## Is the Dedekind sum really a function on SL_2?

Here’s an idle thought I wanted to record before I forgot it.

The Dedekind sum comes up in a bunch of disparate places; it’s how you keep track of the way half-integral weight forms like the eta function aka discriminant to the one-twelfth transforms under SL_2, it shows up in the topology of modular knots, the alternating sum of continued fraction coefficients, etc.  It has a weird definition which I find it hard to get a feel for.  The Dedekind sum also satsfies Rademacher reciprocity:

$D(a,b;c) + D(b,c;a) + D(c,a;b) = \frac{1}{12abc}(a^2 + b^2 + c^2 - 3abc)$

If that right-hand side looks familiar, it’s because it’s the very same cubic form whose vanishing defines the Markoff numbers!  Here’s a nice way to interpret it.  Suppose A,B,C are matrices with ABC = 1 and

(1/3)Tr A = a

(1/3)Tr B = b

(1/3)Tr C = c

(Why 1/3?  See this post from last month.)

Then

$a^2 + b^2 + c^2 - 3abc = (1/9)(2 + \mathbf{Tr}([A,B]))$

(see e.g. this paper of Bowditch.)

The well-known invariance of the Markoff form under moves like (a,b,c) -> (a,b,ab-c) now “lifts” to the fact that (the conjugacy class of) [A,B] is unchanged by the action of the mapping class group Gamma(0,4) on the set of triples (A,B,C) with ABC=1.

The Dedekind sum can be thought of as a function on such triples:

D(A,B,C) = D((1/3)Tr A, (1/3) Tr B; (1/3) Tr C).

Is there an alternate definition or characterization of D(A,B,C) which makes Rademacher reciprocity

$D(A,B,C) + D(B,C,A) + D(C,A,B) = (1/9)(2 + \mathbf{Tr}([A,B]))$

more manifest?