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Author ORCID Identifier
https://orcid.org/0000-0001-8829-8224
AccessType
Open Access Dissertation
Document Type
dissertation
Degree Name
Doctor of Philosophy (PhD)
Degree Program
Mathematics
Year Degree Awarded
2019
Month Degree Awarded
September
First Advisor
Thomas Weston
Subject Categories
Number Theory
Abstract
Let E1 x E2 over Q be a fixed product of two elliptic curves over Q with complex multiplication. I compute the probability that the pth Fourier coefficient of E1 x E2, denoted as ap(E1) + ap(E2), is a square modulo p. The results are 1/4, 7/16, and 1/2 for different imaginary quadratic fields, given a technical independence of the twists. The similar prime densities for cubes and 4th power are 19/54, and 1/4, respectively. I also compute the probabilities without the technical assumption on the twists in various cases. Next, I consider the sum of quadratic residue of ap as primes p and elliptic curves vary. The purpose is to test the conjecture that ap of an elliptic curve is a square modulo p about half of the time across prime numbers so that the sum is expected to be 0. Although the sum turns out to be positively biased, I show, assuming a natural independence result, that the ap are evenly distributed between squares and non-squares modulo p asymptotically.
DOI
https://doi.org/10.7275/15013797
Recommended Citation
Nguyen, Vy Thi Khanh, "Elliptic Curves And Power Residues" (2019). Doctoral Dissertations. 1802.
https://doi.org/10.7275/15013797
https://scholarworks.umass.edu/dissertations_2/1802