4_irrationality.tex

% !TEX root = Daniel-Miller-thesis.tex

\chapter{Irrationality exponents and CM abelian varieties}\label{chapter:irrationality-exponent}





\section{Definitions and first results}

We follow the notation of \cite{laurent-2009}. Fix a dimension 
$d\geqslant 1$, and let $\vx=(x_1,\dots,x_d)\in \bR^d$ be such that the $x_i$ 
are irrational and linearly independent over $\bQ$. If $d = 1$, the 
\emph{irrationality 
exponent} of $x\in \bR$ is the supremum of the set of $w\in \bR^+$ such that 
there infinitely many rational numbers $\frac p q$ with 
$\left| x - \frac p q\right| \leqslant q^{-w}$. If $x$ is rational, then it has 
irrationality exponent $1$. If $x$ is an algebraic irrational, then Roth's 
theorem says its irrationality exponent is $2$. Liouiville constructed 
transcendental numbers with arbitrarily large irrationality exponent. By 
\cite[Th.~E.3]{bugeaud--2012}, only a measure-zero set of reals, for example 
the Louiville number $\sum_{r\geqslant 1} 10^{-r!}$, have infinite 
irrationality exponent. In fact, by the same result, only a measure-zero set of 
reals have irrationality exponent $\ne 2$. In the results below, we will only 
consider reals with finite irrationality exponent. When $d\geqslant 1$, there 
are a $d$ natural measures of irrationality, but we will use only two of them. 

For the remainder of this thesis, let $\langle \cdot,\cdot\rangle$ be the 
standard inner product on $\bR^d$. 

\begin{definition}\label{def:approx-exp}
Let $\omega_0(\vx)$ (resp.~$\omega_{d-1}(\vx)$) be the supremum of the set of 
real numbers $w$ for which there exist infinitely many 
$(n,\vm)\in \bZ\times\bZ^d$ such that 
\begin{align*}
	|n \vx - \vm|_\infty 
		&\leqslant |(n,\vm)|_\infty^{-w} \\
	\text{(resp.~}
	|n +\langle \vm,\vx\rangle| 
		&\leqslant |(n,\vm)|_\infty^{-w} \text{).}
\end{align*}
\end{definition}

It is easy to see that both $\omega_0(\vx)$ and $\omega_{d-1}(\vx)$ are 
nonnegative. Even better, by \cite[Th.~2 Cor]{laurent-2009}, 
$\omega_0(\vx) \geqslant \frac 1 d$ and $\omega_{d-1}(\vx) \geqslant d$. 
These two quantities are related by Khintchine's transference 
principle \cite[Th.~2]{laurent-2009}, namely 
\[
	\frac{\omega_{d-1}(\vx)}{(d-1) \omega_{d-1}(\vx)+d} \leqslant \omega_0(\vx) \leqslant \frac{\omega_{d-1}(\vx)-d+1}{d} .
\]
Moreover, the second of these inequalities is sharp in a very strong sense. 

% For Jarnik's paper online
% http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-aav2i1p1bwm
\begin{theorem}[\cite{jarnik-1936}]\label{thm:jarnik}
Let $w\geqslant 1/d$. Then there exists $\vx\in \bR^d$ such that 
$\omega_0(\vx)=w$ and $\omega_{d-1}(\vx) = d w+d-1$. 
\end{theorem}

We can relate the traditional irrationality exponent and the invariant 
$\omega_0$ in the special case $d = 1$. 

\begin{theorem}\label{thm:omega-irrationality}
If $d=1$, then $\omega_0(x) = \mu-1$, where $\mu$ is the 
traditional irrationality exponent of $x$. 
\end{theorem}
\begin{proof}
Both $\mu$ and $\omega_0$ are invariant under translation by $\bZ$, so without 
loss of generality we may assume $x\in [0,1)$. 

First we show that $\omega_0(x)\geqslant \mu-1$. Suppose there exist infinitely 
many $p/q$ with $\left| x - \frac p q\right| \leqslant q^{-w}$. Since $x<1$ we 
may assume that for infinitely many of the $p/q$, $p<q$. Then 
$| q x - p| \leqslant q^{-(w - 1)} = \max(p,q)^{-(w-1)}$, which tells us that 
$\omega_0(x) \geqslant \mu - 1$. 

Now, we show that $\mu \geqslant \omega_0(x) + 1$. Suppose there exist 
infinitely many $(n,m)$ with $|n x - m| \leqslant \max(|n|,|m|)^{-w}$. By the 
reverse triangle inequality, 
$\left| |n x| - |m|\right| \leqslant \max(|n|,|m|)^{-w}$, and since 
$x<1$, for $n$ sufficiently large this implies $|n| \geqslant |m|$. It follows 
that for infinitely many $\frac m n$, we have 
$\left| x - \frac m n\right| \leqslant n^{-(w + 1)}$, which implies 
$\mu \geqslant \omega_0(x) + 1$. 
\end{proof}

Here is a statement of Roth's theorem in the current context. 

\begin{theorem}[Roth]
Let $x\in (\overline\bQ\cap \bR)\smallsetminus \bQ$. Then 
$\omega_0(x) = 1$. 
\end{theorem}
\begin{proof}
This follows directly from \cite{roth-1955} and Theorem 
\ref{thm:omega-irrationality}.
\end{proof}

Given $\vx\in \bR^d$, write 
$\dd(\vx,\bZ^d)=\min_{\vm\in \bZ^d} |\vx-\vm|_\infty$. Note that 
$\dd(\vx,\bZ^d)=0$ if and only if $\vx\in \bZ^d$. Moreover, $\dd(-,\bZ^d)$ 
is well-defined for elements of $\bT^d = (\bR/\bZ)^d$. 

\begin{lemma}\label{lem:bound-distance}
Let $\vx\in \bR^d$ with $|\vx|_\infty< 1$ and $\omega_0(\vx)$ 
(resp.~$\omega_{d-1}(\vx)$) finite. Then 
\begin{align*}
	\frac{1}{\dd(n \vx,\bZ^d)} 
		&\ll |n|^{\omega_0(\vx)+\epsilon} 
		&&\text{ for $n\in \bZ\smallsetminus 0$} \\
	\text{(resp.~}
	\frac{1}{\dd\left(\langle \vm,\vx\rangle, \bZ\right)} 
		&\ll |\vm|_\infty^{\omega_{d-1}(\vx)+\epsilon} 
		&&\text{ for $\vm\in \bZ^d\smallsetminus \vzero$).}
\end{align*}
\end{lemma}
\begin{proof}
Let $\epsilon>0$. Then there are only finitely many $n\in \bZ$ 
(resp.~$\vm\in \bZ^d$) such that the inequalities in Definition 
\ref{def:approx-exp} hold with $w = \omega_0(x)+\epsilon$ 
(resp.~$w = \omega_{d-1}(\vx)+\epsilon$). In other words, there exist constants 
$C_0, C_{d-1}>0$, depending on $\vx$ and $\epsilon$, such that 
\begin{align*}
	|n \vx - \vm|_\infty 
		&\geqslant C_0 |(n,\vm)|_\infty^{-\omega_0(\vx)-\epsilon} ,\\
	|n + \langle \vm,\vx\rangle| 
		&\geqslant C_{d-1} |(n,\vm)|_\infty^{-\omega_{d-1}(\vx)-\epsilon} 
\end{align*}
for all $(n,\vm)\ne (0,\vzero)$ in $\bZ\times\bZ^d$. 

Start with the first inequality. Fix $n$, and let $\vm$ be a lattice point 
achieving the minimum $|n \vx - \vm|_\infty$; then 
$\dd(n \vx,\bZ^d) \geqslant C_0 |(n,\vm)|_\infty^{-\omega_0(\vx)-\epsilon}$. 
Since $|n\vx - \vm|_\infty < 1$, the reverse triangle inequality gives 
$\left| |n| - \frac{|\vm|_\infty}{|\vx|_\infty}\right| \leqslant \frac{1}{|\vx|_\infty}$. So $|n|$ and $|\vm|$ are bounded above and below by scalar multiples 
of each other, which tells us that 
$\dd(n \vx,\bZ^d) \geqslant C_0' |n|^{-\omega_0(\vx)-\epsilon}$ for $C_0'$ 
depending on $\vx$. Thus 
$\frac{1}{\dd(n \vx,\bZ^d)} \ll |n|^{\omega_0(\vx)+\epsilon}$, the 
implied constant depending on both $\vx$ and $\epsilon$.

Now we consider the second inequality. Note that when $\vm\ne 0$, 
$\dd(\langle \vm,\vx\rangle,\bZ) = |n + \langle \vm,\vx\rangle|$ for 
some $n$ with $|n| \leqslant |\vm|_2\cdot |\vx|_2 + 1$. Thus 
$|(n,\vm)|_\infty \ll |\vm|_2 \ll |\vm|_\infty$ with the implied constants 
depending on $d$ and $\vx$, because any two norms on a 
finite-dimensional Banach space are equivalent. This gives us 
$\dd(\langle\vm,\vx\rangle,\bZ) \geqslant C_{d-1}' |\vm|_\infty^{-\omega_{d-1}(\vx)-\epsilon}$, 
for some constant $C_{d-1}'$, which implies 
\[
	\frac{1}{d(\langle \vm,\vx\rangle,\bZ)} \ll |\vm|_\infty^{\omega_{d-1}(\vx)+\epsilon} ,
\]
the implied constant depending on $\vx$ and $\epsilon$.
\end{proof}





\section{Irrationality exponents and discrepancy}

Let $\vx=(x_1,\dots,x_d)\in \bR^d$. The sequence 
$(\vx\mod \bZ^d,2\vx\mod\bZ^d,\dots)$ will be equidistributed in a subgroup of 
$\bT^d$. We are interested in the case where this sequence is equidistributed 
in the whole torus $\bT^d$, so assume $x_1,\dots,x_d$ are irrational and 
linearly independent over $\bQ$ (this condition also makes sense for elements 
of $\bT^d$). For $\vx\in \bT^d$, we wish to control the discrepancy of the 
sequence $(\vx,2\vx,3\vx,\dots)$ with respect to the Haar measure of $\bT^d$. 

\begin{theorem}[Erd\"os--Tur\'an--Koksma. {\cite[Th.~1.21]{drmota-tichy-1997}}]\label{thm:ETK}
Let $\bvx$ be a sequence in $\bT^d$ and $h$ an arbitrary integer. Then 
\[
	\D_N(\bvx) \ll \frac 1 h + \sum_{0\leqslant |\vm|_\infty \leqslant h} \frac{1}{r(\vm)} \left| \frac 1 N \sum_{n\leqslant N} e^{2\pi i \langle \vm,\vx_n\rangle}\right| ,
\]
where the first sum ranges over $\vm\in \bZ^d$, 
$r(\vm) = \prod \max\{1,|m_i|\}$, and the implied constant depends only on $d$. 
\end{theorem}

\begin{lemma}\label{lem:bound-exp-sum}
Let $x\in \bR\smallsetminus \bZ$. Then 
$\left| \sum_{n\leqslant N} e^{2\pi i n x}\right| \leqslant \frac{2}{\dd(x, \bZ)}$. 
\end{lemma}
\begin{proof}
We begin with an easy bound: 
\[
	\left| \sum_{n\leqslant N} e^{2\pi i n x}\right| = \frac{|e^{2\pi i (N+1) x} - e^{2\pi i x}|}{|e^{2\pi i x} - 1|} \leqslant \frac{2}{|e^{2\pi i x} - 1|} .
\]
Since $|e^{2\pi i x} - 1| = \sqrt{2-2\cos(2\pi x)}$ and 
$\cos(2\theta) = 1-2\sin^2\theta$, we obtain 
\[
	\left|\sum_{n\leqslant N} e^{2\pi i n x}\right| \leqslant \frac{1}{|\sin(\pi x)|} .
\]
It is easy to check that $|\sin(\pi x)| \geqslant \dd(x,\bZ)$, whence the result.  
\end{proof}

\begin{corollary}\label{cor:bound-disc-distance}
Let $\vx$ generate a dense subgroup of $\bT^d$. 
For $\bvx=(\vx,2\vx,3\vx,\dots)$ in $\bT^d$, we have 
\[
	\D_N(\bvx) \ll \frac 1 h + \frac 1 N \sum_{0<|\vm|_\infty \leqslant h} \frac{2}{r(\vm) \dd(\langle \vm,\vx\rangle,\bZ)} 
\]
for any integer $h$, with the implied constant depending only on $d$. 
\end{corollary}
\begin{proof}
Apply the Erd\"os--Tur\'an--Koksma inequality (Theorem \ref{thm:ETK}), and 
bound the exponential sums using Lemma \ref{lem:bound-exp-sum}. 
\end{proof}

We combine the above results to estimate an upper bound on the discrepancy of 
the sequence $\bvx$. 

\begin{theorem}\label{thm:disc-upper-bound}
Let $\vx$ generate a dense subgroup of $\bT^d$, and let 
$\bvx=(\vx,2\vx,3\vx,\dots)$ in $\bT^d$. Then 
$\D_N(\bvx) \ll N^{-\frac{1}{\omega_{d-1}(\vx)+1}+\epsilon}$. 
\end{theorem}
\begin{proof}
Fix $\epsilon>0$ smaller than $\frac{1}{\omega_{d-1}(\vx) - 1}$, and choose 
$\delta>0$ such that 
$\frac{1}{\omega_{d-1}(\vx)+1+\delta} = \frac{1}{\omega_{d-1}(\vx)+1} - \epsilon$. 
By Corollary \ref{cor:bound-disc-distance}, we know that 
\[
	\D_N(\bvx) \ll \frac 1 h + \frac 1 N \sum_{0<|\vm|_\infty \leqslant h} \frac{1}{r(\vm) \dd(\langle \vm,\vx\rangle,\bZ)} ,
\]
and by Lemma \ref{lem:bound-distance}, we know that 
$\dd(\langle \vm,\vx\rangle,\bZ)^{-1} \ll |\vm|_\infty^{\omega_{d-1}(\vx)+\delta}$. 
It follows that 
\[
	\D_N(\bvx) \ll \frac 1 h + \frac 1 N \sum_{0 < |\vm|_\infty \leqslant h} \frac{|\vm|_\infty^{\omega_{d-1}(\vx)+\delta}}{r(\vm)} .
\]
All that remains is to bound the sum. Clearly 
\[
	\sum_{0 < |\vm|_\infty \leqslant h} \frac{|\vm|_\infty^{\omega_{d-1}(\vx) + \delta}}{r(\vm)} \ll \int_1^h \int_1^h \dots \int_1^h \frac{\max(|t_1|,\dots,|t_d|)^{\omega_{d-1}(\vx)+\delta}}{t_1 \dots t_d}\, \dd t_1 \dots \dd t_d .
\]
For each permutation $\sigma$ of $\{1,\dots,d\}$, call $I_\sigma$ the set of 
all $(t_1,\dots,t_d)$ in $[1,h]^d$ with 
$t_{\sigma(1)} \leqslant \dots \leqslant t_{\sigma(d)}$. Then 
$[1,h]^d = \bigcup_{\sigma\in S_d} I_\sigma$, and each integral over 
$I_\sigma$ is easy to bound. For example, the integral over $I_1$ is 
\begin{align*}
	\int_1^h \int_1^{t_d} \dots \int_1^{t_2} \frac{t_d^{\omega_{d-1}(\vx)+\delta}}{t_1 \dots t_d}\, \dd t_1 \dots \dd t_d 
		&\ll \int_1^h t^{\omega_{d-1}(\vx)+\delta-1}\, \dd t \prod_{j=1}^{d-1} \int_1^h \frac{\dd t}{t} \\
		&\ll (\log h)^{d-1} h^{\omega_{d-1}(\vx)+\delta} .
\end{align*}
It follows that 
$\D_N(\bvx) \ll \frac 1 h + \frac 1 N (\log h)^{d-1} h^{\omega_{d-1}(\vx)+\delta}$. 
Setting $h\approx N^{\frac{1}{1+\omega_{d-1}(\vx)+\delta}}$, we see that 
$D_N(\bvx) \ll N^{-\frac{1}{\omega_{d-1}(\vx)+1+\delta}} = N^{-\frac{1}{\omega_{d-1}(\vx)+1} + \epsilon}$. 

For a slightly different proof of a similar result, given as a sequence of 
exercises, see  \cite[Ch.~2, Ex.~3.15, 16, 17]{kuipers-niederreiter-1974}. 
Also, this estimate is quite coarse, but a better one would only have a smaller 
leading coefficient, which no doubt would be useful for computational 
purposes, but does not strengthen any of the results in this thesis.  
\end{proof}

\begin{theorem}\label{thm:disc-lower-bound}
Let $\vx\in \bT^d$ generate a dense subgroup, with $\omega_0(\vx)$, 
$\omega_{d-1}(\vx)$ finite. Let $\bvx=(\vx,2\vx,3\vx,\dots)$. Then 
$\D_N(\bvx) = \Omega\left(N^{-\frac{d}{\omega_0(\vx)}-\epsilon} \right)$. 
\end{theorem}
\begin{proof}
We follow the proof of \cite[Ch.~2, Th.~3.3]{kuipers-niederreiter-1974}, 
modifying it as needed for our context. Given $\epsilon>0$, there exists 
$\delta>0$ such that 
$\frac{d}{\omega_0(\vx)-\delta} = \frac{d}{\omega_0(\vx)} + \epsilon$. 

By the definition of $\omega_0(\vx)$, there exist infinitely many 
$(n,\vm)$ with $n>0$ such that 
$|n \vx - \vm|_\infty \leqslant |(n,\vm)|_\infty^{-\omega_0(\vx)+\delta/2}$. 
For any fixed $n$, where are only finitely many $\vm$ with 
$|n \vx - \vm|_\infty \leqslant 1$. Since $|(n,\vm)|_\infty \geqslant n$, for 
any fixed $n$ there are at most finitely many $\vm$ with 
$|n \vx - \vm|_\infty \leqslant |(n,\vm)|_\infty^{-\omega_0(\vx)+\delta/2}$. 
Thus we derive the seemingly stronger statement that for infinitely many $n$, 
there exists $\vm\in \bZ^d$ such that 
$|n \vx-\vm|_\infty \leqslant n^{-\omega_0(\vx)+\delta/2}$ or, equivalently, 
$|\vx-n^{-1}\vm| \leqslant n^{-1-\omega_0(\vx)+\delta/2}$. Fix one such $n$, 
and let $N=\lfloor n^{\omega_0(\vx)-\delta}\rfloor$. For each $r\leqslant N$, 
we have 
\[
	\left|r \vx - r n^{-1} \vm\right|_\infty 
		= r \left|\vx - n^{-1} \vm\right|_\infty
		\leqslant r n^{-1-\omega_0(\vx)+\delta/2}
		\leqslant n^{-1-\delta/2}. 
\]
Thus, for each $r\leqslant N$, $r \vx$ is within $n^{-1-\delta/2}$ of the 
grid $\frac 1 n \bZ^d\subset \bT^d$. So no element of 
$\{\vx,\dots,N \vx\}$ lies in the half-open box 
$I_n = \left[ n^{-1 - \delta / 3}, n^{-1} - n^{-1 - \delta / 3}\right)^d$. 
Moreover, $I_n$ has volume $\left(n^{-1} - 2n^{-1 - \delta / 3}\right)^d$. 
For $n$ sufficiently large, the volume of $I_n$ is bounded below by 
$2^{-d} n^{-d}$, so the discrepancy $\D_N(\bvx)$ is 
bounded below by $2^{-d} n^{-d}$. Since $n^{\omega_0(\vx)-\delta} \leqslant 2 N$, 
the discrepancy $\D_N(\bvx)$ is bounded below by 
\[
	2^{-d} \left( (2 N)^{\frac{1}{\omega_0(\vx)-\delta}}\right)^{-d} 
		= 2^{-d-\frac{d}{\omega_0(\vx)-\delta}} N^{-\frac{d}{\omega_0(\vx)-\delta}}
		= 2^{-d\left(1+\frac{1}{\omega_0(\vx)}\right)-\epsilon} N^{-\frac{d}{\omega_0(\vx)}-\epsilon} .
\]
Since $\D_N(\bvx)$ can, as $N\to \infty$, be bounded below by a constant 
multiple of $N^{-\frac{d}{\omega_0(\vx)}-\epsilon}$, the proof is complete.
\end{proof}






\section{Pathological Satake parameters for CM abelian varieties}
\label{sec:Satake-CM}

We apply the results of the previous sections to $L$-functions associated to 
CM abelian varieties. For background on the motivic Galois group and Sato--Tate 
group of an abelian variety, see \cite{serre-tate-1968,serre-1994,yu-2015}. 
Recall that for $E$ a non-CM elliptic curve, the 
Akiyama--Tanigawa conjecture implies the Riemann hypothesis for all 
$L(\sym^k E,s)$, $k\geqslant 1$. The appearance of $\sym^k$ is dictated by the 
classification of irreducible representations of $\SU(2)$, the Sato--Tate group 
of $E$. If $A$ is a CM abelian variety, there should be an $L$-function (and 
Galois representation) for each irreducible representation of the Sato--Tate 
group of $A$, which we denote by $\ST(A)$. In the 
CM case, $\ST(A)$ is a real torus, so things can be described relatively 
explicitly. 

Let $K/\bQ$ be a finite Galois extension, $A_{/K}$ a $g$-dimensional abelian 
variety with complex multiplication by $F$, defined over $K$, that is, 
$F = \End_K(A)_\bQ$. Since the action of $F$ commutes with 
$\rho_l\colon G_\bQ \to \GL_{2g}(\bQ_l)$, the Galois representation coming 
from the $l$-adic Tate module of $A$ takes values in $\R_{F/\bQ}\Gm(\bQ_l)$, 
where $R_{F/\bQ} \Gm$ is the Weil restriction of scalars of the multiplicative 
group from $F$ to $\bQ$. The functor of points of $\R_{F/\bQ}\Gm$ is 
$R\mapsto (R\otimes F)^\times$. It follows that the Sato--Tate group of $A$ is 
a subgroup of the maximal compact torus inside $\R_{F/\bQ}\Gm(\bC)$. 

Recall, following \cite{serre-1994}, that the motivic Galois group of $A$ 
should be a subgroup $G_A\subset \R_{F/\bQ}\Gm$ such that for all primes $l$, 
the image $\rho_l(G_\bQ)$ lies inside $G_A(\bQ_l)$, and is open in 
$G_A(\bQ_l)$. For general abelian varieties, the existence of the motivic 
Galois group is a matter of conjecture, but for CM abelian varieties, it can be 
described directly. Let $\fa=\Lie(A)$ and 
$\det_\fa\colon \R_{K/\bQ}\Gm \to \R_{F/\bQ}\Gm$ be the map induced by the 
determinant of the action of $K$ on $\fa$ (viewed as an $F$-vector space). Then 
$G_A = \im(\det_\fa)$ \cite{yu-2015}, and $\ST(A)$ is a maximal compact 
subgroup of $G_A^1(\bC) = G_A^{\N_{F/\bQ} = 1}(\bC)$. So 
$\ST(A) \simeq \bT^d$ for some $1\leqslant d \leqslant g$ (we will use the 
same $d$ when applying Theorem \ref{thm:disc-lower-bound}), and every 
unitary character of $\ST(A)$ is induced by an algebraic character of 
$G_A^1$. Any character of a subtorus extends to the whole torus, so any 
character of $G_A^1$ is the restriction of a character of $\R_{F/\bQ}\Gm$. 

Let $\fp$ be a prime of $K$ at which $A$ has good reduction. Then 
$F = \End(A)_\bQ\hookrightarrow \End(A_{/\bF_\fp})_\bQ$, and the Frobenius 
element $\frob_\fp\in \End(A_{/\bF_\fp})_\bQ$ comes from an element 
$\pi_\fp\in F$. In other words, $\rho_l(\frob_\fp) = \pi_\fp$. The element 
$\pi_\fp\in F$ is $\fp$-Weil of weight $1$, 
i.e.~$|\sigma(\pi_\fp)| = \N(\fp)^{1/2}$ for all embeddings 
$\sigma\colon F\hookrightarrow \bC$. The normalized element 
$\theta_\fp = \frac{\pi_\fp}{\N(\fp)^{1/2}}$ lies in $\ST(A)$, and we call this 
the Satake parameter for $A$ at $\fp$. For the Satake parameters to be 
equidistributed in $\ST(A)$, it is necessary and sufficient for the 
$L$-function $L(r\circ \rho_l,s)$ to have non-vanishing analytic continuation 
to $\Re =1$ for each $r\in \X^\ast(\R_{F/\bQ}\Gm)$ which has nontrivial 
restriction to $\ST(A)$. By the Wiener--Ikehara Tauberian theorem, this is 
equivalent to an estimate 
$\left| \sum_{\N(\fp)\leqslant x} r(\theta_\fp)\right| = o(\pi_K(x))$, where 
$\pi_K(x)$ is the number of primes $\fp$ of $K$ with $\N(\fp)\leqslant x$. 

\begin{theorem}[Shimura--Taniyama, Weil, Hecke]
The elements $\theta_\fp\in \ST(A)$ are equidistributed with respect to the 
Haar measure. 
\end{theorem}
\begin{proof}
By \cite[Th.~10, 11]{serre-tate-1968}, for every 
$r\in \X^\ast(\R_{F/\bQ}\Gm)$ induced by $\sigma\colon F\hookrightarrow \bC$, 
there exists a Hecke character $\omega_r$ of $K$ such that 
$L(r\circ \rho_l,s) = L(s,\omega_r)$. For $r = \sum m_\sigma \sigma$, we have 
$L(r\circ \rho_l,s) = \prod L(\sigma\circ \rho_l,s)^{m_\sigma}$, so the general 
result follows. Moreover $\omega_r$ is nontrivial if and only if 
$\left.r\right|_{\ST(A)}$ is. Since $L$-functions of Hecke characters have the 
desired analytic continuation and nonvanishing, the result follows. 
\end{proof}

Recall that 
$L(r\circ \rho_l,s) = \prod \left(1 - r(\theta_\fp) \N(\fp)^{-s}\right)^{-1}$ 
(this is the normalized $L$-function, not the algebraic $L$-function). 
As in Chapter \ref{ch2:discrepancy}, the choice of an isomorphism
$\bT^d\simeq \ST(A)$ yields a definition of discrepancy 
for sequences in $\ST(A)$. We call the ``Akiyama--Tanigawa conjecture for $A$'' 
the estimate $\D_N(\btheta) \ll N^{-\frac 1 2+\epsilon}$, where 
$\btheta = (\theta_\fp)_\fp$ is the sequence of Satake parameters of $A$. 

\begin{theorem}\label{AT->RH:AB}
The Akiyama--Tanigawa conjecture for $A$ 
implies the Riemann hypothesis for all $L(r\circ \rho_l,s)$ with 
$\left. r\right|_{\ST(A)}$ nontrivial. 
\end{theorem}
\begin{proof}
The Akiyama--Tanigawa estimate implies, via the Koksma--Hlawka inequality, an 
estimate 
$\left| \sum_{\N(\fp)\leqslant N} r(\theta_\fp)\right| \ll N^{\frac 1 2+\epsilon}$. 
By Theorem \ref{thm:AT->RH:gp}, the function $L(r\circ \rho_l,s)$ satisfies 
the Riemann hypothesis. 
\end{proof}

It is natural to ask: does the Riemann hypothesis for all $L(r\circ \rho_l,s)$ 
imply the Akiyama--Tanigawa conjecture for $A$? We proceed to construct 
$L$-functions coming from ``fake Satake parameters'' which provide evidence to 
the contrary for nonmotivic (non-automorphic, in fact) sequences of Satake 
parameters. 

Give $\bT^d$ the Haar measure normalized to have total mass one. 
Recall that for any $f\in L^1(\bT^d)$, the Fourier coefficients of $f$ 
are, for $\vm\in \bZ^d$: 
\[
	\widehat f(\vm) = \int_{\bT^d} e^{2\pi i \langle \vm,\vx\rangle} \, \dd \vx ,
\]
where $\langle \vm,\vx\rangle = m_1 x_1 + \cdots + m_d x_d$ is the usual inner 
product. If $f$ is a continuous function on $\bT^d$ with $\widehat f(\vzero) = 0$ 
and $\bvx = (\vx_1,\vx_2,\dots)$ is equidistributed in $\bT^d$, then sums of 
the form $\sum_{n\leqslant N} f(n \vx)$ will be $o(N)$. When $f$ is a character 
of the torus, and $\bvx$ is the sequence of translates of an element 
generating a dense subgroup, there is a much stronger bound. 

\begin{theorem}
Fix $\vx\in \bT^d$ which generates a dense subgroup, with $\omega_{d-1}(\vx)$ 
finite. Then 
\[
	\left| \sum_{n\leqslant N} e^{2\pi i \langle \vm, n \vx\rangle}\right| \ll |\vm|_\infty^{\omega_{d-1}(\vx) + \epsilon} 
\]
as $\vm$ ranges over $\bZ^d\smallsetminus \vzero$. 
\end{theorem}
\begin{proof}
Since $\vx$ generates a dense subgroup of $\bT^d$, 
$\langle \vm,\vx\rangle\in \bR\smallsetminus \bZ$. Thus Lemma 
\ref{lem:bound-exp-sum} tells us that 
\[
	\left| \sum_{n\leqslant N} e^{2\pi i \langle \vm, n \vx\rangle}\right| 
		= \left| \sum_{n\leqslant N} e^{2\pi i n \langle \vm, \vx\rangle}\right|
		\ll \dd(\langle \vm, \vx\rangle,\bZ)^{-1},
\] 
and from Lemma \ref{lem:bound-distance}, we know that 
$\dd(\langle \vm,\vx\rangle, \bZ)^{-1} \ll |\vm|_\infty^{\omega_{d-1}(x)+\epsilon}$. 
The result follows. 
\end{proof}

By writing any function as a Fourier series, we can apply this result to sums 
of the form $\sum_{n\leqslant N} f(n \vx)$. 

\begin{theorem}\label{thm:translates-bound-sum}
Let $\vx\in \bR^d$ with $\omega_{d-1}(\vx)$ finite. Let 
$r>d+\omega_{d-1}(\vx)$, and fix $f\in C^r(\bT^d)$ with $\widehat f(\vzero)=0$. 
Then $\left| \sum_{n\leqslant N} f(n \vx)\right| \ll 1$. 
\end{theorem}
\begin{proof}
Write $f$ as a Fourier series:
$f(\vx) = \sum_{\vm\in \bZ^d} \widehat f(\vm) e^{2\pi i \langle \vm,\vx\rangle}$. 
Since $\widehat f(\vzero)=0$, we can compute:
\begin{align}
	\left| \sum_{n\leqslant N} f(n \vx)\right| 
		&= \left| \sum_{n\leqslant N} \sum_{\vm\in \bZ^d\smallsetminus \vzero} \widehat f(\vm) e^{2\pi i n \langle \vm,\vx\rangle}\right|\nonumber \\ 
		&\leqslant \sum_{\vm\in \bZ^d\smallsetminus \vzero} \left|\widehat f(\vm)\right|\cdot \left| \sum_{n\leqslant N} e^{2\pi i n \langle \vm,\vx\rangle}\right| \nonumber \\ 
		\label{eq:sum-converges}
		&\ll \sum_{\vm\in \bZ^d\smallsetminus \vzero} \left|\widehat f(\vm)\right|\cdot |\vm|_\infty^{\omega_{d-1}(\vx) + \epsilon} . 
\end{align}
Recall that an integral of $\phi$ over $\bR^d$ can be re-written in spherical 
coordinates as $\int \phi(r,s) r^{d-1} \psi(s)\, \dd r \dd s$, where 
$r$ ranges over $\bR^+$ with the usual Lebesgue measure, $s$ ranges over 
$S^{d-1}$ with its rotation-invariant measure, and $\psi$ is bounded. Thus 
$\int_{[1,\infty)^d} |\vx|_\infty^\alpha\, \dd x$ converges (and hence 
$\sum_{\vm\in \bZ^d\smallsetminus \vzero} |\vm|_\infty^\alpha$ converges)
whenever as $d-1+\alpha < -1$. The sum \eqref{eq:sum-converges} 
converges since the Fourier coefficients $\widehat f(\vm)$ converge to 
zero faster than $|\vm|_\infty^{-r}$ \cite[Th.~8.22]{folland-1999}, 
$\epsilon>0$ is arbitrary, and $d-1-r+\omega_{d-1}(\vx) < -1$. 
\end{proof}

Enumerate the primes of $K$ with increasing norms as $\fp_1,\fp_2,\fp_3,\dots$.  
Let $\vx\in \bT^d$ generate a dense subgroup.  The associated sequence of 
``fake Satake parameters'' is $\bvx = (\vx_\fp)_\fp$, where we put 
$\vx_{\fp_n} = n \vx$. For any fixed $w\geqslant \frac 1 d$, by Theorem 
\ref{thm:jarnik}, we can find $\vx$ with $\omega_0(\vx) = w$ and 
$\omega_{d-1}(\vx) = d w + d - 1$. 

\begin{theorem}
The sequence $\bvx$ is equidistributed in $\bT^d$, with discrepancy 
decaying as $\D_N(\bvx) \ll N^{-\frac{1}{d w+d} + \epsilon}$, and for which 
$\D_N(\bvx) = \Omega\left(N^{-\frac{d}{w} - \epsilon}\right)$. 
However, for any $f\in C^\infty(\bT^d)$ with $\widehat f(\vzero)=0$, the 
Dirichlet series  $L_f(\bx,s)$ satisfies the Riemann hypothesis. 
\end{theorem}
\begin{proof}
The upper bound on discrepancy is Theorem \ref{thm:disc-upper-bound}, and 
the lower bound is Theorem \ref{thm:disc-lower-bound}. For the functions $f$ in 
question, Theorem \ref{thm:translates-bound-sum} gives an estimate (stronger 
than) $\left| \sum_{\N(\fp)\leqslant N} f(\vx_\fp)\right|\ll N^{\frac 1 2}$, and 
Theorem \ref{thm:AT->RH:gp} tells us this estimate implies the Riemann 
hypothesis. 
\end{proof}

This shows that for a sequence $\btheta = (\theta_\fp)$ in $\bT^d$, even if 
each $L(r(\btheta),s)$ satisfies the Riemann hypothesis, we may not conclude 
that the discrepancy of $\btheta$ decays like $N^{-\alpha}$ for any fixed 
$\alpha$. So for CM abelian varieties, the Akiyama--Tanigawa conjecture does 
not follow in a straightforward manner from the generalized Riemann hypothesis 
together with basic facts about Dirichlet series. Note also that Theorem 
\ref{thm:AT->RH:gp} does \emph{not} tell us that $L_f(\bvx,s)$ has analytic 
continuation to $\Re > 0$, or that there are no zeros in $\Re > 0$. For, the 
term $\sum_\fp \sum_{r\geqslant 2} \frac{f(\vx_\fp)^r}{r \N(\fp)^{r s}}$ will 
not converge past $\Re > \frac 1 2$.