Daniel-Miller-thesis.tex

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\title{Counterexamples related to the Sato--Tate conjecture}
\author{Daniel Miller}
\conferraldate{May}{2017}

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\begin{abstract}
Let $E_{/\bQ}$ be an elliptic curve. The Sato--Tate conjecture, now a theorem, 
tells us that the angles $\theta_p =\cos^{-1}\left(\frac{a_p}{2\sqrt p}\right)$ 
are equidistributed in $[0,\pi]$ with respect to the measure 
$\frac{2}{\pi}\sin^2\theta\, \dd\theta$ if $E$ is non-CM
(resp.~$\frac{1}{2\pi} \dd \theta + \frac 1 2 \delta_{\pi/2}$ if $E$ is CM). 
In the non-CM case, Akiyama and Tanigawa conjecture that the discrepancy 
\[
	D_N = \sup_{x\in [0,\pi]} \left| \frac{1}{\pi(N)} \sum_{p\leqslant N} 1_{[0,x]}(\theta_p) - \int_0^x \frac{2}{\pi}\sin^2\theta\, \dd\theta\right| 
\]
asymptotically decays like $N^{-\frac 1 2+\epsilon}$, as is suggested by computational 
evidence and certain reasonable heuristics on the Kolmogorov--Smirnov 
statistic. This conjecture implies the Riemann hypothesis 
for all $L$-functions associated with $E$. It is natural to assume that the 
converse (``generalized Riemann hypothesis implies discrepancy estimate'') holds, 
as is suggested by analogy with Artin $L$-functions. We construct, for compact 
real tori, ``fake Satake parameters'' yielding $L$-functions which satisfy the 
generalized Riemann hypothesis, but for which the discrepancy decays like 
$N^{-\epsilon}$ for any fixed $\epsilon>0$. This provides evidence that for 
CM abelian varieties, the converse to ``Akiyama--Tanigawa conjecture implies 
generalized Riemann hypothesis'' does not follow in a straightforward way from 
the standard analytic methods. 

We also show that there are Galois representations 
$\rho\colon \Gal(\overline \bQ /\bQ) \to \GL_2(\bZ_l)$, ramified at an 
arbitrarily thin (but still infinite) set of primes, whose Satake parameters 
can be made to converge at any specified rate to any fixed measure $\mu$ on 
$[0,\pi]$ for which $\cos_\ast\mu$ is absolutely continuous with bounded 
derivative. 
\end{abstract}





\begin{biosketch}
Daniel Miller was born in St.~Paul, Minnesota. He completed his Bachelor of 
Science at the University of Nebraska Omaha. In addition to his studies there, 
he played the piano competitively and attended Cornell's Summer Mathematics 
Institute. He started his Ph.D.~at Cornell planning on a career in academia. 
Halfway through he had a change of heart, and will be joining Microsoft's 
Analysis and Experimentation team as a data scientist after graduation. He is 
happily married to Ivy Lai Miller, and owns a cute but grumpy cat named Socrates. 
\end{biosketch}





\begin{dedication}
This thesis is dedicated to my undergraduate adviser, Griff Elder. He is the 
reason I considered a career in mathematics, and his infectious enthusiasm 
for number theory has inspired me more than I can say. 
\end{dedication}





\begin{acknowledgements}
I could not have completed this thesis without help and support from many 
people. I would like to offer my sincerest thanks to the following people, and 
my sincerest apologies to anyone whose name I have forgotten to include here. 

My parents Jay and Cindy, for noticing and fostering my mathematical 
interests early on, and for being unfailingly loving and supportive. 

My undergraduate thesis advisor, Griffith Elder. Without his encouragement 
and inspiration I probably would have never considered a career in math. 

Tara Holm, Jason Boynton, and Anthony Weston, for making Cornell's 2011 Summer 
Mathematics Institute the fantastic experience it was. 

My fellow graduate students Sasha Patotski, Bal\'azs Elek, and Sergio Da 
Silva, for sharing my early love of algebraic geometry, laughing with me at the 
absurdities of academic life, and listening to my ramblings about number theory. 

The mathematics department at Cornell, where many professors were generous 
with their time and ideas. I appreciate Yuri Berest, John Hubbard, Farbod 
Shokrieh, Birget Speh, and David Zywina for letting me bounce ideas off them, 
helping me add rigor to half-baked ideas, and pointing my research in new and 
interesting directions. 

My advisor Ravi Ramakrishna. He kindled my first love for number theory, 
stayed supportive as my research bounced all over the place, and kept me 
focused, grounded, and concrete when I needed to be. 

Most importantly, my wife Ivy for being there for me through the highs 
and the lows, when I prematurely thought my thesis was complete, and when I 
thought my results were completely in shambles. I couldn't have done it without 
her. 
\end{acknowledgements}





\contentspage





\symlist

\begin{tabular}{ll}
$1_S$     
	& characteristic function of a set $S$. \\
$l$       
	& rational prime $\geqslant 5$. \\
$f_\ast\mu$
	& pushforward measure $(f_\ast\mu)(S) = \mu(f^{-1}(S))$. \\
$\Re z$
	& real part of $z$. \\
$\Re > \alpha$
	& half-plane $\{z\in \bC : \Re z > \alpha\}$. \\
$f \ll g$ 
	& there is a constant $C>0$ such that $f \leqslant C g$. \\
$f(x) \ll x^{\alpha+\epsilon}$ 
	& for all $\epsilon>0$, $f(x) \ll x^{\alpha+\epsilon}$ (the constant may depend on $\epsilon$). \\
$\delta_x$
	& Dirac measure concentrated on $x$. \\
$\frob_p$
	& conjugacy class of arithmetic Frobenius at $p$. \\
$G^\natural$
	& space of conjugacy classes of a group $G$. \\
$\smat{a}{}{}{b}$
	& shorthand for $\smat{a}{0}{0}{b}$. \\
$\ST$
	& Sato--Tate measure $\frac{2}{\pi} \sin^2\theta\, \dd\theta$ on $[0,\pi]$. \\
$\bx$
	& sequence $(x_1,x_2,x_3,\dots)$ or $(x_2,x_3,x_5,\dots)$. \\
$\bvx$
	& sequence of vectors $(\vx_1,\vx_2,\dots)$ or $(\vx_2,\vx_3,\vx_5,\dots)$. \\
$C(X)$
	& continuous, $\bC$-valued functions on $X$. \\
$f = o(g)$
	& means $\limsup \frac f g = 0$. \\
$|\cdot|_\infty$
	& supremum norm. \\
$[\vx,\vy)$
	& half-open box $[x_1,y_1) \times \cdots \times [x_d,y_d)$. \\
$\mu[a,b]$
	& shorthand for $\mu([a,b])$ if $\mu$ is a measure. \\
$\D(\mu,\nu)$
	& discrepancy between $\mu$ and $\nu$. \\
$P_{\bx,N}$
	& empirical measure associated to the set $\{x_\alpha\}_{\alpha\leqslant N}$. \\
$\D_N(\bx,\mu)$
	& discrepancy between $P_{\bx,N}$ and $\mu$. \\
$\bT^d$
	& $d$-dimensional real torus $(\bR/\bZ)^d$. \\
$\Var(f)$
	& total variation of $f$. \\
$\frac{\dd\mu}{\dd\lambda}$
	& Radon--Nikodym derivative of $\mu$. \\
$\cdf_\mu(x)$
	& cumulative distribution function $x\mapsto \mu[-\infty,x]$. \\	
$\bx\wr\by$
	& interleaved sequence $(x_1,y_1,x_2,y_2,\dots)$. \\
$f = \Theta(g)$
	& there exist constants $0 < C_1 < C_2$ such that $C_1 g \leqslant f \leqslant C_2 g$. \\
$\bx_{\leqslant N} : a^M$
	& shorthand for $(x_1,\dots,x_N,a,\dots,a)$ ($M$ copies of $a$). \\
$U_k(\theta)$
	& $\tr\sym^k\smat{e^{i\theta}}{}{}{e^{-i\theta}} = \frac{\sin((k+1)\theta)}{\sin\theta}$ on $\SU(2)^\natural = [0,\pi]$. \\
$L(\bx,s)$
	& Dirichlet series associated to a sequence $\bx$ in $\bC$. \\
$L(\rho(\bx),s)$
	& Dirichlet series associated to a representation. \\
$\omega_i(\vx)$
	& $i$-th irrationality measure of $\vx$. \\
$\langle \cdot,\cdot\rangle$
	& standard inner product on $\bR^d$. \\
$r(\vm)$
	& shorthand for $\max(1,|m_1|) \cdots \max(1,|m|_d)$. \\
$f = \Omega(g)$
	& means $\limsup \frac f g > 0$ (Hardy--Littlewood convention). \\
$\R_{F/\bQ} \Gm$
	& Weil restriction of scalars of $\Gm$. \\
$\h^i(F,M)$
	& Galois cohomology $\h^i(G_F,M)$. \\
$\Sha^i_S(M)$
	& Tate--Shafarevich group of $M$. \\
$M^\ast$
	& Cartier dual of $M$. \\
$\h_\nr^1(\bQ_p,M)$
	& unramified cohomology classes in $\h^1(\bQ_p,M)$. 
\end{tabular}
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\include{1_introduction}
\include{2_discrepancy}
\include{3_Dirichlet}
\include{4_irrationality}
\include{5_representations}
\include{6_conclusion}





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