{"id":{"repo_id":"uic","oai_identifier":"oai:figshare.com:article/30425038"},"canonical_url":"https://search.dev.ndltd.org/etd/uic/oai:figshare.com:article/30425038","repository":{"repo_id":"uic","name":"University of Illinois - Chicago","base_url":"https://api.figshare.com/v2/oai"},"display":{"title":"Arithmetic Properties Related to Isogeny Criteria for Elliptic Curves and Drinfeld Modules","abstract":"Let E_1 and E_2 be elliptic curves defined over a number field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of norm at most x for which E_1 and E_2 have the same Frobenius trace or Frobenius field is bounded above asymptotically by a function of x. We prove a bound with a log saving unconditionally, prove a bound with a power saving dependent on the Reimann hypothesis and generalized Reimann hypothesis, and prove a bound with a stronger power saving dependent on the generalized Reimann hypothesis, Artin's holomorphy conjecture, and a pair correlation conjecture. Let Phi_1 and Phi_2 be Drinfeld modules of rank r at least two defined over a generic A-field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of degree x for which Phi_1 and Phi_2 have the same Frobenius trace or characteristic polynomial of the Frobenius is bounded above asymptotically by a function of x. We prove bounds with a power saving dependent upon a generalization of the open image theorem for Drinfeld modules of Pink and Rütsche.","abstract_html":"Let E_1 and E_2 be elliptic curves defined over a number field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of norm at most x for which E_1 and E_2 have the same Frobenius trace or Frobenius field is bounded above asymptotically by a function of x. We prove a bound with a log saving unconditionally, prove a bound with a power saving dependent on the Reimann hypothesis and generalized Reimann hypothesis, and prove a bound with a stronger power saving dependent on the generalized Reimann hypothesis, Artin's holomorphy conjecture, and a pair correlation conjecture. Let Phi_1 and Phi_2 be Drinfeld modules of rank r at least two defined over a generic A-field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of degree x for which Phi_1 and Phi_2 have the same Frobenius trace or characteristic polynomial of the Frobenius is bounded above asymptotically by a function of x. We prove bounds with a power saving dependent upon a generalization of the open image theorem for Drinfeld modules of Pink and Rütsche.","abstract_has_math":false,"creators":["Auden McEuen Hinz (22481815)"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-08-01T00:00:00Z","date_published":"2025-08-01T00:00:00Z","updated_at":"2026-07-27T21:34:45Z","subjects":["Elliptic Curves over Global Fields","Drinfeld Modules","Curves over Finite and Local Fields","Distribution of Primes","Generalized Primes and Integers"],"languages":[],"rights":["In Copyright"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.25417/uic.30425038.v1","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Auden McEuen Hinz (22481815)"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-08-01T00:00:00Z"]},{"key":"dc:relation","label":"Dc Relation","values":["https://figshare.com/articles/thesis/Arithmetic_Properties_Related_to_Isogeny_Criteria_for_Elliptic_Curves_and_Drinfeld_Modules/30425038"]},{"key":"dc:type","label":"Dc Type","values":["Text","Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Elliptic Curves over Global Fields","Drinfeld Modules","Curves over Finite and Local Fields","Distribution of Primes","Generalized Primes and Integers"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["10.25417/uic.30425038.v1"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let E_1 and E_2 be elliptic curves defined over a number field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of norm at most x for which E_1 and E_2 have the same Frobenius trace or Frobenius field is bounded above asymptotically by a function of x. We prove a bound with a log saving unconditionally, prove a bound with a power saving dependent on the Reimann hypothesis and generalized Reimann hypothesis, and prove a bound with a stronger power saving dependent on the generalized Reimann hypothesis, Artin's holomorphy conjecture, and a pair correlation conjecture. Let Phi_1 and Phi_2 be Drinfeld modules of rank r at least two defined over a generic A-field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of degree x for which Phi_1 and Phi_2 have the same Frobenius trace or characteristic polynomial of the Frobenius is bounded above asymptotically by a function of x. We prove bounds with a power saving dependent upon a generalization of the open image theorem for Drinfeld modules of Pink and Rütsche."]},{"key":"dc:title","label":"Title","values":["Arithmetic Properties Related to Isogeny Criteria for Elliptic Curves and Drinfeld Modules"]}]}],"canonical_facts":{"dc:creator":["Auden McEuen Hinz (22481815)"],"dc:date":["2025-08-01T00:00:00Z"],"dc:description":["Let E_1 and E_2 be elliptic curves defined over a number field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of norm at most x for which E_1 and E_2 have the same Frobenius trace or Frobenius field is bounded above asymptotically by a function of x. We prove a bound with a log saving unconditionally, prove a bound with a power saving dependent on the Reimann hypothesis and generalized Reimann hypothesis, and prove a bound with a stronger power saving dependent on the generalized Reimann hypothesis, Artin's holomorphy conjecture, and a pair correlation conjecture. Let Phi_1 and Phi_2 be Drinfeld modules of rank r at least two defined over a generic A-field K which have trivial endomorphism ring and are non-isogenous. We prove that the number of primes of K of degree x for which Phi_1 and Phi_2 have the same Frobenius trace or characteristic polynomial of the Frobenius is bounded above asymptotically by a function of x. We prove bounds with a power saving dependent upon a generalization of the open image theorem for Drinfeld modules of Pink and Rütsche."],"dc:identifier":["10.25417/uic.30425038.v1"],"dc:relation":["https://figshare.com/articles/thesis/Arithmetic_Properties_Related_to_Isogeny_Criteria_for_Elliptic_Curves_and_Drinfeld_Modules/30425038"],"dc:rights":["In Copyright"],"dc:subject":["Elliptic Curves over Global Fields","Drinfeld Modules","Curves over Finite and Local Fields","Distribution of Primes","Generalized Primes and Integers"],"dc:title":["Arithmetic Properties Related to Isogeny Criteria for Elliptic Curves and Drinfeld Modules"],"dc:type":["Text","Thesis"]},"updated_at":"2026-07-27T21:34:45Z"}