{"id":{"repo_id":"cornell","oai_identifier":"oai:ecommons.cornell.edu:1813/51667"},"canonical_url":"https://search.dev.ndltd.org/etd/cornell/oai:ecommons.cornell.edu:1813/51667","repository":{"repo_id":"cornell","name":"Cornell University","base_url":"https://ecommons.cornell.edu/server/oai/request"},"display":{"title":"Counterexamples related to the Sato-Tate conjecture","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_html":"Let <span class=\"etd-inline-math\">E<sub>/<strong>Q</strong></sub></span> be an elliptic curve. The Sato--Tate conjecture, now a theorem, tells us that the angles <span class=\"etd-inline-math\">&theta;<sub>p</sub> =\\cos<sup>-1</sup>\\left(\\frac{a<sub>p</sub>}{2\\sqrt p}\\right)</span> are equidistributed in <span class=\"etd-inline-math\">[0,&pi;]</span> with respect to the measure <span class=\"etd-inline-math\">\\frac{2}{&pi;}\\sin<sup>2</sup>&theta;  d&theta;</span> if $E$ is non-CM (resp.~<span class=\"etd-inline-math\">\\frac{1}{2&pi;} d &theta; + \\frac 1 2 &delta;<sub>&pi;/2</sub></span> 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 <span class=\"etd-inline-math\">N<sup>-\\frac 1 2+&epsilon;</sup></span>, 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&#x27;&#x27;) holds, as is suggested by analogy with Artin $L$-functions. We construct, for compact real tori, ``fake Satake parameters&#x27;&#x27; yielding $L$-functions which satisfy the generalized Riemann hypothesis, but for which the discrepancy decays like <span class=\"etd-inline-math\">N<sup>-&epsilon;</sup></span> for any fixed <span class=\"etd-inline-math\">&epsilon;&gt;0</span>. This provides evidence that for CM abelian varieties, the converse to ``Akiyama--Tanigawa conjecture implies generalized Riemann hypothesis&#x27;&#x27; does not follow in a straightforward way from the standard analytic methods. We also show that there are Galois representations <span class=\"etd-inline-math\">\\rho\\colon Gal(\\overline{<strong>Q</strong>} /<strong>Q</strong>) \\to GL<sub>2</sub>(<strong>Z</strong><sub>l</sub>)</span>, 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 <span class=\"etd-inline-math\">&mu;</span> on <span class=\"etd-inline-math\">[0,&pi;]</span> for which <span class=\"etd-inline-math\">\\cos<sub>\\</sub>ast&mu;</span> is absolutely continuous with bounded derivative.","abstract_has_math":true,"creators":["Miller, Daniel Keegan"],"institution":"Cornell University","degree_name":"Ph. D., Mathematics","degree_level":"Doctor of Philosophy","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":["Speh, Birgit E M","Zywina, David J"],"year":2017,"date_issued":"2017-05-30","date_published":"2017-05-30","updated_at":"2026-07-24T01:49:10Z","subjects":["Dirichlet series","discrepancy","Galois representations","Sato-Tate conjecture","Mathematics"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.7298/X4PN93Q3"],"render_values":[{"text":"https://doi.org/10.7298/X4PN93Q3","href":"https://doi.org/10.7298/X4PN93Q3","code":true}]},{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["ProQuest Submission ID: 10258","ProQuest Publication ID: 10276670"],"render_values":[{"text":"ProQuest Submission ID: 10258","href":null,"code":true},{"text":"ProQuest Publication ID: 10276670","href":null,"code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/1813/51667","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Speh, Birgit E M","Zywina, David J"]},{"key":"dc:creator","label":"Author","values":["Miller, Daniel Keegan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-07-07T12:48:48Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-07-07T12:48:48Z"]},{"key":"dc:date.issued","label":"Date","values":["2017-05-30"]},{"key":"dc:type","label":"Dc Type","values":["dissertation or thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctor of Philosophy"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D., Mathematics"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Cornell University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Dirichlet series","discrepancy","Galois representations","Sato-Tate conjecture","Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.7298/X4PN93Q3"]},{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["ProQuest Submission ID: 10258","ProQuest Publication ID: 10276670"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1813/51667"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Counterexamples related to the Sato-Tate conjecture"]}]}],"canonical_facts":{"dc:contributor.committeemember":["Speh, Birgit E M","Zywina, David J"],"dc:creator":["Miller, Daniel Keegan"],"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:format.mimetype":["application/pdf"],"dc:identifier.doi":["https://doi.org/10.7298/X4PN93Q3"],"dc:identifier.other":["ProQuest Submission ID: 10258","ProQuest Publication ID: 10276670"],"dc:identifier.uri":["https://hdl.handle.net/1813/51667"],"dc:language.iso":["en_US"],"dc:subject":["Dirichlet series","discrepancy","Galois representations","Sato-Tate conjecture","Mathematics"],"dc:title":["Counterexamples related to the Sato-Tate conjecture"],"dc:type":["dissertation or thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctor of Philosophy"],"thesis:degree_name":["Ph. D., Mathematics"],"thesis:institution_name":["Cornell University"]},"updated_at":"2026-07-24T01:49:10Z"}