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oai:figshare.com:article/30425038 2025-08-01T00:00:00Z portal_693 item_type_8 month_year_08_2025

Arithmetic Properties Related to Isogeny Criteria for Elliptic Curves and Drinfeld Modules Auden McEuen Hinz (22481815) Elliptic Curves over Global Fields Drinfeld Modules Curves over Finite and Local Fields Distribution of Primes Generalized Primes and Integers 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. 2025-08-01T00:00:00Z Text Thesis 10.25417/uic.30425038.v1 https://figshare.com/articles/thesis/Arithmetic_Properties_Related_to_Isogeny_Criteria_for_Elliptic_Curves_and_Drinfeld_Modules/30425038 In Copyright