2026-09-19T17:27:03Zhttps://ecommons.cornell.edu/server/oai/requestoai:ecommons.cornell.edu:1813/516672026-05-15T19:52:01Zcom_1813_35col_1813_47
Miller, Daniel Keegan
Ramakrishna, Ravi K
Speh, Birgit E M
Zywina, David J
2017-07-07T12:48:48Z
2017-07-07T12:48:48Z
2017-05-30
ProQuest Submission ID: 10258
ProQuest Publication ID: 10276670
https://hdl.handle.net/1813/51667
https://doi.org/10.7298/X4PN93Q3
9948890
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.
en_US
Dirichlet series
discrepancy
Galois representations
Sato-Tate conjecture
Mathematics
Counterexamples related to the Sato-Tate conjecture
dissertation or thesis
application/pdf
Mathematics
Cornell University
Doctor of Philosophy
Ph. D., Mathematics
https://hdl.handle.net/1813/59810
Publication
Cornell Theses and Dissertations
Miller, Daniel Keegan
5893a6ea-7af3-41d7-abc6-04bcd26ab5df
open.access
open.access
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="0dbe751e-6d49-4641-9fa9-08168dfe992e">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Language>en_US</Language>
<Title>Counterexamples related to the Sato-Tate conjecture</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2017-05-30</PublicationDate>
<DOI>https://doi.org/10.7298/X4PN93Q3</DOI>
<Authors>
<Author>
<DisplayName>Miller, Daniel Keegan</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<OrgUnit />
</Publisher>
</Publishers>
<Keyword>Dirichlet series</Keyword>
<Keyword>discrepancy</Keyword>
<Keyword>Galois representations</Keyword>
<Keyword>Sato-Tate conjecture</Keyword>
<Keyword>Mathematics</Keyword>
<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.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>