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## Counterexamples related to the Sato-Tate conjecture

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**Author**

Miller, Daniel Keegan

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**Abstract**

Let $E_{/\mathbf{Q}}$ 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
$\frac{2}{\pi}\sin^2\theta\, d\theta$ if $E$ is non-CM
(resp.~$\frac{1}{2\pi} d \theta + \frac 1 2 \delta_{\pi/2}$ if $E$ is CM).
In the non-CM case, Akiyama and Tanigawa conjecture that the discrepancy
\[
D_N = \sup_{x\in [0,\pi]} \left| \frac{1}{\pi(N)} \sum_{p\leqN} 1_{[0,x]}(\theta_p) - \int_0^x \frac{2}{\pi}\sin^2\theta\, d\theta\right|
\]
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
$\rho\colon Gal(\overline{\mathbf{Q}} /\mathbf{Q}) \to GL_2(\mathbf{Z}_l)$, 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.

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**Date Issued**

2017-05-30#####
**Subject**

Dirichlet series; discrepancy; Galois representations; Sato-Tate conjecture; Mathematics

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**Committee Chair**

Ramakrishna, Ravi K

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**Committee Member**

Speh, Birgit E M; Zywina, David J

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**Degree Discipline**

Mathematics

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**Degree Name**

Ph. D., Mathematics

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**Degree Level**

Doctor of Philosophy

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**Type**

dissertation or thesis