# U(3) Sato-Tate measure.

An undergraduate is performing some computations, related to a Sato-Tate conjecture of $U(3)$ type (a curve over $Q$, for which the roots of local L-functions look like eigenvalues of a random matrix in $U(3)$ and their complex conjugates, scaled appropriately).

There are well-known results in the literature, on random matrices from $U(3)$, with respect to Haar measure. For example, the expected value of the trace is zero (obviously), and the expected value of the square of the trace is one (not-so-obviously, but this is what I recall). The Weyl integration formula allows one to express other expectations as definite integrals. Results of Diaconis and others give a few moments, but most results I find look at $U(N)$ as $N$ grows large.

Does anyone know a precise elementary formula for the distribution of traces of a random matrix in $U(3)$? Or a good reference?

For example, the trace of a random matrix $g$ in $SU(2)$ is between $-2$ and $2$. The probability distribution of $Tr(g) = t$ is given, up to some normalization, by: $$P(t) = \sqrt{1 - \left( \frac{t}{2} \right)^2 },$$

The real part of the trace of a random matrix $g$ in $U(3)$ is between $-3$ and $3$. Can anyone write the probability distribution of $Tr(g) = z$, in as simple a fashion? Of course, here the traces can be complex -- I'd be happy also to find the probability that $Re(Tr(g)) = t$, as an elementary function of $t$.

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I think that Rains (after Diaconis-Shashahani) found the expected value of $|Tr(U)|^{2k}$ for all integers k (not just those up to the dimension $N$), from which you can apply inversion, I guess? See: combinatorics.org/Volume_5/PDF/v5i1r12.pdf I got this from a paper of Tracy (see page 2), who notes work of Gessel. Tracy: arxiv.org/pdf/math.CO/9811154 Gessel: people.brandeis.edu/~gessel/homepage/papers/dfin.pdf Sorry, that's the best I can do for now. –  Junkie May 26 '10 at 7:39
I should have said all positive integers of course. The moments are given in Sloane as A005802. research.att.com/~njas/sequences/A005802 –  Junkie May 26 '10 at 11:52
Thanks for the references! I hadn't found the Rains paper before, which sounds very promising. I'll take a look later today, but it sounds like exactly what I want. –  Marty May 26 '10 at 14:15
I thought that the roots of local L-functions of curves seemed to be drawn from USp(2g), not just a unitary group...? –  David Hansen May 27 '10 at 20:03
Correct, generically. But I'm having the undergraduate work with "Picard curves" -- certain genus three curves whose Jacobian has extra endomorphisms (by the ring of integers in $Q(\sqrt{-3})$). These extra endomorphisms cut down the image of the Galois representation from $USp(6)$ to $U(3)$ (at primes congruent to $1$ mod $3$), and make it an interesting case to study. It's a natural first choice to study, if one wants to go beyond the $SU(2)$ regime. –  Marty May 27 '10 at 23:43