The Normal Distribution as a Limit of Generalized Sato-Tate Measures

We consider this inequality from a naive probabilistic point of view. Suppose g = 1, for example. Then E has a model as a projective cubic curve. Let F (x, y, z) denote the corresponding homogeneous cubic polynomial. There are q + q + 1 Fq-rays through the origin in affine 3-space, and loosely speaking, each ray has a probability 1/q of lying on the zero-locus of F . Therefore, X(Fq) should should have about q+1 points, with an expected error of O( √ q). Thus (0.0.1) gives the right order of magnitude. On the other hand, rather than satisfying a Gaussian distribution law as one might expect, T/ √ q is absolutely bounded by 2. More precisely, in the limit q → ∞, the values of T/√q are uniformly distributed with respect to a certain measure μ1, known as the Sato-Tate measure, if E is drawn at random from the set of isomorphism classes of elliptic curves over Fq. For each value of g, we obtain in this way a measure μg supported on [−2g, 2g]. The main result of this paper is that the limit of these measures is, in fact, μ = (2π)−1/2e−x 2/2 dx. Heuristically speaking, for a random curve of random genus, over a random finite field, T/ √ q is normally distributed. The distribution of |X(Fq)|, as X varies over a family of varieties, can be approached via the cohomological theory of exponential sums, due to Deligne [5]. More generally, to a family of exponential sums (suitably defined) one can associate a compact Lie group G, the geometric monodromy group, and a finite-dimensional representation (ρ, V ) of G with character χ. Under suitable hypotheses, the distribution of values of the sums is the same as the distibution of values of χ(g), as g is drawn randomly from the uniform (Haar) measure on G. For each genus g ≥ 2, we construct such a group Gg and a symplectic representation Vg. It turns out that Gg is the full (compact) symplectic group Sp(2g) and Vg is its standard 2g-dimensional representation. It is not easy to explicitly compute the distribution of χ(g) as g ranges over Sp(2g), but the moments of the distribution can be read off from the invariant theory of Sp(2g). There is a precise sense in which the invariant