数理解析研究所講究録
第 1874 巻 2014 年 210-214

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Continued fractions and Dedekind sums for function fields
Yoshinori Hamahata
Institute for Teaching and Learning
Ritsumeikan University

1 Introduction
For coprime integers$a$and $c>0$ , the classical Dedekind sum $d(a, c)$ is defined by$d(a, c)= \frac{1}{4c}\sum_{k=1}^{c-1}\cot(\frac{\pi k}{c})\cot(\frac{\pi ka}{c})$For coprime positive integers

$a$.

(1)

and , it holds that$c$$d(a, c)+d(c, a)= \frac{1}{12}(\frac{a}{c}+\frac{c}{a}+\frac{1}{ac}-3)$;

this is called the reciprocity law. The value of $d(a, c)$ has been investigated. Rewniting (1) in tenns of the sawtooth function, we can easily see that $d(a, c)$ is a rational
number. Rademacher [4] proved that $d(a, c)$ is not bounded above and below in theneighborhood of each $a/c$ . Rademacher and Grosswald [5] posed the following twoquestions:
1. Is $\{(a/c, d(a, c))|a/c\in \mathbb{Q}^{*}\}$ dense in ?
dense in ?
2. Is
Hickerson [3] answered them using the theory of continued fractions.
As is well known, there is an analogy between algebraic number fields and function$K$, andare similarfields. For example,, and , respectively. Each -lattice is an analog of a lattice in . In [1, 2],
towe introduced Dedekind sums and their higher-dimensional generalization for a given-lattice in a function field, and we established the reciprocity law. The -latticecorresponding to the Carlitz module defines the Dedekind sum $s(a, c)$ (see Section 2),
which is very similar to $d(a, c)$ . In this report, we answer the analogous questions for$s(a, c)$ .
$\mathbb{R}^{2}$$\{d(a, c)|a/c\in \mathbb{Q}^{*}\}$$A$$\mathbb{Z},$$:=\mathbb{F}_{q}(T)$$:=\mathbb{F}_{q}[T],$$\mathbb{R}$$\mathbb{Q}$$\mathbb{R}$$:=\mathbb{F}_{q}((1/T))$$K_{\infty}$$A$$\mathbb{C}$

$A$

$A$

$L$

2 Dedekind sums
2.1$A$

Letrank$r$-lattices and Drinfeld modules
be the completion of an algebraic closure of-lattice is a finitely generated -module of rank

$K_{\infty}$$C_{\infty}$

$A$

$A$

$r$

; it is an analog ofsuch that it is discrete in$\mathbb{C}.$

$A$

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$C_{\infty}$. For such an -lattice$A$$\Lambda$, we define the infinite product$e_{\Lambda}(z)$by.

$e_{\Lambda}(z)=z \prod_{0\neq\lambda\in\Lambda}(1-\frac{z}{\lambda})$This product uniformly converges at a bounded set in, and defines a map. The function
has the following properties:(El)
is entire in the sense ofrigid analysis;(E2):
is surjective -linear, and -periodic;(E3)
has a simple zero at each point in , and no filrther zeros;$C_{\infty}$$C_{\infty}arrow C_{\infty}$$e_{\Lambda}$:$e_{\Lambda}(z)$$e_{\Lambda}(z)$$e_{\Lambda}$$C_{\infty}arrow C_{\infty}$$\Lambda$$\mathbb{F}_{q}$$\Lambda$$e_{\Lambda}$(E4) $de_{\Lambda}(z)/dz=e_{\Lambda}’(z)=1.$

For $a\in A$ , there exists a unique polynomial
such thatholds. Let :be the Frobenius map, and let
bea non-commutative ring in with the commutation rule $c^{q}\tau=\tau c(c\in C_{\infty})$ . Thereexists a unique positive integer such that for any$\phi_{a}(z)=\phi_{a}^{\Lambda}(z)=\sum l_{i}(\phi_{a})z^{q^{i}}$$\phi_{a}(e_{\Lambda}(z))=e_{\Lambda}(az)$$\tau$$z\mapsto z^{q}$$C_{\infty}\{\tau\}$$\tau$$r$$a\in A\backslash \{O\},$$\phi_{a}=\sum_{i=0}^{r\deg a}l_{i}(a)\tau^{i} (l_{0}(a)=a)$

Then, the map:.

is called a rank Drinfeld module over. The map is an -algebra homomorphism; hence, the values $\phi_{a}(a\in A)$ aredetermined by . The rank 1 Drinfeld module with $\rho_{T}(z)=Tz+z^{q}$ is called theCarlitz module. The Carlitz module and a Drinfeld module ofrankare similar to
the multiplicative group
and an elliptic curve, respectively. There exists a bijection
between the set of rank $rA$ -lattices and the set of rank Drinfeld modules overdefined by $\phi_{a}(e_{\Lambda}(z))=e_{\Lambda}(az)(a\in A)$ . The -lattice corresponding to is similar, and each -lattice ofranktois similar to a lattice in$C_{\infty}$$\phi$$\phi$$Aarrow C_{\infty}\{\tau\},$$a\mapsto\phi_{a}$$r$$\mathbb{F}_{q}$$\phi_{T}$$\rho$$\geq 2$$\mathbb{G}_{m}$

$r$

$A$

$C_{\infty},$$L$$\rho$$2\pi i$$A$$\geq 2$$\mathbb{C}.$2.2 Dedekind sumsLet

be the -lattice corresponding to the Carlitz module . For coprime, we define the inhomogeneous Dedekind sum $s(a, c)$ by$L$$A\backslash \{O\}$$A$$\rho$$a,$ $c\in$$s(a, c)= \frac{1}{c}\sum_{0\neq\ell\in L/cL}e_{L}(\frac{a\ell}{c})^{-1}e_{L}(\frac{\ell}{c})^{-1}$

When $L/cL=0,$ $s(a, c)$ is defined to be zero. Using the Galois theory, we see that$s(a, c)\in K$ . By (E2), it holds that $s(a, c)=0$ if $q>3$ . Thus, henceforth,
we assumethat $q=3$ or 2. The reciprocity law for $s(a, c)$ is as follows.

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Theorem 2.1 (Reciprocity law) For coprime$a,$ $c\in

A$ ,we have$s(a, c)+s(c, a)=\{\begin{array}{ll}\frac{1}{T^{3}-T}(\frac{a}{c}+\frac{c}{a}+\frac{1}{ac}) if q=3,\frac{1}{T^{4}+T^{2}}(\frac{a}{c}+\frac{c}{a}+\frac{1}{a}+\frac{1}{c}+\frac{1}{ac}+1) if q=2.\end{array}$This result follows from the fact that the sum of all residues of

2.3$1/(z\rho_{a}(z)\rho_{c}(z))$

is zero.

Continued fractions

Since the value $s(a, c)$ depends on $a/c$ , we write $s(a/c)=s(a, c)$ . Then $s(a/c+b)=$
by $x_{0}=x,$ $x_{n+1}=$$s(a/c)$ is valid. For $x=a/c\in K$ , we define the sequenceof the Laurent expansion$1/(x_{n}-a_{n})$ , whereis the polynomial part. This sequence yields the continued fraction development of :$(x_{n})_{n\geq 0}$$\sum_{i=0}^{k}A_{i}T^{i}$$a_{n}$$x$$x_{n}= \sum_{i=-\infty}^{k}A_{i}T^{i}$$x=[a_{0}, a_{1}, \ldots, a_{n}]:=a_{0}+\frac{1}{a_{1}+\frac{1}{1}},$. . . $+\overline{1}$$a_{n-1}+_{\overline{a_{n}}}$is an infinite continued
where $a_{i}(i\geq 1)$ are non-constant. Note that if
fraction. The following theorem gives us the value of $s(a/c)$ .
$x\in K_{\infty}\backslash K,$$x$

Theorem 2.2 (i) If $q=3_{J}$ then$s([a_{0}, \ldots, a_{r}])=\{\begin{array}{ll}\frac{1}{T^{3}-T}([0, a_{1}, \ldots, a_{r}]+(-1)^{r+1}[0, a_{r}, \ldots, a_{1}] +a_{1}-a_{2}+\cdots+(-1)^{r+1}a_{r}) if r\geq 1,0 if r=0.\end{array}$(ii) If $q=2$, then$s([a_{0}, \ldots, a_{r}])=\{\begin{array}{l}\frac{1}{T^{4}+_{r}T^{2}}([0, a_{1}, \ldots, a_{r}]+(-1)^{r+1}[0, a_{r}, \ldots,a_{1}]+\prod_{i=1}[0, a_{i}, \ldots, a_{r}]+a_{1}-a_{2}+\cdots+(-1)^{r+1}a_{r}+r-1) ifr\geq 1,0\end{array}$if $r=0.$We can prove this by induction on by using Theorem 2.1.
$r$Remark2.3 Hickerson [3] proved the following result for $d(a/c):=d(a, c)$ :$d([a_{0}, \ldots, a_{r}])=\{\begin{array}{ll}\frac{-1+(-1)^{r}}{8}+\frac{1}{12}([0, a_{1}, \ldots, a_{r}]+(-1)^{r+1}[0, a_{r}, \ldots, a_{1}] +a_{1}-a_{2}+\cdots+(-1)^{r+1}a_{r}) if r\geq 1,0 if r=0.\end{array}$

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3 Density theorem
As an analog ofHickerson’s result, the following two theorems are obtained.
Theorem 3.1 $Ifq=3$ or 2, then $\{(a/c, s(a/c))|a/c\in K^{*}\}$ is dense in
Theorem 3.2 If $q=3$ or 2, then $\{s(a/c)|a/c\in K^{*}\}$ is dense in$K_{\infty}^{2}.$$K_{\infty}.$

Outline ofproof of Theorems 3.1, 3.2. We consider the case $q=3$ . Since
is dense in, it suffices to prove that for anyand for, there exists $a/c\in K^{*}$ such that $|x-a/c|<\epsilon,$ $|y-s(a/c)|<2\epsilon$ . We write$x=[b_{0}, b_{1}, \ldots]$ .
Take any element. For any, taking fully large
holds. Similarly, we write $x-(T^{3}-T)y=[d_{0}, d_{1}, \ldots].$
Taking fully large $|x-(T^{3}-T)y-[d_{0}, \ldots, d_{t-1}, \alpha]|<\epsilon$ holds. Suppose that $s+t$
is even. There exits $m,$such that$(K_{\infty}\backslash$$K)\cross K$$(x, y)\in K_{\infty}\backslash K$$K_{\infty}^{2}$$\epsilon>0$$\alpha\in K_{\infty}^{*}$$\epsilon>0$$s,$$|x-[b_{0}, \ldots, b_{s-1}, \alpha]|<\epsilon$$t,$$n\in A\backslash \mathbb{F}_{q}$$-b_{0}+b_{1}-b_{2}+\cdots+(-1)^{s}b_{s-1}+(-1)^{t-1}d_{t-1}+\cdots-d_{1}+d_{0}=(-1)^{s}(m-n)$.

Putting$a/c=[b_{0}, \ldots, b_{s-1}, m, n, d_{t-1}, \ldots, d_{1}], \alpha=[m, n, d_{t-1}, \ldots, d_{1}],$

we have $|x-a/c|<\epsilon$ . By Theorem 2.2 (i), we obtain$s(a/c)= \frac{1}{T^{3}-T}([0, b_{1}, \ldots, b_{s-1}, m, n, d_{t-1}, \ldots, d_{1}]$$-[0, d_{1}, \ldots, d_{t-1}, n, m, b_{s-1}, \ldots, b_{1}]$$+b_{1}-b_{2}+\cdots+(-1)^{s}b_{S}+(-1)^{s+1}m+(-1)^{s+2}n$$+(-1)^{t-1}d_{t-1}+\cdots+-d_{1})$,which yields $|y-s(a/c)|<2\epsilon$ . Theorem 3.2 follows from Theorem 3.1. The case$q=2$ can be proved in the same way.

References[1] A. Bayad and Y Hamahata, Higher dimensional Dedekind sums in function

fields, Acta Arithmetica 152 (2012), 71-80.
[2] Y Hamahata, Dedekind sums in function fields, Monatshefte ffi Mathematik

167 (2012), 461-480.

214[3] D. Hickerson, Continued fractions and density results for Dedekind sums, J.
Reine Angew. Math. 290 (1977), 113-116.
[4] H. Rademacher, Zur Theorie der Dedekindschen Summen, Math. Z. 63 (1956),

445-463.
[5] H. Rademacher and E. Grosswald, Dedekind sums, Math. Assoc. Amer., Wash-ington, DC, 1972.
Institute for Teaching and Leaming
Ritsumeikan University
1-1-1 Noji-higashi, Kusatsu, Shiga 525-8577
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