5) and unwinding

r s2 e 2π |n-α|r P U m (P, s 1 , χ α )e 2π i n-α,z dx dy dr r 3 .

Examining the inner integral, P U m (P, s, χ α ) e 2π i n-α,x+iy dx dy

we will break this sum into two terms, where c = 0 and where c = 0. For the contribution of the elements M ∈ ∞ \ where c = 0, we follow [4]. First notice that when c = 0, we have |a| = |d| = 1. Thus z M = ϕ a (z) + b d where ϕ a : C → C is an orthogonal linear map (as in [4] p. 676). Denote ϕ * a as the dual map of ϕ a with respect to •, • . Then P r s e -2π |m-α|r-2π i ϕ * a (m)-n ,z dz = r s e 2π |m-α|r vol(P)δ ϕ * a (m),n , where δ is the Kronecker delta function. Thus

where

The c = 0 case will be absorbed into R(s).

We now turn to the contribution of the elements M ∈ ∞ \ where c = 0,

× e 2π i n-α,x+iy dx dy dr r 3 .

We first examine the inner integral and denote H(r) by

Since Im(uv) = -Im(ūv), we have

Letting z = z + d/c and then z = z /r we have

where we recall that S ∞ (m, n, c,

Finally we integrate

-(n -α)zr dx dy dr r 3 .

Letting s2 = s 1 + 2 and s 1 = s, then we have

r 2 e -2π |n-α|r e((n -α)rz) (|z| 2 + 1) s dx dy dr r

where

|n-α|r e(-(n -α)rz) (|z| 2 + 1) s dx dy dr r and R m,n (s, c) := ∞ 0 R 2 re -2π |n-α|r e(-(n -α)rz) (|z| 2 + 1) s × exp -2π|m -α| (|z| 2 + 1)r|c| 2 e -(m -α)z c 2 r(|z| 2 + 1) -1 dx dy dr.

We evaluate I, following Proposition 3.10 of [10]. Examining the inner integral, for n = n 1 + in 2 and α = a 1 + ia 2 , we have Thus

as in [10, p.272].

As in [10] Proposition 3.10, R m,n (s, c)

σ -1 for Re(s) = σ and so

Finally, Stirling's formula implies that for Im(s) = t,

as |t| → ∞. From this and Lemmas 6 and 7, we have Theorem 5. where χ is an additive homomorphism on (a). By [3, Lemma 7.6.6] its analytic behaviour is parallel to that of U m (P, s, χ α ) above, and its inner product is computed explicitly in [3,Proposition 7.6.11] in the case χ = 1.

We can now prove the desired bound on our Selberg-Kloosterman sums. Let > 0. By the definition of Z ∞ and the trivial bound on the Kloosterman sum, on Re(s

where the implied constant depends on . Recall that

By the Phragmén-Lindelöf principle, there is an unique affine-linear function g(σ ) with g(1

for all 1 < σ ≤ 2 as |t| → ∞.

where σ := 2 + , and it follows that 1≤|c|≤x

Recall that Z ∞ m, n, 1 + s 2 , χ α has a finite number of simple poles at s = 2(s j -1) := β j with s j ∈ (1, 2) and define

We will now apply the Residue Theorem to the rectangular path of integration along the box

as x → ∞. Now choose T = x and the result follows.

4 Proof of equidistribution 4.1 Coset counting Define the counting function N (X) := * * c d ∈ R : |c| < X , (4.1)

where we recall R is a complete set of double-coset representatives in (1.4). We derive an asymptotic for N (X) using the Dirichlet series φ 0 in the constant term of the Eisenstein series E A (P, s).

For N (X) as in (4.1), we have

Proof The proof follows the same method as that of Theorem 2 (see also [2,Theorem 5]), so we will be brief. In place of the series Z ∞ (m, n, s, χ α ), we consider instead the Dirichlet series φ 0 (s) arising from the constant term of the Eisenstein series E A (P, s), whose analytic properties we have recalled in Section 2.2. We first write φ 0 (s) = Then shifting the line of integration to the line Re(s) = 1 and using the residue theorem, we have

and taking T = X yields the claim.

For the remainder of this paper we fix = PSL 2 (O K ) and ζ = ∞. Proposition 9 specializes to

2)

The proof of the main theorem now follows, as explained in the introduction. That is, returning to the setting of Theorem 1 and the discussion after it, the set under consideration can be rewritten as The length of the sum is precisely N (X), so combining Theorem 2 and (4.2), it follows by (1.2) that the latter equals |c|<X S ∞ ( -r , -r , c, χ α ) = o(X 4 ), as X → ∞, and by the Weyl criterion we conclude the main result.