NT Seminar: A haruspicy

Following JSE’s advice, I will blog on something that I know absolutely nothing about. Apologies in advance for mathematical errors!

SLM gave a number theory seminar this week about the first Betti number of \(\Gamma(n)\) — as \(n\) varies — for certain lattices in \(\mathrm{SU}(2,1)\). In particular, he proved an upper bound of the form:

\(\mathrm{dim} \ H^1(\Gamma(n),\mathbf{Q}) \ll [\Gamma:\Gamma(n)]^{3/8 + \epsilon},\)

which turns out (in certain cases) to be essentially the best possible estimate. As was known to Rogawski, the forms contributing to \(H^1\) all arise via endoscopy. In particular, if \(\Gamma\) is simple in the sense of Kottwitz, then the first cohomology vanishes (this also is due to Rogawski). So assume we are not in that case. The argument proceeds mostly as one would expect: Rogawski classifies the endoscopic forms which contribute to cohomology — they come from certain representations \(\xi \times \mu\) for \(U(2) \times U(1)\). Here I think the choice of Grossencharacter \(\mu\) is almost determined by \(\xi\), so I will drop it from the notation below. The possible packets can be described as follows:

1. Singletons for the split primes.
2. A set \(\{J^{+},D^{-}\}\) for the interesting infinite prime, where \(J^{+}\) contributes (via \((\mathfrak{g},K)\) cohomology) to \(H^1\) and another representation \(D^{-}\) which doesn’t (although it contributes to \(H^2\), I think).
3. A set \(\{\pi_s, \pi_p\}\) consisting of a supercuspidal representation and another representation at the inert primes.
4. Something similar to 3. for the ramified primes.

Using Matsushima’s formula, in order to count the contribution to cohomology one has to deal with the following:

1. The global multiplicity: this is either \(1\) or \(0\) depending on certain signs related to epsilon factors. As one varies \(n\) this should vanish half the time, but one can ignore it as far as an upper bound goes.

2. Suppose that \(p\) divides \(n\), and let \(K\) be a hyperspecial maximal compact at \(p\). Then one has to bound the trace of the characteristic function of \(K(p^k)\) on the representations \(\pi_s\) and \(\pi_p\).

Let \(f\) be such a characteristic function. One would like to write down a corresponding transfer function \(f^H\) on the endoscopic group such that:

\(\mathrm{Tr}(\pi_s,f) + \mathrm{Tr}(\pi_t,f) = \mathrm{Tr}(\xi,f^H)\)

By the Fundamental Lemma, if \(f\) is the characteristic function of the hyperspecial \(K\) itself, then \(f^H\) turns out to be the characteristic function on the maximal compact of \(U(2)\). SML shows that (using some of the same computations required for the fundamental lemma for \(U(3)\)) the same identity holds for the corresponding characteristic function for \(K(p^n)\), that is, the transfer \(f^H\) is the characteristic function of \(U(2)(p^n)\). Is this true for any deeper reason? More generally, to what extent do characteristic functions transfer to characteristic functions?