Counterexamples related to the Sato-Tate conjecture

Let E_/Q be an elliptic curve. The Sato--Tate conjecture, now a theorem, tells us that the angles θ_p =\cos^-1\left(\frac{a_p}{2\sqrt p}\right) are equidistributed in [0,π] with respect to the measure \frac{2}{π}\sin²θ dθ if $E$ is non-CM (resp.~\frac{1}{2π} d θ + \frac 1 2 δ_π/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+ε}, 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^-ε for any fixed ε>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{Q} /Q) \to GL₂(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 μ on [0,π] for which \cos_\astμ is absolutely continuous with bounded derivative.

Name thesis:degree_name

Ph. D., Mathematics

Level thesis:degree_level

Doctor of Philosophy

Discipline thesis:degree_discipline

Mathematics

Grantor

Cornell University

Year dc:date.issued

2017

Author dc:creator

• Miller, Daniel Keegan

Committee members dc:contributor.committeemember

• Speh, Birgit E M
• Zywina, David J

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

Language dc:language.iso

en_US

DOI

https://doi.org/10.7298/X4PN93Q3

Dc Identifier Other

ProQuest Submission ID: 10258
ProQuest Publication ID: 10276670

OAI identifier oai:identifier

oai:ecommons.cornell.edu:1813/51667