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:
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.)
(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