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Author ORCID Identifier

https://orcid.org/0000-0001-8829-8224

Document Type

Open Access 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.

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Number Theory Commons

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