I recently talked with Lynnelle Ye (a soon to be graduating student of Mark Kisin) for a few hours about her thesis and related mathematics. In her thesis, she generalizes (in part) the work Liu-Wan-Xiao on the boundary (halo) of the eigencurve to unitary groups. One of her main results gives a precise asymptotic growth rate of the Newton Polygon of U_p as one moves towards the boundary.
Turning this around, this leads to estimates for the function \(N_{\lambda}(X)\) which counts the number of eigenvalues \(\lambda\) of U_p (with multiplicity) of valuation at most \(X.\)
I have always had a soft spot for counting slopes, although I haven’t really done anything in this business for many years. It is already interesting to estimate this growth function for classical overconvergent modular forms in the centre of weight space. Precise estimates were first obtained by Wan in his work on the Gouvea-Mazur conjectures.
Suppose we fix a tame level \Gamma, and let X = X(\Gamma) denote the relevant modular curve. Then it turns out that, conjecturally at least, that:
\(\displaystyle{N_{\lambda}(X) \sim^{?} \frac{\mathrm{Vol}(X_0(p))}{4 \pi} X.}\)
But this is precisely the growth estimate in Weyl’s law for the Laplacian on X_0(p)! This suggests an analogy between the spectrum of the compact operator U_p in the p-adic case and the spectrum of the Laplacian operator in the complex case which was first suggested to me by Don Blasius and which I always hoped but never quite managed to extract anything from (see section 5 of these notes, which also contain more precise details about Wan’s results and related results towards the conjecture above, as well as many further speculations on Overconvergent P-Adic Quantum Unique Ergodicity, if you were wondering about the title).
What growth rate should one expect for the Unitary group U(n)? Lynnelle exploits the fact (as do Liu-Wan-Xiao) that one can work on a compact form of the group which is zero dimensional. However, the eigenvariety is (or should be) essentially the same as the eigenvariety for other forms of the group. Following the analogy above, we can consider the growth rate of Weyl’s law for U(n-1,1), which, since the Shimura variety for U(n-1,1) has complex dimension n-1, grows like \(X^{n-1}.\) However, the exponent in Lynnelle’s work turns out to be
\(X^{n(n-1)/2}.\)
If I understood correctly, this one can even predict (if not prove) by simply counting the dimension of certain classical spaces of regular algebraic automorphic forms as one ranges over local systems of appropriate weights (proving it requires more work, of course). However, this seems to spoil the very precise (up to the level of constants) analogy for the complex dimension n = 1 case above. Is there something one can do to massage these results so they look more similar or was the n = 1 case simply misleading?
By work of Lapid and Muller, Weyl’s law is known for the cuspidal spectrum of arithmetic quotients $latex \Gamma \backslash SL_n(\mathbb{R})/SO_n(\mathbb{R})$, with exponent given (unsurprisingly) by $latex n(n-1)/2$.
Added: Oops, I bobbled this a bit; the exponent is half the real dimension, and the latter is n(n+1)/2-1=(n+2)(n-1)/2. So one gets exponent (n+2)(n-1)/4 in the Weyl law.
Another complaint is that the arithmetic quotients associated to SL(n) are probably not the right spaces to consider when n > 2. I am starting to suspect that the coincidences for n=2 are just that — the fact that the exponents are the same in that case is pretty slim evidence, and the fact that the constants are the same is also fairly weak, especially given that the volume divided by $latex 2 \pi$ is not so mysterious (being the Euler characteristic).
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