Counterexamples related to the Sato-Tate conjecture
| dc.contributor.author | Miller, Daniel Keegan | |
| dc.contributor.chair | Ramakrishna, Ravi K | |
| dc.contributor.committeeMember | Speh, Birgit E M | |
| dc.contributor.committeeMember | Zywina, David J | |
| dc.date.accessioned | 2017-07-07T12:48:48Z | |
| dc.date.available | 2017-07-07T12:48:48Z | |
| dc.date.issued | 2017-05-30 | |
| dc.description.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. | |
| dc.identifier.doi | https://doi.org/10.7298/X4PN93Q3 | |
| dc.identifier.other | Miller_cornellgrad_0058F_10258 | |
| dc.identifier.other | http://dissertations.umi.com/cornellgrad:10258 | |
| dc.identifier.other | bibid: 9948890 | |
| dc.identifier.uri | https://hdl.handle.net/1813/51667 | |
| dc.language.iso | en_US | |
| dc.subject | Dirichlet series | |
| dc.subject | discrepancy | |
| dc.subject | Galois representations | |
| dc.subject | Sato-Tate conjecture | |
| dc.subject | Mathematics | |
| dc.title | Counterexamples related to the Sato-Tate conjecture | |
| dc.type | dissertation or thesis | |
| dcterms.license | https://hdl.handle.net/1813/59810 | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Cornell University | |
| thesis.degree.level | Doctor of Philosophy | |
| thesis.degree.name | Ph. D., Mathematics |
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