1_introduction.tex

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

\chapter{Introduction}





\section{Motivation from classical analytic number theory}

Start with an old problem central to number theory: counting 
prime numbers. As usual, let $\pi(x)$ be the prime counting function and 
$\Li(x) = \int_2^x \frac{\dd t}{\log t}$ be the Eulerian logarithmic integral. 
The prime number theorem tells us that as $x\to \infty$, 
$\frac{\pi(x)}{\Li(x)} \to 1$. The standard approach 
to proving the prime number theorem is by showing that the Riemann 
$\zeta$-function has non-vanishing meromorphic continuation to $\Re = 1$.

\begin{theorem}
The function $\zeta(s)$ admits a non-vanishing meromorphic continuation to 
$\Re = 1$ with a simple pole at $s=1$, if and only if 
$\lim_{x\to \infty} \frac{\pi(x)}{\Li(x)} = 1$. 
\end{theorem}

Since $\zeta(s)$ does have the desired properties, the prime number 
theorem is true. It is natural to try to bound the difference 
$\pi(x) - \Li(x)$. Numerical experiments dating back to Gauss suggest that 
$|\pi(x) - \Li(x)| \ll x^{\frac 1 2+\epsilon}$. By this we mean that the 
estimate holds for any $\epsilon>0$, though the implied constant may depend on 
$\epsilon$. In fact, we have the following result. 

\begin{theorem}[{\cite[Th., p.~90]{edwards-1974}}]
The Riemann hypothesis is true if and only if 
$|\pi(x) - \Li(x)| \ll x^{\frac 1 2+\epsilon}$. 
\end{theorem}

Neither side of this equivalence is known for certain to be true! 

There is an analogue of the above discussion for Artin $L$-functions. 
Let $K/\bQ$ be a nontrivial finite  Galois extension with group 
$G=\Gal(K/\bQ)$. For any 
rational prime $p$ at which $K$ is unramified, let $\frob_p$ be the conjugacy 
class of the Frobenius at $p$ in $G$. For any complex irreducible representation 
$\rho$ of $G$, there is a corresponding $L$-function defined as 
\[
	L(\rho,s) = \prod_p \det\left(1-\rho(\frob_p) p^{-s}\right)^{-1} ,
\]
where here, and for the remainder of this thesis, we tacitly omit from the 
product those primes at which $\rho$ is ramified. If $\rho$ is the trivial 
representation, then $L(1,s) = \zeta(s)$. For each $p$, let $\delta_{\frob_p}$ 
be the Dirac delta measure concentrated at $\frob_p$ on $G^\natural$, the set 
of conjugacy classes of $G$. Given a cutoff $x$, there is a 
natural empirical measure 
$P_x = \frac{1}{\pi(x)} \sum_{p\leqslant x} \delta_{\frob_p}$ on $G^\natural$.
Let $\mu$ be the normalized Haar measure 
on $G^\natural$ (induced from the uniform measure on $G$), and let 
$\D(P_x) = \max_{S\subset G^\natural} \left| P_x(S) - \mu(S)\right|$. 
Then $P_x$ converges weakly to the Haar measure on $G^\natural$ if and 
only if $\D(P_x) \to 0$. Recall that weak convergence of $P_x$ to $\mu$ means 
$\int f\, \dd P_x \to \int f\, \dd\mu$ for all continuous functions $f$ on 
$G^\natural$. Since $G^\natural$ is a finite set, all functions on $G^\natural$ 
are continuous, but later on we will consider weak convergence on more general 
spaces.  

\begin{theorem}[{\cite[Th.~2 Cor., A.1]{serre-1989}}]
The measures $P_x$ converge weakly to the Haar measure on $G^\natural$ if and 
only if the function $L(\rho,s)$ admits a non-vanishing analytic continuation 
to $\Re = 1$ for all nontrivial $\rho$. 
\end{theorem}

Both sides of this equivalence are true, and known as the Chebotarev density 
theorem. If $K = \bQ$, so that $G$ (and hence $\rho$) are trivial, then the 
``Frobenius elements'' are all the identity, so equidistribution holds 
trivially. However, $\zeta(s)$ does not admit a non-vanishing \emph{analytic} 
continuation to $\Re = 1$, for it has a simple pole at $s = 1$. So the 
result is only true when $K/\bQ$ is a nontrivial extension. Returning to that 
case ($K\ne \bQ$), there is a version of the strong prime number theorem. It is 
known that Artin $L$-functions admit a meromorphic continuation to the complex 
plane, and that this continuation satisfies a functional equation. However, in 
this thesis, we will consider Dirichlet series for which no such continuation 
or functional equation exist---even conjecturally. As a result, in this thesis, 
by the ``Riemann hypothesis'' for a Dirichlet series $L(s)$ we mean the 
statement that $L(s)$ admits a non-vanishing analytic continuation to 
$\Re > \frac 1 2$. 

\begin{theorem}
The bound $\D(P_x) \ll x^{-\frac 1 2+\epsilon}$ holds if and only if each
$L(\rho,s)$, $\rho$ nontrivial, satisfies the Riemann hypothesis. 
\end{theorem}

The forward implication follows from Theorem \ref{thm:AT->RH:gp}, while the reverse implication is a result of Serre \cite[Th.~4]{serre-1981}. 
This whole discussion generalizes to a more complicated set of Galois 
representations---those arising from elliptic curves and more general motives.  





\section{Discrepancy and the Riemann hypothesis for elliptic curves}

For background on the Galois representations and $L$-functions associated to 
elliptic curves, see \cite[III\S7, C\S17]{silverman-2009}. Throughout this 
thesis, what we call the $L$-function of an elliptic curve (motive, etc.) is 
the normalized (i.e.~analytic instead of algebraic) $L$-function. 
Let $E_{/\bQ}$ be a non-CM elliptic curve. For any prime $l$, the $l$-adic Tate 
module of $E$ induces a continuous representation 
$\rho_l \colon G_\bQ \to \GL_2(\bZ_l)$. It is known that for $p$ not dividing 
either $l$ or the conductor of $E$, the quantities 
$a_p = \tr \rho_l(\frob_p)$ lie in $\bZ$, are independent of $l$, and satisfy 
the Hasse bound $|a_p| \leqslant 2\sqrt p$. For each unramified prime $p$, the 
corresponding Satake parameter for $E$ is 
$\theta_p = \cos^{-1}\left(\frac{a_p}{2\sqrt p}\right) \in [0,\pi]$. 
These parameters are packaged into an $L$-function as follows:
\[
	L(E,s) = \prod_p \frac{1}{(1 - e^{i \theta_p} p^{-s})(1- e^{-i \theta_p} p^{-s})} = \prod_p \det\left(1 - \smat{e^{i\theta_p}}{}{}{e^{-i \theta_p}}p^{-s}\right)^{-1}.
\]
More generally we have, for each irreducible representation $\sym^k$ of 
$\SU(2)$, the $k$-th symmetric power $L$-function: 
\[
	L(\sym^k E, s) = \prod_p \prod_{j=0}^k \frac{1}{1 - e^{i (k - 2j) \theta_p} p^{-s}} = \prod_p \det\left(1-\sym^k \smat{e^{i\theta_p}}{}{}{e^{-i \theta_p}}p^{-s}\right)^{-1}.
\]

Numerical experiments suggest that the Satake parameters are equidistributed 
with respect to the Sato--Tate distribution 
$\ST = \frac{2}{\pi} \sin^2\theta\, \dd\theta$. Indeed, for any cutoff $x$, let 
$P_x$ be the empirical measure 
$\frac{1}{\pi(x)} \sum_{p\leqslant x} \delta_{\theta_p}$. 
The convergence of $P_x$ to the Sato--Tate measure is closely related to 
the analytic properties of the $L(\sym^k E,s)$. First, here is the famous 
Sato--Tate conjecture, now a theorem, in our notation. 

\begin{theorem}[{\cite[Cor.~8.9]{bght-2011}}]
The measures $P_x$ converge weakly to $\ST$. 
\end{theorem}

\begin{theorem}[{\cite[Th.~2 Cor.]{serre-1989}}]
The Sato--Tate conjecture holds for $E$ if and only if each of 
the functions $L(\sym^k E,s)$ have analytic continuation to $\Re = 1$. 
\end{theorem}

The stunning recent proof of the Sato--Tate conjecture over totally real fields 
showed that the functions $L(\sym^k E,s)$ have the desired analytic 
continuation. Moreover, it showed that for all $k$, $L(\sym^k E,s)$ has 
meromorphic continuation to the whole complex plane. Even better, when $k$ is 
odd, the $L$-function is potentially automorphic. See Theorem 
\ref{thm:bad-Galois} for a result in this thesis where more can be said about 
odd symmetric power $L$-functions than even ones. 

The Riemann hypothesis, and its analogue for Artin $L$-functions, has a natural 
generalization to elliptic curves. In this context, the discrepancy of the set 
$\{\theta_p\}_{p\leqslant x}$ is 
\[
	\D_x\left(E,\ST\right) = \sup_{t\in [0,\pi]} \left| P_x[0,t) - \ST[0,t)\right| .
\]
The following conjecture is made in \cite{akiyama-tanigawa-1999}.

\begin{conjecture}[Akiyama--Tanigawa]\label{conj:akiyama-tanigawa}
$\D_x\left(E,\ST\right)\ll x^{-\frac 1 2+\epsilon}$.
\end{conjecture}

Akiyama and Tanigawa provide computational evidence for their conjecture, then 
go on to prove a special case of the following theorem, proved in full 
generality by Mazur. 

\begin{theorem}[{\cite[\S3.4]{mazur-2008}}]
If $\D_x\left(E,\ST\right)\ll x^{-\frac 1 2+\epsilon}$, 
then all the functions $L(\sym^k E, s)$ satisfy the Riemann hypothesis. 
\end{theorem}

This discussion also makes sense when $E$ has complex multiplication (for 
simplicity, we consider $E_{/F}$ where $F$ is the field of definition of the 
complex multiplication). The Sato--Tate measure for such $E$ is the Haar 
measure on $\SO(2)$, i.e.~the uniform measure on $[0,\pi]$. Instead of 
symmetric power $L$-functions, there is an $L$-function for each character of 
$\SO(2)$. Once again, there is a theorem ``Akiyama--Tanigawa conjecture implies 
generalized Riemann hypothesis.'' For a precise statement and proof, see Section 
\ref{sec:Satake-CM}.

It is natural to assume that the converse to the implication 
``Akiyama--Tanigawa conjecture implies Riemann hypothesis'' holds. 
In this thesis, we construct a range of counterexamples to the implication 
``generalized Riemann hypothesis implies fast discrepancy decay '' for sequences 
in compact real tori. This suggests that for CM abelian varieties, proving the 
converse to ``Akiyama--Tanigawa implies generalized Riemann hypothesis'' is not 
as straightforward as in the case of Artin $L$-functions. 

Moreover, we generalize the results of \cite{pande-2011} to show that there are 
(infinitely ramified) Galois representations whose Satake parameters exist and 
are equidistributed with respect to essentially arbitrary specified measures. 
Moreover, the rate of decay of discrepancy can be prescribed, and for ``odd'' 
measures, all the odd symmetric-power $L$-functions can be made to satisfy the 
Riemann hypothesis. We also show 
that some of the results of \cite{sarnak-2007} about sums of the form 
$\sum_{p\leqslant x} \frac{a_p}{\sqrt p}$ cannot be generalized to 
general---in particular, infinitely ramified---Galois representations.