The Sato–Tate distribution

Chapter 24 The Sato–Tate distribution

In this chapter, we follow [23], appendix to chapter 1.

Section 24.1 Equidistribution on compact groups

Let \(X\) be a compact topological space. Let \(C(X)\) be the space of continuous functions \(X \to \CC\text{;}\) this is a Banach space under the supremum norm. Let \(\mu\) be a measure on \(X\text{,}\) i.e., a continuous linear map \(C(X) \to \CC\) which is nonnegative (i.e., the integral of a function taking nonnegative real values is and of total measure 1.

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A sequence \(x_1, x_2, \dots\) of elements of \(X\) is equidistributed with respect to \(\mu\) if for any continuous function \(f\text{,}\)

\begin{equation*} \int_X f \,d\mu = \lim_{N \to \infty} \frac{1}{N} \sum_{i=1}^N f(x_i)\text{.} \end{equation*}

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Section 24.2 Topological groups

The key example for us is when \(X\) is a compact Lie group (e.g., a finite group), and \(K\) is the space of conjugacy classes of \(X\) (viewed with the quotient topology from \(G\)). In this case, \(K\) has a unique translation-invariant measure with total measure 1, called the Haar measure; we use this measure on \(X\) and on \(K\text{.}\)

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Theorem 24.1. Peter–Weyl.

With notation as above, the sequence \(x_1, x_2, \dots\) is equidistributed with respect to the Haar measure \(\mu\) if and only if for any irreducible character \(\chi: G \to \CC\) of \(G\text{,}\)

\begin{equation*} \lim_{N \to \infty} \frac{1}{N} \sum_{i=1}^N \chi(x_i) = \int_X \chi\,d\mu\text{.} \end{equation*}

Note that the integral on the right is 1 if \(\chi\) is the trivial character and 0 otherwise (orthogonality of characters).

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Section 24.3 \(L\)-functions and equidistribution

Here is a big generalization of our approach to Chebotaryov’s density theorem. Take \(K\) and \(X\) as in the previous example. Let \(x_1, x_2, \dots\) be a sequence of elements of \(X\text{,}\) and let \(x_i \to N(x_i)\) be a function whose values are all integers at least 2. We make the following additional hypotheses.

Assume that the Euler product
\begin{equation*} \prod_i (1 - N(x_i)^{-s})^{-1} \end{equation*}
converges absolutely for \(\Real(s) > 1\text{,}\) and extends to a meromorphic function on a neighborhood of \(\Real(s) \geq 1\) with no zeroes or poles in \(\Real(s) \geq 1\) except for a simple pole at \(s=1\text{.}\)

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Let \(\rho\) be any irreducible representation of \(K\) with character \(\chi\text{.}\) Put
\begin{equation*} L(s, \rho) := \prod_i \det (1 - \rho(x_i)N(x_i)^{-s})^{-1}\text{.} \end{equation*}
(Note that \(\rho(x_i)\) is only defined up to conjugation.) Then \(L(s,\rho)\) converges absolutely for \(\Real(s) > 1\text{,}\) and extends to a meromorphic function on a neighborhood of \(\Real(s) \geq 1\) with no zeroes or poles in \(\Real(s) \geq 1\) except possibly at \(s=1\text{.}\)

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Theorem 24.2.

The number of \(x_i\) with \(N(x_i) \leq n\) is asymptotic to \(n/\log n\) as \(n \to \infty\text{.}\) Moreover, for any irreducible character \(\chi\) of \(G\text{,}\)

\begin{equation*} \sum_{i\colon N(x_i) \leq n} \chi(x_i) = c(\chi) n/\log n + o(n/\log n)\text{,} \end{equation*}

where \(-c(\chi)\) is the order of vanishing of \(L(s,\rho)\) at \(s=1\text{.}\)

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Proof.

Yet another straightforward generalization of our original proof of the prime number theorem.

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Conjecture 24.3.

Assume that there exists \(c\) such that for any \(n \in \ZZ\text{,}\) there are at most \(c\) values of \(i\) with \(N(x_i) \leq c\text{.}\) Then the \(x_i\) are equidistributed for Haar measure if and only if \(c(\chi) = 0\) for every nontrivial irreducible character at \(\chi\text{.}\)

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This reproduces the Chebotaryov density theorem from the previous chapter.

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Section 24.4 The Sato–Tate conjecture

The following is a rather nonobvious application of the above formalism.

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Conjecture 24.4. Sato–Tate.

Suppose \(E\) does not have complex multiplication. Let \(\alpha_p\) be the root of \(x^2 - a_p x + p\) with nonnegative imaginary part. Then \(\arg(\alpha_p/\sqrt{p})\) is equidistributed in \([0, \pi]\) for the measure \(\frac{2}{\pi} \sin^2 \theta d\theta\text{.}\)

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What does the condition that \(E\) does not have complex multiplication mean? The points of \(E\) naturally form an abelian group, in which three points add to 0 if and only if they are collinear. We say \(E\) has complex multiplication if the only endomorphisms of \(E\) as an algebraic group are multiplication by integers. (Over \(\CC\text{,}\) \(E\) forms a Riemann surface which looks like the quotient of \(\CC\) by a lattice; an endomorphism of \(E\) corresponds to a complex number which multiplies the lattice into itself.)

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Theorem 24.5. Clozel–Harris–Taylor.

The Sato–Tate conjecture holds if \(j(E) \notin \ZZ\text{.}\) (This implies that \(E\) does not have complex multiplication.)

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I’ll skip the definition of the \(j\)-invariant \(E\) for now; see [24].

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Section 24.5 Equidistribution and Sato–Tate

How does the elliptic curve example relate to Sato–Tate? Put \(K = SU(2)\text{,}\) the group of \(2 \times 2\) unitary matrices of determinant 1. Any class in \(X\) contains a unique matrix of the form

\begin{equation*} \begin{pmatrix} e^{i \theta} \amp 0 \\ 0 \amp e^{-i \theta} \end{pmatrix} \qquad 0 \leq \theta \leq \pi\text{.} \end{equation*}

Thus we may use the \(\alpha_p\)’s to generate elements \(x_p\) of \(X\) by taking \(\theta = \arg(\alpha_p/\sqrt{p})\text{.}\) The Haar measure on \(X\) is precisely the Sato–Tate measure, so we are reduced to asking whether the \(x_p\) are equidistributed.

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The irreducible representations of \(K\) are just the symmetric powers of the standard 2-dimensional representation. Hence Sato–Tate reduces to the following, which is the real hard content in the work of Clozel–Harris–Taylor. (Note that you have to shift the abscissa of absolute convergence by \(1/2\text{.}\))

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Theorem 24.6.

Let \(P_n(T)\) be the polynomial with constant coefficient \(1\) and roots \(\alpha_p^n, \alpha_p^{n-1} \overline{\alpha_p}, \dots, \overline{\alpha_p}^n\text{.}\) If \(j(E) \notin \ZZ\text{,}\) then the Euler product

\begin{equation*} \prod_p P_n(p^{-s})^{-1} \end{equation*}

extends to a holomorphic function on \(\CC\text{.}\) (Since the Euler product converges absolutely for \(\Real(s) \gt 3/2\text{,}\) the product cannot vanish for \(\Real(s) \geq 3/2\text{.}\))

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Exercises 24.6 Exercises

1.

Let \(\alpha_1, \dots, \alpha_m\) be real numbers such that \(1, \alpha_1, \dots, \alpha_m\) are linearly independent over \(\QQ\text{.}\) Apply Weyl’s criterion to prove that the sequence \(x_n = (n \alpha_1, \dots, n \alpha_m) \in (\RR/\ZZ)^m\) is equidistributed for the usual measure.

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2.

Prove that the sequence \(\log n\) is not uniformly distributed for any measure on \(\RR/\ZZ\text{.}\)

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